Average Error: 36.9 → 0.4
Time: 6.5s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\mathsf{fma}\left(\sin x, \log \left(e^{\frac{\mathsf{fma}\left({\left(\cos \varepsilon\right)}^{3}, 1, -1\right)}{\mathsf{fma}\left(\cos \varepsilon, \cos \varepsilon + 1, 1\right)}}\right), \cos x \cdot \sin \varepsilon\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\mathsf{fma}\left(\sin x, \log \left(e^{\frac{\mathsf{fma}\left({\left(\cos \varepsilon\right)}^{3}, 1, -1\right)}{\mathsf{fma}\left(\cos \varepsilon, \cos \varepsilon + 1, 1\right)}}\right), \cos x \cdot \sin \varepsilon\right)
double f(double x, double eps) {
        double r132632 = x;
        double r132633 = eps;
        double r132634 = r132632 + r132633;
        double r132635 = sin(r132634);
        double r132636 = sin(r132632);
        double r132637 = r132635 - r132636;
        return r132637;
}
↓
double f(double x, double eps) {
        double r132638 = x;
        double r132639 = sin(r132638);
        double r132640 = eps;
        double r132641 = cos(r132640);
        double r132642 = 3.0;
        double r132643 = pow(r132641, r132642);
        double r132644 = 1.0;
        double r132645 = -1.0;
        double r132646 = fma(r132643, r132644, r132645);
        double r132647 = r132641 + r132644;
        double r132648 = fma(r132641, r132647, r132644);
        double r132649 = r132646 / r132648;
        double r132650 = exp(r132649);
        double r132651 = log(r132650);
        double r132652 = cos(r132638);
        double r132653 = sin(r132640);
        double r132654 = r132652 * r132653;
        double r132655 = fma(r132639, r132651, r132654);
        return r132655;
}

Error

Bits error versus x

Bits error versus eps

Target

Original36.9
Target15.0
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.9

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

  3. Applied sin-sum21.7

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

  4. Taylor expanded around inf 21.7

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

  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-log0.4

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

  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 flip3--0.4

    \[\leadsto \mathsf{fma}\left(\sin x, \log \left(e^{\color{blue}{\frac{{\left(\cos \varepsilon\right)}^{3} - {1}^{3}}{\cos \varepsilon \cdot \cos \varepsilon + \left(1 \cdot 1 + \cos \varepsilon \cdot 1\right)}}}\right), \cos x \cdot \sin \varepsilon\right)\]

  13. Simplified0.4

    \[\leadsto \mathsf{fma}\left(\sin x, \log \left(e^{\frac{\color{blue}{\mathsf{fma}\left({\left(\cos \varepsilon\right)}^{3}, 1, -1\right)}}{\cos \varepsilon \cdot \cos \varepsilon + \left(1 \cdot 1 + \cos \varepsilon \cdot 1\right)}}\right), \cos x \cdot \sin \varepsilon\right)\]

  14. Simplified0.4

    \[\leadsto \mathsf{fma}\left(\sin x, \log \left(e^{\frac{\mathsf{fma}\left({\left(\cos \varepsilon\right)}^{3}, 1, -1\right)}{\color{blue}{\mathsf{fma}\left(\cos \varepsilon, \cos \varepsilon + 1, 1\right)}}}\right), \cos x \cdot \sin \varepsilon\right)\]

  15. Final simplification0.4

    \[\leadsto \mathsf{fma}\left(\sin x, \log \left(e^{\frac{\mathsf{fma}\left({\left(\cos \varepsilon\right)}^{3}, 1, -1\right)}{\mathsf{fma}\left(\cos \varepsilon, \cos \varepsilon + 1, 1\right)}}\right), \cos x \cdot \sin \varepsilon\right)\]

Reproduce

herbie shell --seed 2020001 +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)))