Average Error: 37.6 → 0.4
Time: 9.6s
Precision: binary64
\[\sin \left(x + \varepsilon\right) - \sin x \]
↓
\[\mathsf{fma}\left(\cos x, \sin \varepsilon, \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \varepsilon\right)\right) - 1\right) \cdot \sin x\right) \]
\sin \left(x + \varepsilon\right) - \sin x
↓
\mathsf{fma}\left(\cos x, \sin \varepsilon, \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \varepsilon\right)\right) - 1\right) \cdot \sin x\right)

(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))
↓
(FPCore (x eps)
 :precision binary64
 (fma (cos x) (sin eps) (* (- (log1p (expm1 (cos eps))) 1.0) (sin x))))
double code(double x, double eps) {
	return sin(x + eps) - sin(x);
}
↓
double code(double x, double eps) {
	return fma(cos(x), sin(eps), ((log1p(expm1(cos(eps))) - 1.0) * sin(x)));
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.6
Target15.4
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \]

Derivation

  1. Initial program 37.6

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

  2. Applied sin-sum_binary6422.2

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

  3. Applied add-cube-cbrt_binary6422.9

    \[\leadsto \color{blue}{\left(\sqrt[3]{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \cdot \sqrt[3]{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x}\right) \cdot \sqrt[3]{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x}} \]

  4. Simplified22.8

    \[\leadsto \color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \left(\cos \varepsilon + -1\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \left(\cos \varepsilon + -1\right)\right)}\right)} \cdot \sqrt[3]{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \]

  5. Simplified1.6

    \[\leadsto \left(\sqrt[3]{\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \left(\cos \varepsilon + -1\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \left(\cos \varepsilon + -1\right)\right)}\right) \cdot \color{blue}{\sqrt[3]{\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \left(\cos \varepsilon + -1\right)\right)}} \]

  6. Taylor expanded in eps around inf 22.2

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

  7. Simplified0.4

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

  8. Applied log1p-expm1-u_binary640.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2022020 
(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)))