Average Error: 37.6 → 0.4
Time: 5.9s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\sin x \cdot \sqrt[3]{{\left(\cos \varepsilon - 1\right)}^{3}} + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\sin x \cdot \sqrt[3]{{\left(\cos \varepsilon - 1\right)}^{3}} + \cos x \cdot \sin \varepsilon
double f(double x, double eps) {
        double r113974 = x;
        double r113975 = eps;
        double r113976 = r113974 + r113975;
        double r113977 = sin(r113976);
        double r113978 = sin(r113974);
        double r113979 = r113977 - r113978;
        return r113979;
}
↓
double f(double x, double eps) {
        double r113980 = x;
        double r113981 = sin(r113980);
        double r113982 = eps;
        double r113983 = cos(r113982);
        double r113984 = 1.0;
        double r113985 = r113983 - r113984;
        double r113986 = 3.0;
        double r113987 = pow(r113985, r113986);
        double r113988 = cbrt(r113987);
        double r113989 = r113981 * r113988;
        double r113990 = cos(r113980);
        double r113991 = sin(r113982);
        double r113992 = r113990 * r113991;
        double r113993 = r113989 + r113992;
        return r113993;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.6
Target15.4
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.6

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

  3. Applied sin-sum22.1

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

  4. Taylor expanded around inf 22.1

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

  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 add-cbrt-cube0.4

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

  8. Simplified0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2020025 
(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)))