Average Error: 37.6 → 0.5
Time: 6.3s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\left(\log \left(e^{\sin x \cdot \left(\cos \varepsilon - 1\right)}\right) + \cos x \cdot \sin \varepsilon\right) + \mathsf{fma}\left(-\sin x, 1, \sin x\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\left(\log \left(e^{\sin x \cdot \left(\cos \varepsilon - 1\right)}\right) + \cos x \cdot \sin \varepsilon\right) + \mathsf{fma}\left(-\sin x, 1, \sin x\right)
double f(double x, double eps) {
        double r110075 = x;
        double r110076 = eps;
        double r110077 = r110075 + r110076;
        double r110078 = sin(r110077);
        double r110079 = sin(r110075);
        double r110080 = r110078 - r110079;
        return r110080;
}
↓
double f(double x, double eps) {
        double r110081 = x;
        double r110082 = sin(r110081);
        double r110083 = eps;
        double r110084 = cos(r110083);
        double r110085 = 1.0;
        double r110086 = r110084 - r110085;
        double r110087 = r110082 * r110086;
        double r110088 = exp(r110087);
        double r110089 = log(r110088);
        double r110090 = cos(r110081);
        double r110091 = sin(r110083);
        double r110092 = r110090 * r110091;
        double r110093 = r110089 + r110092;
        double r110094 = -r110082;
        double r110095 = fma(r110094, r110085, r110082);
        double r110096 = r110093 + r110095;
        return r110096;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.6
Target15.4
Herbie0.5
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.6

    \[\sin \left(x + \varepsilon\right) - \sin x\]
  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-cbrt22.6

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

  6. Applied add-sqr-sqrt44.1

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

  7. Applied prod-diff44.1

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

  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 fma-udef0.4

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

  13. Applied add-log-exp0.5

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

  14. Final simplification0.5

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

Reproduce

herbie shell --seed 2020025 +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)))