Average Error: 37.0 → 0.4
Time: 23.9s
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 r3482732 = x;
        double r3482733 = eps;
        double r3482734 = r3482732 + r3482733;
        double r3482735 = sin(r3482734);
        double r3482736 = sin(r3482732);
        double r3482737 = r3482735 - r3482736;
        return r3482737;
}
↓
double f(double x, double eps) {
        double r3482738 = x;
        double r3482739 = cos(r3482738);
        double r3482740 = 0.5;
        double r3482741 = eps;
        double r3482742 = r3482740 * r3482741;
        double r3482743 = cos(r3482742);
        double r3482744 = r3482739 * r3482743;
        double r3482745 = sin(r3482738);
        double r3482746 = sin(r3482742);
        double r3482747 = r3482745 * r3482746;
        double r3482748 = r3482747 * r3482747;
        double r3482749 = r3482747 * r3482748;
        double r3482750 = cbrt(r3482749);
        double r3482751 = r3482744 - r3482750;
        double r3482752 = r3482751 * r3482746;
        double r3482753 = 2.0;
        double r3482754 = r3482752 * r3482753;
        return r3482754;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.0
Target15.0
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.0

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

  3. Applied diff-sin37.3

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

  4. Simplified15.0

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

  5. Taylor expanded around -inf 15.0

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

  6. Simplified15.0

    \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\varepsilon \cdot \frac{1}{2} + x\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)}\]
  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 2019144 
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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