Average Error: 58.5 → 0.7
Time: 14.9s
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 r42636 = 1.0;
        double r42637 = 2.0;
        double r42638 = r42636 / r42637;
        double r42639 = x;
        double r42640 = r42636 + r42639;
        double r42641 = r42636 - r42639;
        double r42642 = r42640 / r42641;
        double r42643 = log(r42642);
        double r42644 = r42638 * r42643;
        return r42644;
}
↓
double f(double x) {
        double r42645 = 2.0;
        double r42646 = x;
        double r42647 = r42646 * r42646;
        double r42648 = r42646 + r42647;
        double r42649 = 1.0;
        double r42650 = r42646 / r42649;
        double r42651 = r42650 * r42650;
        double r42652 = r42648 - r42651;
        double r42653 = r42645 * r42652;
        double r42654 = log(r42649);
        double r42655 = r42653 + r42654;
        double r42656 = r42645 / r42649;
        double r42657 = r42655 / r42656;
        return r42657;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.5

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

  2. Simplified58.5

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

  3. Taylor expanded around 0 0.7

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

  4. Simplified0.7

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

  5. Final simplification0.7

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

Reproduce

herbie shell --seed 2019195 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))