Average Error: 37.1 → 0.3
Time: 27.3s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot (\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\cos x\right) + \left(\left(-\sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \sin x\right))_*\right)\]
double f(double x, double eps) {
        double r6985304 = x;
        double r6985305 = eps;
        double r6985306 = r6985304 + r6985305;
        double r6985307 = sin(r6985306);
        double r6985308 = sin(r6985304);
        double r6985309 = r6985307 - r6985308;
        return r6985309;
}
↓
double f(double x, double eps) {
        double r6985310 = 2.0;
        double r6985311 = 0.5;
        double r6985312 = eps;
        double r6985313 = r6985311 * r6985312;
        double r6985314 = sin(r6985313);
        double r6985315 = cos(r6985313);
        double r6985316 = x;
        double r6985317 = cos(r6985316);
        double r6985318 = -r6985314;
        double r6985319 = sin(r6985316);
        double r6985320 = r6985318 * r6985319;
        double r6985321 = fma(r6985315, r6985317, r6985320);
        double r6985322 = r6985314 * r6985321;
        double r6985323 = r6985310 * r6985322;
        return r6985323;
}


\sin \left(x + \varepsilon\right) - \sin x
↓
2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot (\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\cos x\right) + \left(\left(-\sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \sin x\right))_*\right)

Error

Bits error versus x

Bits error versus eps

Target

Original37.1
Target15.1
Herbie0.3
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.1

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

  3. Applied diff-sin37.3

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

  4. Simplified15.1

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

  5. Taylor expanded around inf 15.1

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

  6. Simplified15.1

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

  8. Applied fma-udef15.1

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

  9. Applied cos-sum0.3

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

  11. Applied fma-neg0.3

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

  12. Final simplification0.3

    \[\leadsto 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot (\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\cos x\right) + \left(\left(-\sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \sin x\right))_*\right)\]

Reproduce

herbie shell --seed 2019101 +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)))