Average Error: 58.6 → 0.6
Time: 19.3s
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 r48354 = 1.0;
        double r48355 = 2.0;
        double r48356 = r48354 / r48355;
        double r48357 = x;
        double r48358 = r48354 + r48357;
        double r48359 = r48354 - r48357;
        double r48360 = r48358 / r48359;
        double r48361 = log(r48360);
        double r48362 = r48356 * r48361;
        return r48362;
}
↓
double f(double x) {
        double r48363 = 1.0;
        double r48364 = 2.0;
        double r48365 = r48363 / r48364;
        double r48366 = x;
        double r48367 = r48363 * r48363;
        double r48368 = r48366 / r48367;
        double r48369 = r48366 - r48368;
        double r48370 = r48366 * r48369;
        double r48371 = r48366 + r48370;
        double r48372 = r48364 * r48371;
        double r48373 = log(r48363);
        double r48374 = r48372 + r48373;
        double r48375 = r48365 * r48374;
        return r48375;
}

Error

Bits error versus x

Try it out

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

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

  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 2019323 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))