Average Error: 36.9 → 0.3
Time: 1.5m
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[2 \cdot \left(\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right), \cos x, \sin x \cdot \left(-\sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
2 \cdot \left(\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right), \cos x, \sin x \cdot \left(-\sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)
double f(double x, double eps) {
        double r4014590 = x;
        double r4014591 = eps;
        double r4014592 = r4014590 + r4014591;
        double r4014593 = sin(r4014592);
        double r4014594 = sin(r4014590);
        double r4014595 = r4014593 - r4014594;
        return r4014595;
}
↓
double f(double x, double eps) {
        double r4014596 = 2.0;
        double r4014597 = 0.5;
        double r4014598 = eps;
        double r4014599 = r4014597 * r4014598;
        double r4014600 = cos(r4014599);
        double r4014601 = x;
        double r4014602 = cos(r4014601);
        double r4014603 = sin(r4014601);
        double r4014604 = sin(r4014599);
        double r4014605 = -r4014604;
        double r4014606 = r4014603 * r4014605;
        double r4014607 = fma(r4014600, r4014602, r4014606);
        double r4014608 = r4014607 * r4014604;
        double r4014609 = r4014596 * r4014608;
        return r4014609;
}

Error

Bits error versus x

Bits error versus eps

Target

Original36.9
Target14.8
Herbie0.3
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.9

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

  3. Applied diff-sin37.2

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

  4. Simplified14.8

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

  5. Taylor expanded around inf 14.8

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

  6. Simplified14.8

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

  8. Applied fma-udef14.8

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

  9. Applied cos-sum0.3

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

  11. Applied fma-neg0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019138 +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)))