Average Error: 36.5 → 0.4
Time: 6.4s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\mathsf{fma}\left(\sin x, \log \left(\sqrt{e^{\cos \varepsilon - 1}}\right) + \log \left(\sqrt{e^{\cos \varepsilon - 1}}\right), \cos x \cdot \sin \varepsilon\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\mathsf{fma}\left(\sin x, \log \left(\sqrt{e^{\cos \varepsilon - 1}}\right) + \log \left(\sqrt{e^{\cos \varepsilon - 1}}\right), \cos x \cdot \sin \varepsilon\right)
double f(double x, double eps) {
        double r177164 = x;
        double r177165 = eps;
        double r177166 = r177164 + r177165;
        double r177167 = sin(r177166);
        double r177168 = sin(r177164);
        double r177169 = r177167 - r177168;
        return r177169;
}
↓
double f(double x, double eps) {
        double r177170 = x;
        double r177171 = sin(r177170);
        double r177172 = eps;
        double r177173 = cos(r177172);
        double r177174 = 1.0;
        double r177175 = r177173 - r177174;
        double r177176 = exp(r177175);
        double r177177 = sqrt(r177176);
        double r177178 = log(r177177);
        double r177179 = r177178 + r177178;
        double r177180 = cos(r177170);
        double r177181 = sin(r177172);
        double r177182 = r177180 * r177181;
        double r177183 = fma(r177171, r177179, r177182);
        return r177183;
}

Error

Bits error versus x

Bits error versus eps

Target

Original36.5
Target15.2
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.5

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

  3. Applied sin-sum21.3

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

  4. Taylor expanded around inf 21.3

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

  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 add-sqr-sqrt0.5

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

  13. Applied log-prod0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2020034 +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)))