Average Error: 37.0 → 0.5
Time: 6.2s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\mathsf{fma}\left(\sin x, \cos \varepsilon - 1, \mathsf{log1p}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(\cos x \cdot \sin \varepsilon\right)\right)\right)\right)\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\mathsf{fma}\left(\sin x, \cos \varepsilon - 1, \mathsf{log1p}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(\cos x \cdot \sin \varepsilon\right)\right)\right)\right)\right)
double f(double x, double eps) {
        double r112704 = x;
        double r112705 = eps;
        double r112706 = r112704 + r112705;
        double r112707 = sin(r112706);
        double r112708 = sin(r112704);
        double r112709 = r112707 - r112708;
        return r112709;
}
↓
double f(double x, double eps) {
        double r112710 = x;
        double r112711 = sin(r112710);
        double r112712 = eps;
        double r112713 = cos(r112712);
        double r112714 = 1.0;
        double r112715 = r112713 - r112714;
        double r112716 = cos(r112710);
        double r112717 = sin(r112712);
        double r112718 = r112716 * r112717;
        double r112719 = expm1(r112718);
        double r112720 = expm1(r112719);
        double r112721 = log1p(r112720);
        double r112722 = log1p(r112721);
        double r112723 = fma(r112711, r112715, r112722);
        return r112723;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.0
Target15.5
Herbie0.5
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.0

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

  3. Applied sin-sum21.4

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

  5. Applied *-un-lft-identity21.4

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

  6. Applied *-un-lft-identity21.4

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

  7. Applied distribute-lft-out--21.4

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

  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

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

  12. Applied log1p-expm1-u0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2020060 +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)))