Average Error: 61.4 → 0.4
Time: 8.9s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\log \left(\frac{e^{\frac{\log 1}{\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}}}{e^{\frac{1 \cdot x + \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}{\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}}}\right)\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\log \left(\frac{e^{\frac{\log 1}{\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}}}{e^{\frac{1 \cdot x + \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}{\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}}}\right)
double f(double x) {
        double r75129 = 1.0;
        double r75130 = x;
        double r75131 = r75129 - r75130;
        double r75132 = log(r75131);
        double r75133 = r75129 + r75130;
        double r75134 = log(r75133);
        double r75135 = r75132 / r75134;
        return r75135;
}
↓
double f(double x) {
        double r75136 = 1.0;
        double r75137 = log(r75136);
        double r75138 = x;
        double r75139 = r75136 * r75138;
        double r75140 = r75139 + r75137;
        double r75141 = 0.5;
        double r75142 = 2.0;
        double r75143 = pow(r75138, r75142);
        double r75144 = pow(r75136, r75142);
        double r75145 = r75143 / r75144;
        double r75146 = r75141 * r75145;
        double r75147 = r75140 - r75146;
        double r75148 = r75137 / r75147;
        double r75149 = exp(r75148);
        double r75150 = r75139 + r75146;
        double r75151 = r75150 / r75147;
        double r75152 = exp(r75151);
        double r75153 = r75149 / r75152;
        double r75154 = log(r75153);
        return r75154;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.4
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 61.4

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

  2. Taylor expanded around 0 60.6

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

  3. Taylor expanded around 0 0.4

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

  5. Applied add-log-exp0.4

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

  7. Applied div-sub0.4

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

  8. Applied exp-diff0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2020059 
(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))))