Average Error: 36.3 → 0.5
Time: 41.8s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[2 \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right) - \log \left(e^{\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x}\right)\right)\right)\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
2 \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right) - \log \left(e^{\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x}\right)\right)\right)\right)
double f(double x, double eps) {
        double r6088199 = x;
        double r6088200 = eps;
        double r6088201 = r6088199 + r6088200;
        double r6088202 = sin(r6088201);
        double r6088203 = sin(r6088199);
        double r6088204 = r6088202 - r6088203;
        return r6088204;
}
↓
double f(double x, double eps) {
        double r6088205 = 2.0;
        double r6088206 = eps;
        double r6088207 = 0.5;
        double r6088208 = r6088206 * r6088207;
        double r6088209 = sin(r6088208);
        double r6088210 = x;
        double r6088211 = cos(r6088210);
        double r6088212 = cos(r6088208);
        double r6088213 = r6088211 * r6088212;
        double r6088214 = sin(r6088210);
        double r6088215 = r6088209 * r6088214;
        double r6088216 = exp(r6088215);
        double r6088217 = log(r6088216);
        double r6088218 = r6088213 - r6088217;
        double r6088219 = expm1(r6088218);
        double r6088220 = log1p(r6088219);
        double r6088221 = r6088209 * r6088220;
        double r6088222 = r6088205 * r6088221;
        return r6088222;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.3
Target15.2
Herbie0.5
\[2.0 \cdot \left(\cos \left(x + \frac{\varepsilon}{2.0}\right) \cdot \sin \left(\frac{\varepsilon}{2.0}\right)\right)\]

Derivation

  1. Initial program 36.3

    \[\sin \left(x + \varepsilon\right) - \sin x\]
  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. Simplified15.2

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

  6. Applied log1p-expm1-u15.3

    \[\leadsto 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\right)}\right)\]

  7. Simplified15.3

    \[\leadsto 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)\right)}\right)\right)\]
  8. Using strategy rm

  9. Applied fma-udef15.3

    \[\leadsto 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \color{blue}{\left(\frac{1}{2} \cdot \varepsilon + x\right)}\right)\right)\right)\]

  10. Applied cos-sum0.4

    \[\leadsto 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x}\right)\right)\right)\]
  11. Using strategy rm

  12. Applied add-log-exp0.5

    \[\leadsto 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \color{blue}{\log \left(e^{\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x}\right)}\right)\right)\right)\]

  13. Final simplification0.5

    \[\leadsto 2 \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right) - \log \left(e^{\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x}\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"

  :herbie-target
  (* 2.0 (* (cos (+ x (/ eps 2.0))) (sin (/ eps 2.0))))

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