Average Error: 37.2 → 0.3
Time: 16.0s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\mathsf{fma}\left(-\sin x, \sin \left(\varepsilon \cdot \frac{1}{2}\right), \sin x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) + \mathsf{fma}\left(\cos \left(\varepsilon \cdot \frac{1}{2}\right), \cos x, \left(-\sin x\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right)\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\mathsf{fma}\left(-\sin x, \sin \left(\varepsilon \cdot \frac{1}{2}\right), \sin x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) + \mathsf{fma}\left(\cos \left(\varepsilon \cdot \frac{1}{2}\right), \cos x, \left(-\sin x\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right)\right)
double f(double x, double eps) {
        double r2548354 = x;
        double r2548355 = eps;
        double r2548356 = r2548354 + r2548355;
        double r2548357 = sin(r2548356);
        double r2548358 = sin(r2548354);
        double r2548359 = r2548357 - r2548358;
        return r2548359;
}
↓
double f(double x, double eps) {
        double r2548360 = 2.0;
        double r2548361 = eps;
        double r2548362 = r2548361 / r2548360;
        double r2548363 = sin(r2548362);
        double r2548364 = x;
        double r2548365 = sin(r2548364);
        double r2548366 = -r2548365;
        double r2548367 = 0.5;
        double r2548368 = r2548361 * r2548367;
        double r2548369 = sin(r2548368);
        double r2548370 = r2548365 * r2548369;
        double r2548371 = fma(r2548366, r2548369, r2548370);
        double r2548372 = cos(r2548368);
        double r2548373 = cos(r2548364);
        double r2548374 = r2548366 * r2548369;
        double r2548375 = fma(r2548372, r2548373, r2548374);
        double r2548376 = r2548371 + r2548375;
        double r2548377 = r2548363 * r2548376;
        double r2548378 = r2548360 * r2548377;
        return r2548378;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.2
Target15.0
Herbie0.3
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.2

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

  3. Applied diff-sin37.5

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

  4. Simplified15.0

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

  5. Taylor expanded around inf 15.0

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

  6. Simplified15.0

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

  8. Applied fma-udef15.0

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

  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 prod-diff0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019153 +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)))