Average Error: 58.6 → 0.6
Time: 17.2s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\left(\log 1 + \left(\left(x + x \cdot x\right) - \frac{x \cdot x}{1 \cdot 1}\right) \cdot 2\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\left(\log 1 + \left(\left(x + x \cdot x\right) - \frac{x \cdot x}{1 \cdot 1}\right) \cdot 2\right) \cdot \frac{1}{2}
double f(double x) {
        double r2560130 = 1.0;
        double r2560131 = 2.0;
        double r2560132 = r2560130 / r2560131;
        double r2560133 = x;
        double r2560134 = r2560130 + r2560133;
        double r2560135 = r2560130 - r2560133;
        double r2560136 = r2560134 / r2560135;
        double r2560137 = log(r2560136);
        double r2560138 = r2560132 * r2560137;
        return r2560138;
}
↓
double f(double x) {
        double r2560139 = 1.0;
        double r2560140 = log(r2560139);
        double r2560141 = x;
        double r2560142 = r2560141 * r2560141;
        double r2560143 = r2560141 + r2560142;
        double r2560144 = r2560139 * r2560139;
        double r2560145 = r2560142 / r2560144;
        double r2560146 = r2560143 - r2560145;
        double r2560147 = 2.0;
        double r2560148 = r2560146 * r2560147;
        double r2560149 = r2560140 + r2560148;
        double r2560150 = r2560139 / r2560147;
        double r2560151 = r2560149 * r2560150;
        return r2560151;
}

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. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019172 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))