Average Error: 37.3 → 0.5
Time: 16.9s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\mathsf{fma}\left(\cos x, \sin \varepsilon, \log \left(e^{\sin x \cdot \cos \varepsilon - \sin x}\right)\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\mathsf{fma}\left(\cos x, \sin \varepsilon, \log \left(e^{\sin x \cdot \cos \varepsilon - \sin x}\right)\right)
double f(double x, double eps) {
        double r4877426 = x;
        double r4877427 = eps;
        double r4877428 = r4877426 + r4877427;
        double r4877429 = sin(r4877428);
        double r4877430 = sin(r4877426);
        double r4877431 = r4877429 - r4877430;
        return r4877431;
}
↓
double f(double x, double eps) {
        double r4877432 = x;
        double r4877433 = cos(r4877432);
        double r4877434 = eps;
        double r4877435 = sin(r4877434);
        double r4877436 = sin(r4877432);
        double r4877437 = cos(r4877434);
        double r4877438 = r4877436 * r4877437;
        double r4877439 = r4877438 - r4877436;
        double r4877440 = exp(r4877439);
        double r4877441 = log(r4877440);
        double r4877442 = fma(r4877433, r4877435, r4877441);
        return r4877442;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.3
Target15.5
Herbie0.5
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.3

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

  3. Applied sin-sum21.6

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

  4. Taylor expanded around inf 21.6

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

  5. Simplified21.6

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

  6. Taylor expanded around inf 21.6

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

  7. Simplified0.4

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

  9. Applied add-log-exp14.8

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

  10. Applied add-log-exp0.5

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

  11. Applied diff-log0.5

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

  12. Simplified0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"

  :herbie-target
  (* 2.0 (* (cos (+ x (/ eps 2.0))) (sin (/ eps 2.0))))

  (- (sin (+ x eps)) (sin x)))