Average Error: 58.5 → 0.2
Time: 12.8s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right), \frac{-2}{3}, \mathsf{fma}\left(\frac{-2}{5}, {\varepsilon}^{5}, -2 \cdot \varepsilon\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right), \frac{-2}{3}, \mathsf{fma}\left(\frac{-2}{5}, {\varepsilon}^{5}, -2 \cdot \varepsilon\right)\right)
double f(double eps) {
        double r3523524 = 1.0;
        double r3523525 = eps;
        double r3523526 = r3523524 - r3523525;
        double r3523527 = r3523524 + r3523525;
        double r3523528 = r3523526 / r3523527;
        double r3523529 = log(r3523528);
        return r3523529;
}
↓
double f(double eps) {
        double r3523530 = eps;
        double r3523531 = r3523530 * r3523530;
        double r3523532 = r3523530 * r3523531;
        double r3523533 = -0.6666666666666666;
        double r3523534 = -0.4;
        double r3523535 = 5.0;
        double r3523536 = pow(r3523530, r3523535);
        double r3523537 = -2.0;
        double r3523538 = r3523537 * r3523530;
        double r3523539 = fma(r3523534, r3523536, r3523538);
        double r3523540 = fma(r3523532, r3523533, r3523539);
        return r3523540;
}

Error

Bits error versus eps

Target

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

Derivation

  1. Initial program 58.5

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

  2. Taylor expanded around 0 0.2

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

  3. Simplified0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019158 +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))))