Average Error: 58.6 → 0.6
Time: 13.8s
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 r2486041 = 1.0;
        double r2486042 = 2.0;
        double r2486043 = r2486041 / r2486042;
        double r2486044 = x;
        double r2486045 = r2486041 + r2486044;
        double r2486046 = r2486041 - r2486044;
        double r2486047 = r2486045 / r2486046;
        double r2486048 = log(r2486047);
        double r2486049 = r2486043 * r2486048;
        return r2486049;
}
↓
double f(double x) {
        double r2486050 = 1.0;
        double r2486051 = log(r2486050);
        double r2486052 = x;
        double r2486053 = r2486052 * r2486052;
        double r2486054 = r2486052 + r2486053;
        double r2486055 = r2486050 * r2486050;
        double r2486056 = r2486053 / r2486055;
        double r2486057 = r2486054 - r2486056;
        double r2486058 = 2.0;
        double r2486059 = r2486057 * r2486058;
        double r2486060 = r2486051 + r2486059;
        double r2486061 = r2486050 / r2486058;
        double r2486062 = r2486060 * r2486061;
        return r2486062;
}

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)))))