Average Error: 36.9 → 0.5
Time: 6.0s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\mathsf{fma}\left(\sin x, \log \left(\sqrt{e^{\cos \varepsilon - 1}}\right) + \log \left(\sqrt{e^{\cos \varepsilon - 1}}\right), \cos x \cdot \sin \varepsilon\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\mathsf{fma}\left(\sin x, \log \left(\sqrt{e^{\cos \varepsilon - 1}}\right) + \log \left(\sqrt{e^{\cos \varepsilon - 1}}\right), \cos x \cdot \sin \varepsilon\right)
double code(double x, double eps) {
	return (sin((x + eps)) - sin(x));
}
↓
double code(double x, double eps) {
	return fma(sin(x), (log(sqrt(exp((cos(eps) - 1.0)))) + log(sqrt(exp((cos(eps) - 1.0))))), (cos(x) * sin(eps)));
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.9
Target15.1
Herbie0.5
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.9

    \[\sin \left(x + \varepsilon\right) - \sin x\]
  2. Using strategy rm

  3. Applied sin-sum21.7

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

  4. Applied associate--l+21.7

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

  5. Taylor expanded around inf 21.7

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

  6. Simplified0.4

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

  8. Applied add-log-exp0.4

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

  9. Applied add-log-exp0.4

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

  10. Applied diff-log0.4

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

  11. Simplified0.4

    \[\leadsto \mathsf{fma}\left(\sin x, \log \color{blue}{\left(e^{\cos \varepsilon - 1}\right)}, \cos x \cdot \sin \varepsilon\right)\]
  12. Using strategy rm

  13. Applied add-sqr-sqrt0.5

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

  14. Applied log-prod0.5

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

  15. Final simplification0.5

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

Reproduce

herbie shell --seed 2020058 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

  :herbie-target
  (* 2 (* (cos (+ x (/ eps 2))) (sin (/ eps 2))))

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