Average Error: 37.1 → 0.5
Time: 6.9s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\sin \varepsilon \cdot \cos x + \frac{\sin x \cdot \frac{\sin x}{\frac{\mathsf{fma}\left(\sin x, {\left(\cos \varepsilon\right)}^{2}, \sin x\right)}{\sin x \cdot {\left(\cos \varepsilon\right)}^{4} - \sin x}}}{\mathsf{fma}\left(\cos \varepsilon, \sin x, \sin x\right)}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\sin \varepsilon \cdot \cos x + \frac{\sin x \cdot \frac{\sin x}{\frac{\mathsf{fma}\left(\sin x, {\left(\cos \varepsilon\right)}^{2}, \sin x\right)}{\sin x \cdot {\left(\cos \varepsilon\right)}^{4} - \sin x}}}{\mathsf{fma}\left(\cos \varepsilon, \sin x, \sin x\right)}
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) * (sin(x) / (fma(sin(x), pow(cos(eps), 2.0), sin(x)) / ((sin(x) * pow(cos(eps), 4.0)) - sin(x))))) / fma(cos(eps), sin(x), sin(x))));
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.1
Target15.3
Herbie0.5
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.1

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

  3. Applied +-commutative37.1

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

  4. Applied sin-sum21.6

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

  5. Applied associate--l+0.4

    \[\leadsto \color{blue}{\sin \varepsilon \cdot \cos x + \left(\cos \varepsilon \cdot \sin x - \sin x\right)}\]
  6. Using strategy rm

  7. Applied flip--0.6

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

  8. Simplified0.6

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

  9. Simplified0.6

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

  11. Applied flip--0.6

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

  12. Simplified0.5

    \[\leadsto \sin \varepsilon \cdot \cos x + \frac{\sin x \cdot \frac{\color{blue}{\sin x \cdot \left(\left(\sin x \cdot {\left(\cos \varepsilon\right)}^{2}\right) \cdot {\left(\cos \varepsilon\right)}^{2} - \sin x\right)}}{\left(\cos \varepsilon \cdot \sin x\right) \cdot \cos \varepsilon + \sin x}}{\mathsf{fma}\left(\cos \varepsilon, \sin x, \sin x\right)}\]

  13. Simplified0.6

    \[\leadsto \sin \varepsilon \cdot \cos x + \frac{\sin x \cdot \frac{\sin x \cdot \left(\left(\sin x \cdot {\left(\cos \varepsilon\right)}^{2}\right) \cdot {\left(\cos \varepsilon\right)}^{2} - \sin x\right)}{\color{blue}{\mathsf{fma}\left({\left(\cos \varepsilon\right)}^{2}, \sin x, \sin x\right)}}}{\mathsf{fma}\left(\cos \varepsilon, \sin x, \sin x\right)}\]
  14. Using strategy rm

  15. Applied associate-/l*0.6

    \[\leadsto \sin \varepsilon \cdot \cos x + \frac{\sin x \cdot \color{blue}{\frac{\sin x}{\frac{\mathsf{fma}\left({\left(\cos \varepsilon\right)}^{2}, \sin x, \sin x\right)}{\left(\sin x \cdot {\left(\cos \varepsilon\right)}^{2}\right) \cdot {\left(\cos \varepsilon\right)}^{2} - \sin x}}}}{\mathsf{fma}\left(\cos \varepsilon, \sin x, \sin x\right)}\]

  16. Simplified0.5

    \[\leadsto \sin \varepsilon \cdot \cos x + \frac{\sin x \cdot \frac{\sin x}{\color{blue}{\frac{\mathsf{fma}\left(\sin x, {\left(\cos \varepsilon\right)}^{2}, \sin x\right)}{\sin x \cdot {\left(\cos \varepsilon\right)}^{4} - \sin x}}}}{\mathsf{fma}\left(\cos \varepsilon, \sin x, \sin x\right)}\]

  17. Final simplification0.5

    \[\leadsto \sin \varepsilon \cdot \cos x + \frac{\sin x \cdot \frac{\sin x}{\frac{\mathsf{fma}\left(\sin x, {\left(\cos \varepsilon\right)}^{2}, \sin x\right)}{\sin x \cdot {\left(\cos \varepsilon\right)}^{4} - \sin x}}}{\mathsf{fma}\left(\cos \varepsilon, \sin x, \sin x\right)}\]

Reproduce

herbie shell --seed 2020078 +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)))