Average Error: 36.9 → 0.5
Time: 6.4s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\sin \varepsilon \cdot \cos x + \sin x \cdot \sqrt[3]{{\left(\cos \varepsilon - 1\right)}^{3}}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\sin \varepsilon \cdot \cos x + \sin x \cdot \sqrt[3]{{\left(\cos \varepsilon - 1\right)}^{3}}
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) sin(x)) * ((double) cbrt(((double) pow(((double) (((double) cos(eps)) - 1.0)), 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

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. Simplified0.4

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

  8. Applied add-cbrt-cube0.5

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

  9. Simplified0.5

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

  10. Final simplification0.5

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

Reproduce

herbie shell --seed 2020114 +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)))