Average Error: 37.4 → 0.4
Time: 8.5s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\left(\sin x \cdot \cos \varepsilon + \left(-\sin x\right)\right) + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\left(\sin x \cdot \cos \varepsilon + \left(-\sin x\right)\right) + \cos x \cdot \sin \varepsilon
double f(double x, double eps) {
        double r162791 = x;
        double r162792 = eps;
        double r162793 = r162791 + r162792;
        double r162794 = sin(r162793);
        double r162795 = sin(r162791);
        double r162796 = r162794 - r162795;
        return r162796;
}
↓
double f(double x, double eps) {
        double r162797 = x;
        double r162798 = sin(r162797);
        double r162799 = eps;
        double r162800 = cos(r162799);
        double r162801 = r162798 * r162800;
        double r162802 = -r162798;
        double r162803 = r162801 + r162802;
        double r162804 = cos(r162797);
        double r162805 = sin(r162799);
        double r162806 = r162804 * r162805;
        double r162807 = r162803 + r162806;
        return r162807;
}

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
Target15.3
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.4

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

  3. Applied sin-sum21.9

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

  4. Taylor expanded around inf 21.9

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

  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 sub-neg0.4

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

  8. Applied distribute-lft-in0.4

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019322 
(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)))