Average Error: 36.9 → 0.4
Time: 6.4s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\mathsf{fma}\left(\sin x, \frac{\log \left(e^{{\left(\cos \varepsilon\right)}^{3}}\right) - 1}{\mathsf{fma}\left(\cos \varepsilon, \cos \varepsilon + 1, 1\right)}, \cos x \cdot \sin \varepsilon\right) + \mathsf{fma}\left(-\sin x, 1, \sin x \cdot 1\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\mathsf{fma}\left(\sin x, \frac{\log \left(e^{{\left(\cos \varepsilon\right)}^{3}}\right) - 1}{\mathsf{fma}\left(\cos \varepsilon, \cos \varepsilon + 1, 1\right)}, \cos x \cdot \sin \varepsilon\right) + \mathsf{fma}\left(-\sin x, 1, \sin x \cdot 1\right)
double f(double x, double eps) {
        double r105820 = x;
        double r105821 = eps;
        double r105822 = r105820 + r105821;
        double r105823 = sin(r105822);
        double r105824 = sin(r105820);
        double r105825 = r105823 - r105824;
        return r105825;
}
↓
double f(double x, double eps) {
        double r105826 = x;
        double r105827 = sin(r105826);
        double r105828 = eps;
        double r105829 = cos(r105828);
        double r105830 = 3.0;
        double r105831 = pow(r105829, r105830);
        double r105832 = exp(r105831);
        double r105833 = log(r105832);
        double r105834 = 1.0;
        double r105835 = r105833 - r105834;
        double r105836 = r105829 + r105834;
        double r105837 = fma(r105829, r105836, r105834);
        double r105838 = r105835 / r105837;
        double r105839 = cos(r105826);
        double r105840 = sin(r105828);
        double r105841 = r105839 * r105840;
        double r105842 = fma(r105827, r105838, r105841);
        double r105843 = -r105827;
        double r105844 = r105827 * r105834;
        double r105845 = fma(r105843, r105834, r105844);
        double r105846 = r105842 + r105845;
        return r105846;
}

Error

Bits error versus x

Bits error versus eps

Target

Original36.9
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 36.9

    \[\sin \left(x + \varepsilon\right) - \sin x\]
  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. Applied associate--l+21.4

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

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

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

  7. Applied prod-diff21.4

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

  8. Applied associate-+r+21.4

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

  9. Simplified0.4

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

  11. Applied flip3--0.4

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

  12. Simplified0.4

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

  13. Simplified0.4

    \[\leadsto \mathsf{fma}\left(\sin x, \frac{{\left(\cos \varepsilon\right)}^{3} - 1}{\color{blue}{\mathsf{fma}\left(\cos \varepsilon, \cos \varepsilon + 1, 1\right)}}, \cos x \cdot \sin \varepsilon\right) + \mathsf{fma}\left(-\sin x, 1, \sin x \cdot 1\right)\]
  14. Using strategy rm

  15. Applied add-log-exp0.4

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

  16. Final simplification0.4

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

Reproduce

herbie shell --seed 2020065 +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)))