Average Error: 36.6 → 0.5
Time: 6.0s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\sin x \cdot \frac{\left(\sqrt[3]{{\left(\cos \varepsilon\right)}^{3} - 1} \cdot \sqrt[3]{{\left(\cos \varepsilon\right)}^{3} - 1}\right) \cdot \sqrt[3]{{\left(\cos \varepsilon\right)}^{3} - 1}}{\cos \varepsilon \cdot \left(\cos \varepsilon + 1\right) + 1} + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\sin x \cdot \frac{\left(\sqrt[3]{{\left(\cos \varepsilon\right)}^{3} - 1} \cdot \sqrt[3]{{\left(\cos \varepsilon\right)}^{3} - 1}\right) \cdot \sqrt[3]{{\left(\cos \varepsilon\right)}^{3} - 1}}{\cos \varepsilon \cdot \left(\cos \varepsilon + 1\right) + 1} + \cos x \cdot \sin \varepsilon
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(x)) * ((double) (((double) (((double) (((double) cbrt(((double) (((double) pow(((double) cos(eps)), 3.0)) - 1.0)))) * ((double) cbrt(((double) (((double) pow(((double) cos(eps)), 3.0)) - 1.0)))))) * ((double) cbrt(((double) (((double) pow(((double) cos(eps)), 3.0)) - 1.0)))))) / ((double) (((double) (((double) cos(eps)) * ((double) (((double) cos(eps)) + 1.0)))) + 1.0)))))) + ((double) (((double) cos(x)) * ((double) 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

Original36.6
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 36.6

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

  3. Applied sin-sum21.6

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

  4. Applied associate--l+21.6

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

  5. Taylor expanded around inf 21.6

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

  6. Simplified0.4

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

  8. Applied flip3--0.4

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

  9. Simplified0.4

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

  10. Simplified0.4

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

  12. Applied add-cube-cbrt0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2020140 
(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)))