Average Error: 36.9 → 0.5
Time: 6.2s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\sin \varepsilon \cdot \cos x + \left(\sin x \cdot \frac{\sqrt[3]{\cos \varepsilon - 1} \cdot \sqrt[3]{{\left(\cos \varepsilon\right)}^{3} - 1}}{\sqrt[3]{\cos \varepsilon \cdot \cos \varepsilon + \left(1 \cdot 1 + \cos \varepsilon \cdot 1\right)}}\right) \cdot \sqrt[3]{\cos \varepsilon - 1}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\sin \varepsilon \cdot \cos x + \left(\sin x \cdot \frac{\sqrt[3]{\cos \varepsilon - 1} \cdot \sqrt[3]{{\left(\cos \varepsilon\right)}^{3} - 1}}{\sqrt[3]{\cos \varepsilon \cdot \cos \varepsilon + \left(1 \cdot 1 + \cos \varepsilon \cdot 1\right)}}\right) \cdot \sqrt[3]{\cos \varepsilon - 1}
double code(double x, double eps) {
	return ((double) (((double) sin(((double) (x + eps)))) - ((double) sin(x))));
}
↓
double code(double x, double eps) {
	return ((double) (((double) (((double) sin(eps)) * ((double) cos(x)))) + ((double) (((double) (((double) sin(x)) * ((double) (((double) (((double) cbrt(((double) (((double) cos(eps)) - 1.0)))) * ((double) cbrt(((double) (((double) pow(((double) cos(eps)), 3.0)) - 1.0)))))) / ((double) cbrt(((double) (((double) (((double) cos(eps)) * ((double) cos(eps)))) + ((double) (((double) (1.0 * 1.0)) + ((double) (((double) cos(eps)) * 1.0)))))))))))) * ((double) cbrt(((double) (((double) cos(eps)) - 1.0))))))));
}

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
Target14.9
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 +-commutative36.9

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

  4. Applied sin-sum21.9

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

  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 *-un-lft-identity0.4

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

  8. Applied distribute-rgt-out--0.4

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

  10. Applied add-cube-cbrt0.6

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

  11. Applied associate-*r*0.6

    \[\leadsto \sin \varepsilon \cdot \cos x + \color{blue}{\left(\sin x \cdot \left(\sqrt[3]{\cos \varepsilon - 1} \cdot \sqrt[3]{\cos \varepsilon - 1}\right)\right) \cdot \sqrt[3]{\cos \varepsilon - 1}}\]
  12. Using strategy rm

  13. Applied flip3--0.6

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

  14. Applied cbrt-div0.5

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

  15. Applied associate-*r/0.5

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

  16. Simplified0.5

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

  17. Final simplification0.5

    \[\leadsto \sin \varepsilon \cdot \cos x + \left(\sin x \cdot \frac{\sqrt[3]{\cos \varepsilon - 1} \cdot \sqrt[3]{{\left(\cos \varepsilon\right)}^{3} - 1}}{\sqrt[3]{\cos \varepsilon \cdot \cos \varepsilon + \left(1 \cdot 1 + \cos \varepsilon \cdot 1\right)}}\right) \cdot \sqrt[3]{\cos \varepsilon - 1}\]

Reproduce

herbie shell --seed 2020114 
(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)))