Average Error: 61.4 → 0.4
Time: 20.7s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\frac{\frac{1}{\mathsf{fma}\left(\frac{x}{1} \cdot \frac{x}{1}, \frac{-1}{2}, \mathsf{fma}\left(1, x, \log 1\right)\right)}}{\frac{1}{\log 1 - \mathsf{fma}\left(1, x, \frac{\frac{1}{2}}{\frac{1}{x} \cdot \frac{1}{x}}\right)}}\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\frac{\frac{1}{\mathsf{fma}\left(\frac{x}{1} \cdot \frac{x}{1}, \frac{-1}{2}, \mathsf{fma}\left(1, x, \log 1\right)\right)}}{\frac{1}{\log 1 - \mathsf{fma}\left(1, x, \frac{\frac{1}{2}}{\frac{1}{x} \cdot \frac{1}{x}}\right)}}
double f(double x) {
        double r3921186 = 1.0;
        double r3921187 = x;
        double r3921188 = r3921186 - r3921187;
        double r3921189 = log(r3921188);
        double r3921190 = r3921186 + r3921187;
        double r3921191 = log(r3921190);
        double r3921192 = r3921189 / r3921191;
        return r3921192;
}
↓
double f(double x) {
        double r3921193 = 1.0;
        double r3921194 = x;
        double r3921195 = 1.0;
        double r3921196 = r3921194 / r3921195;
        double r3921197 = r3921196 * r3921196;
        double r3921198 = -0.5;
        double r3921199 = log(r3921195);
        double r3921200 = fma(r3921195, r3921194, r3921199);
        double r3921201 = fma(r3921197, r3921198, r3921200);
        double r3921202 = r3921193 / r3921201;
        double r3921203 = 0.5;
        double r3921204 = r3921195 / r3921194;
        double r3921205 = r3921204 * r3921204;
        double r3921206 = r3921203 / r3921205;
        double r3921207 = fma(r3921195, r3921194, r3921206);
        double r3921208 = r3921199 - r3921207;
        double r3921209 = r3921193 / r3921208;
        double r3921210 = r3921202 / r3921209;
        return r3921210;
}

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

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

  4. Taylor expanded around 0 0.4

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

  5. Simplified0.4

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

  7. Applied clear-num0.4

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

  9. Applied div-inv0.6

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

  10. Applied associate-/r*0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019171 +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))))