Average Error: 37.0 → 0.5
Time: 6.4s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\log \left(e^{\sin x \cdot \left(\cos \varepsilon - 1\right)}\right) + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\log \left(e^{\sin x \cdot \left(\cos \varepsilon - 1\right)}\right) + \cos x \cdot \sin \varepsilon
double f(double x, double eps) {
        double r95831 = x;
        double r95832 = eps;
        double r95833 = r95831 + r95832;
        double r95834 = sin(r95833);
        double r95835 = sin(r95831);
        double r95836 = r95834 - r95835;
        return r95836;
}
↓
double f(double x, double eps) {
        double r95837 = x;
        double r95838 = sin(r95837);
        double r95839 = eps;
        double r95840 = cos(r95839);
        double r95841 = 1.0;
        double r95842 = r95840 - r95841;
        double r95843 = r95838 * r95842;
        double r95844 = exp(r95843);
        double r95845 = log(r95844);
        double r95846 = cos(r95837);
        double r95847 = sin(r95839);
        double r95848 = r95846 * r95847;
        double r95849 = r95845 + r95848;
        return r95849;
}

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
Target15.5
Herbie0.5
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.0

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

  3. Applied sin-sum21.3

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

  4. Applied associate--l+21.3

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

  5. Taylor expanded around inf 21.3

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

  6. Simplified0.4

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

  8. Applied add-log-exp0.5

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

  9. Final simplification0.5

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

Reproduce

herbie shell --seed 2020033 
(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)))