Average Error: 36.9 → 1.5
Time: 6.0s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\sin x \cdot \left(\cos \varepsilon - 1\right) + \left(\cos x \cdot \left(\sqrt[3]{\sin \varepsilon} \cdot \sqrt[3]{\sin \varepsilon}\right)\right) \cdot \sqrt[3]{\sin \varepsilon}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\sin x \cdot \left(\cos \varepsilon - 1\right) + \left(\cos x \cdot \left(\sqrt[3]{\sin \varepsilon} \cdot \sqrt[3]{\sin \varepsilon}\right)\right) \cdot \sqrt[3]{\sin \varepsilon}
double code(double x, double eps) {
	return (sin((x + eps)) - sin(x));
}
↓
double code(double x, double eps) {
	return ((sin(x) * (cos(eps) - 1.0)) + ((cos(x) * (cbrt(sin(eps)) * cbrt(sin(eps)))) * cbrt(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

Original36.9
Target15.1
Herbie1.5
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.9

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

  3. Applied sin-sum21.7

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

  4. Applied associate--l+21.7

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

  5. Taylor expanded around inf 21.7

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

  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 add-cube-cbrt1.5

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

  9. Applied associate-*r*1.5

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

  10. Final simplification1.5

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

Reproduce

herbie shell --seed 2020058 
(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)))