Average Error: 58.5 → 0.6
Time: 18.6s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \left(2 \cdot \left(x + x \cdot \left(x - \frac{x}{1 \cdot 1}\right)\right) + \log 1\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \left(2 \cdot \left(x + x \cdot \left(x - \frac{x}{1 \cdot 1}\right)\right) + \log 1\right)
double f(double x) {
        double r78182 = 1.0;
        double r78183 = 2.0;
        double r78184 = r78182 / r78183;
        double r78185 = x;
        double r78186 = r78182 + r78185;
        double r78187 = r78182 - r78185;
        double r78188 = r78186 / r78187;
        double r78189 = log(r78188);
        double r78190 = r78184 * r78189;
        return r78190;
}
↓
double f(double x) {
        double r78191 = 1.0;
        double r78192 = 2.0;
        double r78193 = r78191 / r78192;
        double r78194 = x;
        double r78195 = r78191 * r78191;
        double r78196 = r78194 / r78195;
        double r78197 = r78194 - r78196;
        double r78198 = r78194 * r78197;
        double r78199 = r78194 + r78198;
        double r78200 = r78192 * r78199;
        double r78201 = log(r78191);
        double r78202 = r78200 + r78201;
        double r78203 = r78193 * r78202;
        return r78203;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.5

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019325 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))