Average Error: 36.6 → 0.4
Time: 6.0s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\mathsf{fma}\left(\sin x, \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \varepsilon - 1\right)\right), \cos x \cdot \sin \varepsilon\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\mathsf{fma}\left(\sin x, \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \varepsilon - 1\right)\right), \cos x \cdot \sin \varepsilon\right)
double f(double x, double eps) {
        double r162574 = x;
        double r162575 = eps;
        double r162576 = r162574 + r162575;
        double r162577 = sin(r162576);
        double r162578 = sin(r162574);
        double r162579 = r162577 - r162578;
        return r162579;
}
↓
double f(double x, double eps) {
        double r162580 = x;
        double r162581 = sin(r162580);
        double r162582 = eps;
        double r162583 = cos(r162582);
        double r162584 = 1.0;
        double r162585 = r162583 - r162584;
        double r162586 = expm1(r162585);
        double r162587 = log1p(r162586);
        double r162588 = cos(r162580);
        double r162589 = sin(r162582);
        double r162590 = r162588 * r162589;
        double r162591 = fma(r162581, r162587, r162590);
        return r162591;
}

Error

Bits error versus x

Bits error versus eps

Target

Original36.6
Target15.3
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.6

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

  3. Applied sin-sum21.2

    \[\leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
  4. Using strategy rm

  5. Applied *-un-lft-identity21.2

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

  6. Applied *-un-lft-identity21.2

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

  7. Applied distribute-lft-out--21.2

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

  8. Simplified0.4

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

  10. Applied log1p-expm1-u0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

  :herbie-target
  (* 2 (* (cos (+ x (/ eps 2))) (sin (/ eps 2))))

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