Average Error: 58.4 → 0.7
Time: 14.9s
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 r1629545 = 1.0;
        double r1629546 = 2.0;
        double r1629547 = r1629545 / r1629546;
        double r1629548 = x;
        double r1629549 = r1629545 + r1629548;
        double r1629550 = r1629545 - r1629548;
        double r1629551 = r1629549 / r1629550;
        double r1629552 = log(r1629551);
        double r1629553 = r1629547 * r1629552;
        return r1629553;
}
↓
double f(double x) {
        double r1629554 = 1.0;
        double r1629555 = log(r1629554);
        double r1629556 = x;
        double r1629557 = r1629556 * r1629556;
        double r1629558 = r1629556 + r1629557;
        double r1629559 = r1629554 * r1629554;
        double r1629560 = r1629557 / r1629559;
        double r1629561 = r1629558 - r1629560;
        double r1629562 = 2.0;
        double r1629563 = r1629561 * r1629562;
        double r1629564 = r1629555 + r1629563;
        double r1629565 = r1629554 / r1629562;
        double r1629566 = r1629564 * r1629565;
        return r1629566;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.4

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019174 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))