Average Error: 37.0 → 0.3
Time: 26.2s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right), \cos x, -\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right) \cdot \left(2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right), \cos x, -\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right) \cdot \left(2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)
double f(double x, double eps) {
        double r12278429 = x;
        double r12278430 = eps;
        double r12278431 = r12278429 + r12278430;
        double r12278432 = sin(r12278431);
        double r12278433 = sin(r12278429);
        double r12278434 = r12278432 - r12278433;
        return r12278434;
}
↓
double f(double x, double eps) {
        double r12278435 = 0.5;
        double r12278436 = eps;
        double r12278437 = r12278435 * r12278436;
        double r12278438 = cos(r12278437);
        double r12278439 = x;
        double r12278440 = cos(r12278439);
        double r12278441 = sin(r12278437);
        double r12278442 = sin(r12278439);
        double r12278443 = r12278441 * r12278442;
        double r12278444 = -r12278443;
        double r12278445 = fma(r12278438, r12278440, r12278444);
        double r12278446 = 2.0;
        double r12278447 = r12278446 * r12278441;
        double r12278448 = r12278445 * r12278447;
        return r12278448;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.0
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 37.0

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

  3. Applied diff-sin37.4

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

  4. Simplified15.0

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

  5. Taylor expanded around inf 15.0

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

  6. Simplified15.0

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

  8. Applied fma-udef15.0

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

  9. Applied cos-sum0.3

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

  11. Applied fma-neg0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019173 +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)))