Average Error: 58.6 → 0.6
Time: 15.2s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{2 \cdot \left(\left(x + x \cdot x\right) - \frac{x}{1} \cdot \frac{x}{1}\right) + \log 1}{\frac{2}{1}}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{2 \cdot \left(\left(x + x \cdot x\right) - \frac{x}{1} \cdot \frac{x}{1}\right) + \log 1}{\frac{2}{1}}
double f(double x) {
        double r67122 = 1.0;
        double r67123 = 2.0;
        double r67124 = r67122 / r67123;
        double r67125 = x;
        double r67126 = r67122 + r67125;
        double r67127 = r67122 - r67125;
        double r67128 = r67126 / r67127;
        double r67129 = log(r67128);
        double r67130 = r67124 * r67129;
        return r67130;
}
↓
double f(double x) {
        double r67131 = 2.0;
        double r67132 = x;
        double r67133 = r67132 * r67132;
        double r67134 = r67132 + r67133;
        double r67135 = 1.0;
        double r67136 = r67132 / r67135;
        double r67137 = r67136 * r67136;
        double r67138 = r67134 - r67137;
        double r67139 = r67131 * r67138;
        double r67140 = log(r67135);
        double r67141 = r67139 + r67140;
        double r67142 = r67131 / r67135;
        double r67143 = r67141 / r67142;
        return r67143;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.6

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

  2. Simplified58.6

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

  3. Taylor expanded around 0 0.6

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

  4. Simplified0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2019179 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))