Average Error: 37.6 → 0.6
Time: 6.0s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right) \cdot \left(\sqrt[3]{\sin x} \cdot \left(\cos \varepsilon - 1\right)\right) + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right) \cdot \left(\sqrt[3]{\sin x} \cdot \left(\cos \varepsilon - 1\right)\right) + \cos x \cdot \sin \varepsilon
double f(double x, double eps) {
        double r157709 = x;
        double r157710 = eps;
        double r157711 = r157709 + r157710;
        double r157712 = sin(r157711);
        double r157713 = sin(r157709);
        double r157714 = r157712 - r157713;
        return r157714;
}
↓
double f(double x, double eps) {
        double r157715 = x;
        double r157716 = sin(r157715);
        double r157717 = cbrt(r157716);
        double r157718 = r157717 * r157717;
        double r157719 = eps;
        double r157720 = cos(r157719);
        double r157721 = 1.0;
        double r157722 = r157720 - r157721;
        double r157723 = r157717 * r157722;
        double r157724 = r157718 * r157723;
        double r157725 = cos(r157715);
        double r157726 = sin(r157719);
        double r157727 = r157725 * r157726;
        double r157728 = r157724 + r157727;
        return r157728;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.6
Target15.4
Herbie0.6
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.6

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

  3. Applied sin-sum22.1

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

  4. Applied associate--l+22.1

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

  5. Taylor expanded around inf 22.1

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

  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-cbrt0.6

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

  9. Applied associate-*l*0.6

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

  10. Final simplification0.6

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

Reproduce

herbie shell --seed 2020036 
(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)))