Average Error: 37.2 → 0.4
Time: 14.9s
Precision: binary64
\[\sin \left(x + \varepsilon\right) - \sin x \]
↓
\[\mathsf{fma}\left(-1 + \cos \varepsilon, \sin x, \cos x \cdot \sin \varepsilon\right) \]
\sin \left(x + \varepsilon\right) - \sin x
↓
\mathsf{fma}\left(-1 + \cos \varepsilon, \sin x, \cos x \cdot \sin \varepsilon\right)

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

Error

Bits error versus x

Bits error versus eps

Target

Original37.2
Target15.1
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \]

Derivation

  1. Initial program 37.2

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

  2. Applied sin-sum_binary6422.0

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

  3. Applied log1p-expm1-u_binary6422.1

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\right)\right)} \]

  4. Simplified0.5

    \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \left(\cos \varepsilon + -1\right)\right)\right)}\right) \]

  5. Applied add-log-exp_binary640.5

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \left(\cos \varepsilon + \color{blue}{\log \left(e^{-1}\right)}\right)\right)\right)\right) \]

  6. Applied add-log-exp_binary640.5

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \left(\color{blue}{\log \left(e^{\cos \varepsilon}\right)} + \log \left(e^{-1}\right)\right)\right)\right)\right) \]

  7. Applied sum-log_binary640.6

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \color{blue}{\log \left(e^{\cos \varepsilon} \cdot e^{-1}\right)}\right)\right)\right) \]

  8. Taylor expanded in eps around inf 30.2

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2021224 
(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)))