Average Error: 31.0 → 31.1
Time: 23.5s
Precision: binary64
\[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
↓
\[\left(\left(2 \cdot \mathsf{fma}\left(b, b, -a \cdot a\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \]
\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)
↓
\left(\left(2 \cdot \mathsf{fma}\left(b, b, -a \cdot a\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)

(FPCore (a b angle)
 :precision binary64
 (*
  (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* PI (/ angle 180.0))))
  (cos (* PI (/ angle 180.0)))))
↓
(FPCore (a b angle)
 :precision binary64
 (*
  (* (* 2.0 (fma b b (- (* a a)))) (sin (* PI (/ angle 180.0))))
  (cos (* 0.005555555555555556 (* PI angle)))))
double code(double a, double b, double angle) {
	return ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(((double) M_PI) * (angle / 180.0))) * cos(((double) M_PI) * (angle / 180.0));
}
↓
double code(double a, double b, double angle) {
	return ((2.0 * fma(b, b, -(a * a))) * sin(((double) M_PI) * (angle / 180.0))) * cos(0.005555555555555556 * (((double) M_PI) * angle));
}

Error

Bits error versus a

Bits error versus b

Bits error versus angle

Derivation

  1. Initial program 31.0

    \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

  2. Taylor expanded in angle around inf 31.1

    \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \]

  3. Applied unpow2_binary6431.1

    \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \]

  4. Applied fma-neg_binary6431.1

    \[\leadsto \left(\left(2 \cdot \color{blue}{\mathsf{fma}\left(b, b, -{a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \]

  5. Simplified31.1

    \[\leadsto \left(\left(2 \cdot \mathsf{fma}\left(b, b, \color{blue}{-a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \]

  6. Final simplification31.1

    \[\leadsto \left(\left(2 \cdot \mathsf{fma}\left(b, b, -a \cdot a\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \]

Reproduce

herbie shell --seed 2021275 
(FPCore (a b angle)
  :name "ab-angle->ABCF B"
  :precision binary64
  (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* PI (/ angle 180.0)))) (cos (* PI (/ angle 180.0)))))