Average Error: 37.1 → 0.4
Time: 6.0s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\frac{\sin x \cdot \left(-\sin \varepsilon \cdot \sin \varepsilon\right)}{\cos \varepsilon + 1} + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\frac{\sin x \cdot \left(-\sin \varepsilon \cdot \sin \varepsilon\right)}{\cos \varepsilon + 1} + \cos x \cdot \sin \varepsilon
double f(double x, double eps) {
        double r177736 = x;
        double r177737 = eps;
        double r177738 = r177736 + r177737;
        double r177739 = sin(r177738);
        double r177740 = sin(r177736);
        double r177741 = r177739 - r177740;
        return r177741;
}
↓
double f(double x, double eps) {
        double r177742 = x;
        double r177743 = sin(r177742);
        double r177744 = eps;
        double r177745 = sin(r177744);
        double r177746 = r177745 * r177745;
        double r177747 = -r177746;
        double r177748 = r177743 * r177747;
        double r177749 = cos(r177744);
        double r177750 = 1.0;
        double r177751 = r177749 + r177750;
        double r177752 = r177748 / r177751;
        double r177753 = cos(r177742);
        double r177754 = r177753 * r177745;
        double r177755 = r177752 + r177754;
        return r177755;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.1
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 37.1

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

  3. Applied sin-sum22.0

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

  4. Taylor expanded around inf 22.0

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

  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 flip--0.5

    \[\leadsto \sin x \cdot \color{blue}{\frac{\cos \varepsilon \cdot \cos \varepsilon - 1 \cdot 1}{\cos \varepsilon + 1}} + \cos x \cdot \sin \varepsilon\]

  8. Applied associate-*r/0.5

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

  9. Simplified0.5

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

  11. Applied sub-1-cos0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019347 
(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)))