Average Error: 37.3 → 0.4
Time: 7.3s
Precision: binary64
\[\sin \left(x + \varepsilon\right) - \sin x \]
↓
\[\left(\sin \varepsilon \cdot \cos x + \sin x \cdot \left(\cos \varepsilon + -1\right)\right) + \mathsf{fma}\left(-\sin x, 1, \sin x\right) \]
\sin \left(x + \varepsilon\right) - \sin x
↓
\left(\sin \varepsilon \cdot \cos x + \sin x \cdot \left(\cos \varepsilon + -1\right)\right) + \mathsf{fma}\left(-\sin x, 1, \sin x\right)

(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))
↓
(FPCore (x eps)
 :precision binary64
 (+
  (+ (* (sin eps) (cos x)) (* (sin x) (+ (cos eps) -1.0)))
  (fma (- (sin x)) 1.0 (sin x))))
double code(double x, double eps) {
	return sin(x + eps) - sin(x);
}
↓
double code(double x, double eps) {
	return ((sin(eps) * cos(x)) + (sin(x) * (cos(eps) + -1.0))) + fma(-sin(x), 1.0, sin(x));
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.3
Target15.0
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \]

Derivation

  1. Initial program 37.3

    \[\sin \left(x + \varepsilon\right) - \sin x \]

  2. Applied sin-sum_binary6422.1

    \[\leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x \]

  3. Applied associate--l+_binary6422.1

    \[\leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)} \]

  4. Applied *-un-lft-identity_binary6422.1

    \[\leadsto \sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \color{blue}{1 \cdot \sin x}\right) \]

  5. Applied prod-diff_binary6422.1

    \[\leadsto \sin x \cdot \cos \varepsilon + \color{blue}{\left(\mathsf{fma}\left(\cos x, \sin \varepsilon, -\sin x \cdot 1\right) + \mathsf{fma}\left(-\sin x, 1, \sin x \cdot 1\right)\right)} \]

  6. Applied associate-+r+_binary6422.1

    \[\leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \mathsf{fma}\left(\cos x, \sin \varepsilon, -\sin x \cdot 1\right)\right) + \mathsf{fma}\left(-\sin x, 1, \sin x \cdot 1\right)} \]

  7. Simplified0.4

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \left(\cos \varepsilon + -1\right)\right)} + \mathsf{fma}\left(-\sin x, 1, \sin x \cdot 1\right) \]

  8. Applied fma-udef_binary640.4

    \[\leadsto \color{blue}{\left(\sin \varepsilon \cdot \cos x + \sin x \cdot \left(\cos \varepsilon + -1\right)\right)} + \mathsf{fma}\left(-\sin x, 1, \sin x \cdot 1\right) \]

  9. Final simplification0.4

    \[\leadsto \left(\sin \varepsilon \cdot \cos x + \sin x \cdot \left(\cos \varepsilon + -1\right)\right) + \mathsf{fma}\left(-\sin x, 1, \sin x\right) \]

Reproduce

herbie shell --seed 2022081 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

  :herbie-target
  (* 2.0 (* (cos (+ x (/ eps 2.0))) (sin (/ eps 2.0))))

  (- (sin (+ x eps)) (sin x)))