Average Error: 58.7 → 0.2
Time: 25.4s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\mathsf{fma}\left(\frac{-2}{3}, \left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon, \mathsf{fma}\left(-2, \varepsilon, {\varepsilon}^{5} \cdot \frac{-2}{5}\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\mathsf{fma}\left(\frac{-2}{3}, \left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon, \mathsf{fma}\left(-2, \varepsilon, {\varepsilon}^{5} \cdot \frac{-2}{5}\right)\right)
double f(double eps) {
        double r2682735 = 1.0;
        double r2682736 = eps;
        double r2682737 = r2682735 - r2682736;
        double r2682738 = r2682735 + r2682736;
        double r2682739 = r2682737 / r2682738;
        double r2682740 = log(r2682739);
        return r2682740;
}
↓
double f(double eps) {
        double r2682741 = -0.6666666666666666;
        double r2682742 = eps;
        double r2682743 = r2682742 * r2682742;
        double r2682744 = r2682743 * r2682742;
        double r2682745 = -2.0;
        double r2682746 = 5.0;
        double r2682747 = pow(r2682742, r2682746);
        double r2682748 = -0.4;
        double r2682749 = r2682747 * r2682748;
        double r2682750 = fma(r2682745, r2682742, r2682749);
        double r2682751 = fma(r2682741, r2682744, r2682750);
        return r2682751;
}

Error

Bits error versus eps

Target

Original58.7
Target0.2
Herbie0.2
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.7

    \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]

  2. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{-\left(\frac{2}{3} \cdot {\varepsilon}^{3} + \left(\frac{2}{5} \cdot {\varepsilon}^{5} + 2 \cdot \varepsilon\right)\right)}\]

  3. Simplified0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-2}{3}, \varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right), \mathsf{fma}\left(-2, \varepsilon, \frac{-2}{5} \cdot {\varepsilon}^{5}\right)\right)}\]

  4. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(\frac{-2}{3}, \left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon, \mathsf{fma}\left(-2, \varepsilon, {\varepsilon}^{5} \cdot \frac{-2}{5}\right)\right)\]

Reproduce

herbie shell --seed 2019142 +o rules:numerics
(FPCore (eps)
  :name "logq (problem 3.4.3)"

  :herbie-target
  (* -2 (+ (+ eps (/ (pow eps 3) 3)) (/ (pow eps 5) 5)))

  (log (/ (- 1 eps) (+ 1 eps))))