Average Error: 36.7 → 0.4
Time: 18.3s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\sin x \cdot \sqrt[3]{{\left(\cos \varepsilon - 1\right)}^{3}} + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\sin x \cdot \sqrt[3]{{\left(\cos \varepsilon - 1\right)}^{3}} + \cos x \cdot \sin \varepsilon
double f(double x, double eps) {
        double r105558 = x;
        double r105559 = eps;
        double r105560 = r105558 + r105559;
        double r105561 = sin(r105560);
        double r105562 = sin(r105558);
        double r105563 = r105561 - r105562;
        return r105563;
}
↓
double f(double x, double eps) {
        double r105564 = x;
        double r105565 = sin(r105564);
        double r105566 = eps;
        double r105567 = cos(r105566);
        double r105568 = 1.0;
        double r105569 = r105567 - r105568;
        double r105570 = 3.0;
        double r105571 = pow(r105569, r105570);
        double r105572 = cbrt(r105571);
        double r105573 = r105565 * r105572;
        double r105574 = cos(r105564);
        double r105575 = sin(r105566);
        double r105576 = r105574 * r105575;
        double r105577 = r105573 + r105576;
        return r105577;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.7
Target14.9
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.7

    \[\sin \left(x + \varepsilon\right) - \sin x\]

  2. Simplified36.7

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

  4. Applied sin-sum21.7

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

  5. Applied associate--l+0.4

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

  7. Applied *-un-lft-identity0.4

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

  8. Applied distribute-rgt-out--0.4

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

  10. Applied add-cbrt-cube0.4

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

  11. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019195 
(FPCore (x eps)
  :name "2sin (example 3.3)"

  :herbie-target
  (* 2.0 (* (cos (+ x (/ eps 2.0))) (sin (/ eps 2.0))))

  (- (sin (+ x eps)) (sin x)))