Average Error: 37.1 → 0.4
Time: 6.1s
Precision: binary64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\cos x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right) - \sqrt[3]{{\left(\sin x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)}^{3}}\right)\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\cos x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right) - \sqrt[3]{{\left(\sin x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)}^{3}}\right)\right)
double code(double x, double eps) {
	return ((double) (((double) sin(((double) (x + eps)))) - ((double) sin(x))));
}
↓
double code(double x, double eps) {
	return ((double) (2.0 * ((double) (((double) sin(((double) (eps / 2.0)))) * ((double) (((double) (((double) cos(x)) * ((double) cos(((double) (eps * 0.5)))))) - ((double) cbrt(((double) pow(((double) (((double) sin(x)) * ((double) sin(((double) (eps * 0.5)))))), 3.0))))))))));
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.1
Target15.2
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.1

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

  3. Applied diff-sin37.4

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

  4. Simplified15.3

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

  5. Taylor expanded around inf 15.3

    \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \left(2 \cdot x + \varepsilon\right)\right)}\right)\]

  6. Simplified15.2

    \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\cos \left(x + \varepsilon \cdot \frac{1}{2}\right)}\right)\]
  7. Using strategy rm

  8. Applied cos-sum0.3

    \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\left(\cos x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right) - \sin x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)}\right)\]
  9. Using strategy rm

  10. Applied add-cbrt-cube0.4

    \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\cos x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right) - \sin x \cdot \color{blue}{\sqrt[3]{\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)}}\right)\right)\]

  11. Applied add-cbrt-cube0.4

    \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\cos x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right) - \color{blue}{\sqrt[3]{\left(\sin x \cdot \sin x\right) \cdot \sin x}} \cdot \sqrt[3]{\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)}\right)\right)\]

  12. Applied cbrt-unprod0.4

    \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\cos x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right) - \color{blue}{\sqrt[3]{\left(\left(\sin x \cdot \sin x\right) \cdot \sin x\right) \cdot \left(\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)}}\right)\right)\]

  13. Simplified0.4

    \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\cos x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right) - \sqrt[3]{\color{blue}{{\left(\sin x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)}^{3}}}\right)\right)\]

  14. Final simplification0.4

    \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\cos x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right) - \sqrt[3]{{\left(\sin x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)}^{3}}\right)\right)\]

Reproduce

herbie shell --seed 2020184 
(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)))