Average Error: 58.7 → 0.6
Time: 15.4s
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 r3236651 = 1.0;
        double r3236652 = 2.0;
        double r3236653 = r3236651 / r3236652;
        double r3236654 = x;
        double r3236655 = r3236651 + r3236654;
        double r3236656 = r3236651 - r3236654;
        double r3236657 = r3236655 / r3236656;
        double r3236658 = log(r3236657);
        double r3236659 = r3236653 * r3236658;
        return r3236659;
}
↓
double f(double x) {
        double r3236660 = 1.0;
        double r3236661 = log(r3236660);
        double r3236662 = x;
        double r3236663 = r3236662 * r3236662;
        double r3236664 = r3236662 + r3236663;
        double r3236665 = r3236660 * r3236660;
        double r3236666 = r3236663 / r3236665;
        double r3236667 = r3236664 - r3236666;
        double r3236668 = 2.0;
        double r3236669 = r3236667 * r3236668;
        double r3236670 = r3236661 + r3236669;
        double r3236671 = r3236660 / r3236668;
        double r3236672 = r3236670 * r3236671;
        return r3236672;
}

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 2019169 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))