Average Error: 58.7 → 0.6
Time: 18.8s
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 r1671573 = 1.0;
        double r1671574 = 2.0;
        double r1671575 = r1671573 / r1671574;
        double r1671576 = x;
        double r1671577 = r1671573 + r1671576;
        double r1671578 = r1671573 - r1671576;
        double r1671579 = r1671577 / r1671578;
        double r1671580 = log(r1671579);
        double r1671581 = r1671575 * r1671580;
        return r1671581;
}
↓
double f(double x) {
        double r1671582 = 1.0;
        double r1671583 = log(r1671582);
        double r1671584 = x;
        double r1671585 = r1671584 * r1671584;
        double r1671586 = r1671584 + r1671585;
        double r1671587 = r1671582 * r1671582;
        double r1671588 = r1671585 / r1671587;
        double r1671589 = r1671586 - r1671588;
        double r1671590 = 2.0;
        double r1671591 = r1671589 * r1671590;
        double r1671592 = r1671583 + r1671591;
        double r1671593 = r1671582 / r1671590;
        double r1671594 = r1671592 * r1671593;
        return r1671594;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.7

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

  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 2019171 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))