Average Error: 36.7 → 0.5
Time: 6.6s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\mathsf{fma}\left(\sin x, \mathsf{fma}\left(\sqrt[3]{\cos \varepsilon} \cdot \sqrt[3]{\cos \varepsilon}, \sqrt[3]{\cos \varepsilon}, -1\right), \cos x \cdot \sin \varepsilon\right) + \mathsf{fma}\left(-\sin x, 1, \sin x\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\mathsf{fma}\left(\sin x, \mathsf{fma}\left(\sqrt[3]{\cos \varepsilon} \cdot \sqrt[3]{\cos \varepsilon}, \sqrt[3]{\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 r95110 = x;
        double r95111 = eps;
        double r95112 = r95110 + r95111;
        double r95113 = sin(r95112);
        double r95114 = sin(r95110);
        double r95115 = r95113 - r95114;
        return r95115;
}
↓
double f(double x, double eps) {
        double r95116 = x;
        double r95117 = sin(r95116);
        double r95118 = eps;
        double r95119 = cos(r95118);
        double r95120 = cbrt(r95119);
        double r95121 = r95120 * r95120;
        double r95122 = 1.0;
        double r95123 = -r95122;
        double r95124 = fma(r95121, r95120, r95123);
        double r95125 = cos(r95116);
        double r95126 = sin(r95118);
        double r95127 = r95125 * r95126;
        double r95128 = fma(r95117, r95124, r95127);
        double r95129 = -r95117;
        double r95130 = fma(r95129, r95122, r95117);
        double r95131 = r95128 + r95130;
        return r95131;
}

Error

Bits error versus x

Bits error versus eps

Target

Original36.7
Target14.8
Herbie0.5
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.7

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

  3. Applied sin-sum21.8

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

  5. Applied add-cube-cbrt22.3

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

  6. Applied add-sqr-sqrt43.5

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

  7. Applied prod-diff43.5

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

  8. Simplified22.1

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

  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-cube-cbrt0.5

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

  12. Applied fma-neg0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2019356 +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)))