Average Error: 36.8 → 0.4
Time: 6.2s
Precision: binary64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\sqrt[3]{{\left(\sin x \cdot \left(\cos \varepsilon - 1\right)\right)}^{3}} + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\sqrt[3]{{\left(\sin x \cdot \left(\cos \varepsilon - 1\right)\right)}^{3}} + \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) cbrt(((double) pow(((double) (((double) sin(x)) * ((double) (((double) cos(eps)) - 1.0)))), 3.0)))) + ((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

Original36.8
Target15.1
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.8

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

  3. Applied sin-sum21.5

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

  4. Taylor expanded around inf 21.5

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

  5. Simplified0.4

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

  7. Applied add-cbrt-cube0.4

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

  8. Applied add-cbrt-cube0.5

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

  9. Applied cbrt-unprod0.4

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2020155 
(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)))