Average Error: 58.5 → 0.7
Time: 6.1s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[2 \cdot \left({\varepsilon}^{2} - \mathsf{fma}\left(\frac{\varepsilon}{1}, \frac{\varepsilon}{1}, \varepsilon\right)\right) + \log 1\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
2 \cdot \left({\varepsilon}^{2} - \mathsf{fma}\left(\frac{\varepsilon}{1}, \frac{\varepsilon}{1}, \varepsilon\right)\right) + \log 1
double f(double eps) {
        double r126112 = 1.0;
        double r126113 = eps;
        double r126114 = r126112 - r126113;
        double r126115 = r126112 + r126113;
        double r126116 = r126114 / r126115;
        double r126117 = log(r126116);
        return r126117;
}
↓
double f(double eps) {
        double r126118 = 2.0;
        double r126119 = eps;
        double r126120 = 2.0;
        double r126121 = pow(r126119, r126120);
        double r126122 = 1.0;
        double r126123 = r126119 / r126122;
        double r126124 = fma(r126123, r126123, r126119);
        double r126125 = r126121 - r126124;
        double r126126 = r126118 * r126125;
        double r126127 = log(r126122);
        double r126128 = r126126 + r126127;
        return r126128;
}

Error

Bits error versus eps

Target

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

Derivation

  1. Initial program 58.5

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

  2. Taylor expanded around 0 0.7

    \[\leadsto \color{blue}{\left(2 \cdot {\varepsilon}^{2} + \log 1\right) - \left(2 \cdot \frac{{\varepsilon}^{2}}{{1}^{2}} + 2 \cdot \varepsilon\right)}\]

  3. Simplified0.7

    \[\leadsto \color{blue}{2 \cdot \left({\varepsilon}^{2} - \mathsf{fma}\left(\frac{\varepsilon}{1}, \frac{\varepsilon}{1}, \varepsilon\right)\right) + \log 1}\]

  4. Final simplification0.7

    \[\leadsto 2 \cdot \left({\varepsilon}^{2} - \mathsf{fma}\left(\frac{\varepsilon}{1}, \frac{\varepsilon}{1}, \varepsilon\right)\right) + \log 1\]

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (eps)
  :name "logq (problem 3.4.3)"
  :precision binary64

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

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