Average Error: 37.2 → 0.5
Time: 6.6s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\mathsf{fma}\left(\sin x, \log \left(e^{\frac{\mathsf{fma}\left(\cos \varepsilon, \cos \varepsilon, -1\right)}{\cos \varepsilon + 1}}\right), \cos x \cdot \sin \varepsilon\right) + \mathsf{fma}\left(-\sin x, 1, \sin x\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\mathsf{fma}\left(\sin x, \log \left(e^{\frac{\mathsf{fma}\left(\cos \varepsilon, \cos \varepsilon, -1\right)}{\cos \varepsilon + 1}}\right), \cos x \cdot \sin \varepsilon\right) + \mathsf{fma}\left(-\sin x, 1, \sin x\right)
double f(double x, double eps) {
        double r87872 = x;
        double r87873 = eps;
        double r87874 = r87872 + r87873;
        double r87875 = sin(r87874);
        double r87876 = sin(r87872);
        double r87877 = r87875 - r87876;
        return r87877;
}
↓
double f(double x, double eps) {
        double r87878 = x;
        double r87879 = sin(r87878);
        double r87880 = eps;
        double r87881 = cos(r87880);
        double r87882 = 1.0;
        double r87883 = -r87882;
        double r87884 = fma(r87881, r87881, r87883);
        double r87885 = r87881 + r87882;
        double r87886 = r87884 / r87885;
        double r87887 = exp(r87886);
        double r87888 = log(r87887);
        double r87889 = cos(r87878);
        double r87890 = sin(r87880);
        double r87891 = r87889 * r87890;
        double r87892 = fma(r87879, r87888, r87891);
        double r87893 = -r87879;
        double r87894 = fma(r87893, r87882, r87879);
        double r87895 = r87892 + r87894;
        return r87895;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.2
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 37.2

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

  3. Applied sin-sum22.1

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

  5. Applied add-cube-cbrt22.7

    \[\leadsto \left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \color{blue}{\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right) \cdot \sqrt[3]{\sin x}}\]

  6. Applied add-sqr-sqrt42.9

    \[\leadsto \color{blue}{\sqrt{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon} \cdot \sqrt{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}} - \left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right) \cdot \sqrt[3]{\sin x}\]

  7. Applied prod-diff43.0

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

  8. Simplified22.4

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

  9. Simplified0.4

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

  11. Applied add-log-exp0.4

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

  12. Applied add-log-exp0.5

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

  13. Applied diff-log0.5

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

  14. Simplified0.4

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

  16. Applied flip--0.5

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

  17. Simplified0.5

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

  18. Final simplification0.5

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

Reproduce

herbie shell --seed 2019352 +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)))