Average Error: 37.0 → 0.8
Time: 6.4s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\mathsf{fma}\left(\sin x, \cos \varepsilon - 1, \left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \sin \varepsilon\right)\right) + \mathsf{fma}\left(-\sin x, 1, \sin x\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\mathsf{fma}\left(\sin x, \cos \varepsilon - 1, \left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \sin \varepsilon\right)\right) + \mathsf{fma}\left(-\sin x, 1, \sin x\right)
double f(double x, double eps) {
        double r129182 = x;
        double r129183 = eps;
        double r129184 = r129182 + r129183;
        double r129185 = sin(r129184);
        double r129186 = sin(r129182);
        double r129187 = r129185 - r129186;
        return r129187;
}
↓
double f(double x, double eps) {
        double r129188 = x;
        double r129189 = sin(r129188);
        double r129190 = eps;
        double r129191 = cos(r129190);
        double r129192 = 1.0;
        double r129193 = r129191 - r129192;
        double r129194 = cos(r129188);
        double r129195 = cbrt(r129194);
        double r129196 = r129195 * r129195;
        double r129197 = sin(r129190);
        double r129198 = r129195 * r129197;
        double r129199 = r129196 * r129198;
        double r129200 = fma(r129189, r129193, r129199);
        double r129201 = -r129189;
        double r129202 = fma(r129201, r129192, r129189);
        double r129203 = r129200 + r129202;
        return r129203;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.0
Target15.5
Herbie0.8
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.0

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

  5. Applied add-cube-cbrt22.0

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

  6. Applied add-sqr-sqrt42.7

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

  7. Applied prod-diff42.7

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

  8. Simplified21.7

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

  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-cbrt0.8

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

  12. Applied associate-*l*0.8

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

  13. Final simplification0.8

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

Reproduce

herbie shell --seed 2020060 +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)))