Average Error: 36.9 → 0.4
Time: 18.6s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\left(\left(\cos x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right) - \sqrt[3]{\left(\sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\left(\sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right)}\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\left(\left(\cos x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right) - \sqrt[3]{\left(\sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\left(\sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right)}\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2
double f(double x, double eps) {
        double r5574446 = x;
        double r5574447 = eps;
        double r5574448 = r5574446 + r5574447;
        double r5574449 = sin(r5574448);
        double r5574450 = sin(r5574446);
        double r5574451 = r5574449 - r5574450;
        return r5574451;
}
↓
double f(double x, double eps) {
        double r5574452 = x;
        double r5574453 = cos(r5574452);
        double r5574454 = 0.5;
        double r5574455 = eps;
        double r5574456 = r5574454 * r5574455;
        double r5574457 = cos(r5574456);
        double r5574458 = r5574453 * r5574457;
        double r5574459 = sin(r5574452);
        double r5574460 = sin(r5574456);
        double r5574461 = r5574459 * r5574460;
        double r5574462 = r5574461 * r5574461;
        double r5574463 = r5574461 * r5574462;
        double r5574464 = cbrt(r5574463);
        double r5574465 = r5574458 - r5574464;
        double r5574466 = r5574465 * r5574460;
        double r5574467 = 2.0;
        double r5574468 = r5574466 * r5574467;
        return r5574468;
}

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
Target14.8
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.9

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

  3. Applied diff-sin37.2

    \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]

  4. Simplified14.8

    \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)}\]

  5. Taylor expanded around -inf 14.8

    \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot \left(2 \cdot x + \varepsilon\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)}\]

  6. Simplified14.8

    \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\varepsilon \cdot \frac{1}{2} + x\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)}\]
  7. Using strategy rm

  8. Applied cos-sum0.3

    \[\leadsto 2 \cdot \left(\color{blue}{\left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x - \sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right)} \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\]
  9. Using strategy rm

  10. Applied add-cbrt-cube0.4

    \[\leadsto 2 \cdot \left(\left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x - \color{blue}{\sqrt[3]{\left(\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right) \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right)\right) \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right)}}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\]

  11. Final simplification0.4

    \[\leadsto \left(\left(\cos x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right) - \sqrt[3]{\left(\sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\left(\sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right)}\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2\]

Reproduce

herbie shell --seed 2019138 
(FPCore (x eps)
  :name "2sin (example 3.3)"

  :herbie-target
  (* 2 (* (cos (+ x (/ eps 2))) (sin (/ eps 2))))

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