Average Error: 37.4 → 0.5
Time: 5.8s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\sqrt[3]{{\left(\sin x \cdot \left(\cos \varepsilon - 1\right)\right)}^{3}} + \cos x \cdot \sin \varepsilon\]
\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 f(double x, double eps) {
        double r131701 = x;
        double r131702 = eps;
        double r131703 = r131701 + r131702;
        double r131704 = sin(r131703);
        double r131705 = sin(r131701);
        double r131706 = r131704 - r131705;
        return r131706;
}
↓
double f(double x, double eps) {
        double r131707 = x;
        double r131708 = sin(r131707);
        double r131709 = eps;
        double r131710 = cos(r131709);
        double r131711 = 1.0;
        double r131712 = r131710 - r131711;
        double r131713 = r131708 * r131712;
        double r131714 = 3.0;
        double r131715 = pow(r131713, r131714);
        double r131716 = cbrt(r131715);
        double r131717 = cos(r131707);
        double r131718 = sin(r131709);
        double r131719 = r131717 * r131718;
        double r131720 = r131716 + r131719;
        return r131720;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.4
Target15.1
Herbie0.5
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.4

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

  3. Applied sin-sum22.2

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

  4. Taylor expanded around inf 22.2

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

  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-cube0.5

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

  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-unprod0.5

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

  10. Simplified0.5

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

  11. Final simplification0.5

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

Reproduce

herbie shell --seed 2020047 
(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)))