Average Error: 58.8 → 0.2
Time: 20.0s
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 r2930005 = 1.0;
        double r2930006 = eps;
        double r2930007 = r2930005 - r2930006;
        double r2930008 = r2930005 + r2930006;
        double r2930009 = r2930007 / r2930008;
        double r2930010 = log(r2930009);
        return r2930010;
}
↓
double f(double eps) {
        double r2930011 = -0.6666666666666666;
        double r2930012 = eps;
        double r2930013 = r2930012 * r2930012;
        double r2930014 = r2930013 * r2930012;
        double r2930015 = -2.0;
        double r2930016 = 5.0;
        double r2930017 = pow(r2930012, r2930016);
        double r2930018 = -0.4;
        double r2930019 = r2930017 * r2930018;
        double r2930020 = fma(r2930015, r2930012, r2930019);
        double r2930021 = fma(r2930011, r2930014, r2930020);
        return r2930021;
}

Error

Bits error versus eps

Target

Original58.8
Target0.2
Herbie0.2
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.8

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

  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 2019149 +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))))