Average Error: 37.1 → 0.5
Time: 6.1s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\mathsf{fma}\left(\sin x, \log \left(e^{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \varepsilon - 1\right)\right)}\right), \cos x \cdot \sin \varepsilon\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\mathsf{fma}\left(\sin x, \log \left(e^{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \varepsilon - 1\right)\right)}\right), \cos x \cdot \sin \varepsilon\right)
double f(double x, double eps) {
        double r112693 = x;
        double r112694 = eps;
        double r112695 = r112693 + r112694;
        double r112696 = sin(r112695);
        double r112697 = sin(r112693);
        double r112698 = r112696 - r112697;
        return r112698;
}
↓
double f(double x, double eps) {
        double r112699 = x;
        double r112700 = sin(r112699);
        double r112701 = eps;
        double r112702 = cos(r112701);
        double r112703 = 1.0;
        double r112704 = r112702 - r112703;
        double r112705 = expm1(r112704);
        double r112706 = log1p(r112705);
        double r112707 = exp(r112706);
        double r112708 = log(r112707);
        double r112709 = cos(r112699);
        double r112710 = sin(r112701);
        double r112711 = r112709 * r112710;
        double r112712 = fma(r112700, r112708, r112711);
        return r112712;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.1
Target15.0
Herbie0.5
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.1

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

  3. Applied sin-sum22.0

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

  4. Taylor expanded around inf 22.0

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

  5. Simplified0.4

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

  7. Applied add-log-exp0.4

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

  8. Applied add-log-exp0.4

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

  9. Applied diff-log0.5

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

  10. Simplified0.4

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

  12. Applied log1p-expm1-u0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2020062 +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)))