Average Error: 37.5 → 0.5
Time: 6.1s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\frac{\sin x \cdot \left(\log \left(e^{{\left(\cos \varepsilon\right)}^{3}}\right) - 1\right)}{\cos \varepsilon \cdot \cos \varepsilon + \left(1 + \cos \varepsilon \cdot 1\right)} + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\frac{\sin x \cdot \left(\log \left(e^{{\left(\cos \varepsilon\right)}^{3}}\right) - 1\right)}{\cos \varepsilon \cdot \cos \varepsilon + \left(1 + \cos \varepsilon \cdot 1\right)} + \cos x \cdot \sin \varepsilon
double f(double x, double eps) {
        double r111022 = x;
        double r111023 = eps;
        double r111024 = r111022 + r111023;
        double r111025 = sin(r111024);
        double r111026 = sin(r111022);
        double r111027 = r111025 - r111026;
        return r111027;
}
↓
double f(double x, double eps) {
        double r111028 = x;
        double r111029 = sin(r111028);
        double r111030 = eps;
        double r111031 = cos(r111030);
        double r111032 = 3.0;
        double r111033 = pow(r111031, r111032);
        double r111034 = exp(r111033);
        double r111035 = log(r111034);
        double r111036 = 1.0;
        double r111037 = r111035 - r111036;
        double r111038 = r111029 * r111037;
        double r111039 = r111031 * r111031;
        double r111040 = r111031 * r111036;
        double r111041 = r111036 + r111040;
        double r111042 = r111039 + r111041;
        double r111043 = r111038 / r111042;
        double r111044 = cos(r111028);
        double r111045 = sin(r111030);
        double r111046 = r111044 * r111045;
        double r111047 = r111043 + r111046;
        return r111047;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.5
Target14.9
Herbie0.5
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.5

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

  3. Applied sin-sum22.5

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

  4. Taylor expanded around inf 22.5

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

  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 flip3--0.5

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

  8. Applied associate-*r/0.5

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

  9. Simplified0.5

    \[\leadsto \frac{\color{blue}{\sin x \cdot \left({\left(\cos \varepsilon\right)}^{3} - 1\right)}}{\cos \varepsilon \cdot \cos \varepsilon + \left(1 \cdot 1 + \cos \varepsilon \cdot 1\right)} + \cos x \cdot \sin \varepsilon\]
  10. Using strategy rm

  11. Applied add-log-exp0.5

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

  12. Final simplification0.5

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

Reproduce

herbie shell --seed 2019353 
(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)))