Implement atan

Author: richard-uk1

src/lib.rs

@@ -374,7 +374,6 @@ pub trait F64Ext: private::Sealed + Sized {
 
     fn acos(self) -> Self;
 
-    #[cfg(todo)]
     fn atan(self) -> Self;
 
     #[cfg(todo)]
@@ -541,7 +540,6 @@ impl F64Ext for f64 {
         acos(self)
     }
 
-    #[cfg(todo)]
     #[inline]
     fn atan(self) -> Self {
         atan(self)

src/math/atan.rs

@@ -0,0 +1,170 @@
+/* atan(x)
+ * Method
+ *   1. Reduce x to positive by atan(x) = -atan(-x).
+ *   2. According to the integer k=4t+0.25 chopped, t=x, the argument
+ *      is further reduced to one of the following intervals and the
+ *      arctangent of t is evaluated by the corresponding formula:
+ *
+ *      [0,7/16]      atan(x) = t-t^3*(a1+t^2*(a2+...(a10+t^2*a11)...)
+ *      [7/16,11/16]  atan(x) = atan(1/2) + atan( (t-0.5)/(1+t/2) )
+ *      [11/16.19/16] atan(x) = atan( 1 ) + atan( (t-1)/(1+t) )
+ *      [19/16,39/16] atan(x) = atan(3/2) + atan( (t-1.5)/(1+1.5t) )
+ *      [39/16,INF]   atan(x) = atan(INF) + atan( -1/t )
+ *
+ * Constants:
+ * The hexadecimal values are the intended ones for the following
+ * constants. The decimal values may be used, provided that the
+ * compiler will convert from decimal to binary accurately enough
+ * to produce the hexadecimal values shown.
+ */
+
+use super::fabs;
+use core::f64;
+
+const ATANHI: [f64; 4] = [
+    4.63647609000806093515e-01, /* atan(0.5)hi 0x3FDDAC67, 0x0561BB4F */
+    7.85398163397448278999e-01, /* atan(1.0)hi 0x3FE921FB, 0x54442D18 */
+    9.82793723247329054082e-01, /* atan(1.5)hi 0x3FEF730B, 0xD281F69B */
+    1.57079632679489655800e+00, /* atan(inf)hi 0x3FF921FB, 0x54442D18 */
+];
+
+const ATANLO: [f64; 4] = [
+    2.26987774529616870924e-17, /* atan(0.5)lo 0x3C7A2B7F, 0x222F65E2 */
+    3.06161699786838301793e-17, /* atan(1.0)lo 0x3C81A626, 0x33145C07 */
+    1.39033110312309984516e-17, /* atan(1.5)lo 0x3C700788, 0x7AF0CBBD */
+    6.12323399573676603587e-17, /* atan(inf)lo 0x3C91A626, 0x33145C07 */
+];
+
+const AT: [f64; 11] = [
+    3.33333333333329318027e-01,  /* 0x3FD55555, 0x5555550D */
+    -1.99999999998764832476e-01, /* 0xBFC99999, 0x9998EBC4 */
+    1.42857142725034663711e-01,  /* 0x3FC24924, 0x920083FF */
+    -1.11111104054623557880e-01, /* 0xBFBC71C6, 0xFE231671 */
+    9.09088713343650656196e-02,  /* 0x3FB745CD, 0xC54C206E */
+    -7.69187620504482999495e-02, /* 0xBFB3B0F2, 0xAF749A6D */
+    6.66107313738753120669e-02,  /* 0x3FB10D66, 0xA0D03D51 */
+    -5.83357013379057348645e-02, /* 0xBFADDE2D, 0x52DEFD9A */
+    4.97687799461593236017e-02,  /* 0x3FA97B4B, 0x24760DEB */
+    -3.65315727442169155270e-02, /* 0xBFA2B444, 0x2C6A6C2F */
+    1.62858201153657823623e-02,  /* 0x3F90AD3A, 0xE322DA11 */
+];
+
+#[inline]
+pub fn atan(x: f64) -> f64 {
+    let mut x = x;
+    let mut ix = (x.to_bits() >> 32) as u32;
+    let sign = ix >> 31;
+    ix &= 0x7fff_ffff;
+    if ix >= 0x4410_0000 {
+        if x.is_nan() {
+            return x;
+        }
+
+        let z = ATANHI[3] + f64::from_bits(0x0380_0000); // 0x1p-120f
+        return if sign != 0 { -z } else { z };
+    }
+
+    let id = if ix < 0x3fdc_0000 {
