Average Error: 61.2 → 0.4
Time: 8.7s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\frac{\log 1 - \left(1 \cdot {x}^{2} + \frac{1}{2} \cdot \frac{{x}^{4}}{{1}^{2}}\right)}{\mathsf{fma}\left(x, 1, \log 1 - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)} - 1\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\frac{\log 1 - \left(1 \cdot {x}^{2} + \frac{1}{2} \cdot \frac{{x}^{4}}{{1}^{2}}\right)}{\mathsf{fma}\left(x, 1, \log 1 - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)} - 1
double f(double x) {
        double r91199 = 1.0;
        double r91200 = x;
        double r91201 = r91199 - r91200;
        double r91202 = log(r91201);
        double r91203 = r91199 + r91200;
        double r91204 = log(r91203);
        double r91205 = r91202 / r91204;
        return r91205;
}
↓
double f(double x) {
        double r91206 = 1.0;
        double r91207 = log(r91206);
        double r91208 = x;
        double r91209 = 2.0;
        double r91210 = pow(r91208, r91209);
        double r91211 = r91206 * r91210;
        double r91212 = 0.5;
        double r91213 = 4.0;
        double r91214 = pow(r91208, r91213);
        double r91215 = pow(r91206, r91209);
        double r91216 = r91214 / r91215;
        double r91217 = r91212 * r91216;
        double r91218 = r91211 + r91217;
        double r91219 = r91207 - r91218;
        double r91220 = r91210 / r91215;
        double r91221 = r91212 * r91220;
        double r91222 = r91207 - r91221;
        double r91223 = fma(r91208, r91206, r91222);
        double r91224 = r91219 / r91223;
        double r91225 = 1.0;
        double r91226 = r91224 - r91225;
        return r91226;
}

Error

Bits error versus x

Target

Original61.2
Target0.3
Herbie0.4
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + 0.416666666666666685 \cdot {x}^{3}\right)\]

Derivation

  1. Initial program 61.2

    \[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
  2. Using strategy rm

  3. Applied flip--60.8

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

  4. Applied log-div61.0

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

  5. Applied div-sub61.0

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

  6. Simplified61.0

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

  7. Taylor expanded around 0 1.0

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

  8. Simplified1.0

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

  9. Taylor expanded around 0 0.4

    \[\leadsto \frac{\color{blue}{\log 1 - \left(1 \cdot {x}^{2} + \frac{1}{2} \cdot \frac{{x}^{4}}{{1}^{2}}\right)}}{\mathsf{fma}\left(x, 1, \log 1 - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)} - 1\]

  10. Final simplification0.4

    \[\leadsto \frac{\log 1 - \left(1 \cdot {x}^{2} + \frac{1}{2} \cdot \frac{{x}^{4}}{{1}^{2}}\right)}{\mathsf{fma}\left(x, 1, \log 1 - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)} - 1\]

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (x)
  :name "qlog (example 3.10)"
  :precision binary64
  :pre (and (< -1 x) (< x 1))

  :herbie-target
  (- (+ (+ (+ 1 x) (/ (* x x) 2)) (* 0.4166666666666667 (pow x 3))))

  (/ (log (- 1 x)) (log (+ 1 x))))