Average Error: 61.4 → 0.4
Time: 18.6s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\log \left(e^{\frac{\log 1 - \mathsf{fma}\left(\frac{x}{1} \cdot \frac{x}{1}, \frac{1}{2}, 1 \cdot x\right)}{\mathsf{fma}\left(\frac{x}{1} \cdot \frac{x}{1}, \frac{-1}{2}, \mathsf{fma}\left(x, 1, \log 1\right)\right)}}\right)\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\log \left(e^{\frac{\log 1 - \mathsf{fma}\left(\frac{x}{1} \cdot \frac{x}{1}, \frac{1}{2}, 1 \cdot x\right)}{\mathsf{fma}\left(\frac{x}{1} \cdot \frac{x}{1}, \frac{-1}{2}, \mathsf{fma}\left(x, 1, \log 1\right)\right)}}\right)
double f(double x) {
        double r3721800 = 1.0;
        double r3721801 = x;
        double r3721802 = r3721800 - r3721801;
        double r3721803 = log(r3721802);
        double r3721804 = r3721800 + r3721801;
        double r3721805 = log(r3721804);
        double r3721806 = r3721803 / r3721805;
        return r3721806;
}
↓
double f(double x) {
        double r3721807 = 1.0;
        double r3721808 = log(r3721807);
        double r3721809 = x;
        double r3721810 = r3721809 / r3721807;
        double r3721811 = r3721810 * r3721810;
        double r3721812 = 0.5;
        double r3721813 = r3721807 * r3721809;
        double r3721814 = fma(r3721811, r3721812, r3721813);
        double r3721815 = r3721808 - r3721814;
        double r3721816 = -0.5;
        double r3721817 = fma(r3721809, r3721807, r3721808);
        double r3721818 = fma(r3721811, r3721816, r3721817);
        double r3721819 = r3721815 / r3721818;
        double r3721820 = exp(r3721819);
        double r3721821 = log(r3721820);
        return r3721821;
}

Error

Bits error versus x

Target

Original61.4
Target0.3
Herbie0.4
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + 0.4166666666666666851703837437526090070605 \cdot {x}^{3}\right)\]

Derivation

  1. Initial program 61.4

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

  2. Taylor expanded around 0 60.5

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

  3. Simplified60.5

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

  4. Taylor expanded around 0 0.4

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

  5. Simplified0.4

    \[\leadsto \frac{\color{blue}{\log 1 - \mathsf{fma}\left(\frac{x}{1} \cdot \frac{x}{1}, \frac{1}{2}, 1 \cdot x\right)}}{\mathsf{fma}\left(\frac{x}{1} \cdot \frac{x}{1}, \frac{-1}{2}, \mathsf{fma}\left(x, 1, \log 1\right)\right)}\]
  6. Using strategy rm

  7. Applied add-log-exp0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (x)
  :name "qlog (example 3.10)"
  :pre (and (< -1.0 x) (< x 1.0))

  :herbie-target
  (- (+ (+ (+ 1.0 x) (/ (* x x) 2.0)) (* 0.4166666666666667 (pow x 3.0))))

  (/ (log (- 1.0 x)) (log (+ 1.0 x))))