+        /* |x| < 0.4375 */
+        if ix < 0x3e40_0000 {
+            /* |x| < 2^-27 */
+            if ix < 0x0010_0000 {
+                /* raise underflow for subnormal x */
+                force_eval!(x as f32);
+            }
+
+            return x;
+        }
+
+        -1
+    } else {
+        x = fabs(x);
+        if ix < 0x3ff30000 {
+            /* |x| < 1.1875 */
+            if ix < 0x3fe60000 {
+                /* 7/16 <= |x| < 11/16 */
+                x = (2. * x - 1.) / (2. + x);
+                0
+            } else {
+                /* 11/16 <= |x| < 19/16 */
+                x = (x - 1.) / (x + 1.);
+                1
+            }
+        } else {
+            if ix < 0x40038000 {
+                /* |x| < 2.4375 */
+                x = (x - 1.5) / (1. + 1.5 * x);
+                2
+            } else {
+                /* 2.4375 <= |x| < 2^66 */
+                x = -1. / x;
+                3
+            }
+        }
+    };
+
+    let z = x * x;
+    let w = z * z;
+    /* break sum from i=0 to 10 AT[i]z**(i+1) into odd and even poly */
+    let s1 = z * (AT[0] + w * (AT[2] + w * (AT[4] + w * (AT[6] + w * (AT[8] + w * AT[10])))));
+    let s2 = w * (AT[1] + w * (AT[3] + w * (AT[5] + w * (AT[7] + w * AT[9]))));
+
+    if id < 0 {
+        return x - x * (s1 + s2);
+    }
+
+    let z = ATANHI[id as usize] - (x * (s1 + s2) - ATANLO[id as usize] - x);
+
+    if sign != 0 {
+        -z
+    } else {
+        z
+    }
+}
+
+#[cfg(test)]
+mod tests {
+    use super::atan;
+    use core::f64;
+
+    #[test]
+    fn sanity_check() {
+        for (input, answer) in [
+            (3.0_f64.sqrt() / 3.0, f64::consts::FRAC_PI_6),
+            (1.0, f64::consts::FRAC_PI_4),
+            (3.0_f64.sqrt(), f64::consts::FRAC_PI_3),
+            (-3.0_f64.sqrt() / 3.0, -f64::consts::FRAC_PI_6),
+            (-1.0, -f64::consts::FRAC_PI_4),
+            (-3.0_f64.sqrt(), -f64::consts::FRAC_PI_3),
+        ].iter()
+        {
+            assert!(
+                (atan(*input) - answer) / answer < 1e-5,
+                "\natan({:.4}/16) = {:.4}, actual: {}",
+                input * 16.0,
+                answer,
+                atan(*input)
+            );
+        }
+    }
+
+    #[test]
+    fn zero() {
+        assert_eq!(atan(0.0), 0.0);
+    }
+
+    #[test]
+    fn infinity() {
+        assert_eq!(atan(f64::INFINITY), f64::consts::FRAC_PI_2);
+    }
+
+    #[test]
+    fn minus_infinity() {
+        assert_eq!(atan(f64::NEG_INFINITY), -f64::consts::FRAC_PI_2);
+    }
+
+    #[test]
+    fn nan() {
+        assert!(atan(f64::NAN).is_nan());
+    }
+}

src/math/mod.rs

@@ -11,6 +11,7 @@ mod acos;
 mod acosf;
 mod asin;
 mod asinf;
+mod atan;
 mod atan2f;
 mod atanf;
 mod cbrt;
@@ -70,6 +71,7 @@ pub use self::acos::acos;
 pub use self::acosf::acosf;
 pub use self::asin::asin;
 pub use self::asinf::asinf;
+pub use self::atan::atan;
 pub use self::atan2f::atan2f;
 pub use self::atanf::atanf;
 pub use self::cbrt::cbrt;

test-generator/src/main.rs

@@ -745,7 +745,7 @@ f32i32_f32! {
 f64_f64! {
     acos,
     asin,
-    // atan,
+    atan,
     cbrt,
     ceil,
     cos,