Average Error: 58.4 → 0.3
Time: 18.4s
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 r2559539 = 1.0;
        double r2559540 = eps;
        double r2559541 = r2559539 - r2559540;
        double r2559542 = r2559539 + r2559540;
        double r2559543 = r2559541 / r2559542;
        double r2559544 = log(r2559543);
        return r2559544;
}
↓
double f(double eps) {
        double r2559545 = eps;
        double r2559546 = r2559545 * r2559545;
        double r2559547 = r2559545 * r2559546;
        double r2559548 = -0.6666666666666666;
        double r2559549 = -0.4;
        double r2559550 = 5.0;
        double r2559551 = pow(r2559545, r2559550);
        double r2559552 = -2.0;
        double r2559553 = r2559552 * r2559545;
        double r2559554 = fma(r2559549, r2559551, r2559553);
        double r2559555 = fma(r2559547, r2559548, r2559554);
        return r2559555;
}

Error

Bits error versus eps

Target

Original58.4
Target0.3
Herbie0.3
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.4

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

  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

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019163 +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))))