Average Error: 58.5 → 0.3
Time: 23.0s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right), \frac{-2}{3}, \mathsf{fma}\left(\varepsilon, -2, {\varepsilon}^{5} \cdot \frac{-2}{5}\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right), \frac{-2}{3}, \mathsf{fma}\left(\varepsilon, -2, {\varepsilon}^{5} \cdot \frac{-2}{5}\right)\right)
double f(double eps) {
        double r2206776 = 1.0;
        double r2206777 = eps;
        double r2206778 = r2206776 - r2206777;
        double r2206779 = r2206776 + r2206777;
        double r2206780 = r2206778 / r2206779;
        double r2206781 = log(r2206780);
        return r2206781;
}
↓
double f(double eps) {
        double r2206782 = eps;
        double r2206783 = r2206782 * r2206782;
        double r2206784 = r2206782 * r2206783;
        double r2206785 = -0.6666666666666666;
        double r2206786 = -2.0;
        double r2206787 = 5.0;
        double r2206788 = pow(r2206782, r2206787);
        double r2206789 = -0.4;
        double r2206790 = r2206788 * r2206789;
        double r2206791 = fma(r2206782, r2206786, r2206790);
        double r2206792 = fma(r2206784, r2206785, r2206791);
        return r2206792;
}

Error

Bits error versus eps

Target

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

Derivation

  1. Initial program 58.5

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

  2. Taylor expanded around 0 0.3

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

  3. Simplified0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019141 +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))))