Average Error: 36.4 → 0.4
Time: 21.4s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[2 \cdot \left(\left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x - \log \left(e^{\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x}\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
2 \cdot \left(\left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x - \log \left(e^{\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x}\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)
double f(double x, double eps) {
        double r4382184 = x;
        double r4382185 = eps;
        double r4382186 = r4382184 + r4382185;
        double r4382187 = sin(r4382186);
        double r4382188 = sin(r4382184);
        double r4382189 = r4382187 - r4382188;
        return r4382189;
}
↓
double f(double x, double eps) {
        double r4382190 = 2.0;
        double r4382191 = eps;
        double r4382192 = 0.5;
        double r4382193 = r4382191 * r4382192;
        double r4382194 = cos(r4382193);
        double r4382195 = x;
        double r4382196 = cos(r4382195);
        double r4382197 = r4382194 * r4382196;
        double r4382198 = sin(r4382193);
        double r4382199 = sin(r4382195);
        double r4382200 = r4382198 * r4382199;
        double r4382201 = exp(r4382200);
        double r4382202 = log(r4382201);
        double r4382203 = r4382197 - r4382202;
        double r4382204 = r4382191 / r4382190;
        double r4382205 = sin(r4382204);
        double r4382206 = r4382203 * r4382205;
        double r4382207 = r4382190 * r4382206;
        return r4382207;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.4
Target14.7
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.4

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

  3. Applied diff-sin36.7

    \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]

  4. Simplified14.7

    \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)}\]

  5. Taylor expanded around 0 14.7

    \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \color{blue}{\left(x + \frac{1}{2} \cdot \varepsilon\right)}\right)\]

  6. Simplified14.7

    \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(\varepsilon, \frac{1}{2}, x\right)\right)}\right)\]
  7. Using strategy rm

  8. Applied fma-udef14.7

    \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \color{blue}{\left(\varepsilon \cdot \frac{1}{2} + x\right)}\right)\]

  9. Applied cos-sum0.3

    \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x - \sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right)}\right)\]
  10. Using strategy rm

  11. Applied add-log-exp0.4

    \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x - \color{blue}{\log \left(e^{\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x}\right)}\right)\right)\]

  12. Final simplification0.4

    \[\leadsto 2 \cdot \left(\left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x - \log \left(e^{\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x}\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"

  :herbie-target
  (* 2 (* (cos (+ x (/ eps 2))) (sin (/ eps 2))))

  (- (sin (+ x eps)) (sin x)))