Average Error: 58.5 → 0.2
Time: 21.3s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \left(\frac{x \cdot \left(4 - \left(x \cdot \left(x \cdot \frac{2}{3}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{2}{3}\right)\right)\right)}{2 - x \cdot \left(x \cdot \frac{2}{3}\right)} + \frac{2}{5} \cdot {x}^{5}\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \left(\frac{x \cdot \left(4 - \left(x \cdot \left(x \cdot \frac{2}{3}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{2}{3}\right)\right)\right)}{2 - x \cdot \left(x \cdot \frac{2}{3}\right)} + \frac{2}{5} \cdot {x}^{5}\right)
double f(double x) {
        double r2767368 = 1.0;
        double r2767369 = 2.0;
        double r2767370 = r2767368 / r2767369;
        double r2767371 = x;
        double r2767372 = r2767368 + r2767371;
        double r2767373 = r2767368 - r2767371;
        double r2767374 = r2767372 / r2767373;
        double r2767375 = log(r2767374);
        double r2767376 = r2767370 * r2767375;
        return r2767376;
}
↓
double f(double x) {
        double r2767377 = 0.5;
        double r2767378 = x;
        double r2767379 = 4.0;
        double r2767380 = 0.6666666666666666;
        double r2767381 = r2767378 * r2767380;
        double r2767382 = r2767378 * r2767381;
        double r2767383 = r2767382 * r2767382;
        double r2767384 = r2767379 - r2767383;
        double r2767385 = r2767378 * r2767384;
        double r2767386 = 2.0;
        double r2767387 = r2767386 - r2767382;
        double r2767388 = r2767385 / r2767387;
        double r2767389 = 0.4;
        double r2767390 = 5.0;
        double r2767391 = pow(r2767378, r2767390);
        double r2767392 = r2767389 * r2767391;
        double r2767393 = r2767388 + r2767392;
        double r2767394 = r2767377 * r2767393;
        return r2767394;
}

Error

Bits error versus x

Try it out

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. Simplified58.5

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

  3. Taylor expanded around 0 0.2

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(2 \cdot x + \left(\frac{2}{3} \cdot {x}^{3} + \frac{2}{5} \cdot {x}^{5}\right)\right)}\]

  4. Simplified0.2

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\frac{2}{5} \cdot {x}^{5} + x \cdot \left(2 + \left(x \cdot \frac{2}{3}\right) \cdot x\right)\right)}\]
  5. Using strategy rm

  6. Applied flip-+0.2

    \[\leadsto \frac{1}{2} \cdot \left(\frac{2}{5} \cdot {x}^{5} + x \cdot \color{blue}{\frac{2 \cdot 2 - \left(\left(x \cdot \frac{2}{3}\right) \cdot x\right) \cdot \left(\left(x \cdot \frac{2}{3}\right) \cdot x\right)}{2 - \left(x \cdot \frac{2}{3}\right) \cdot x}}\right)\]

  7. Applied associate-*r/0.2

    \[\leadsto \frac{1}{2} \cdot \left(\frac{2}{5} \cdot {x}^{5} + \color{blue}{\frac{x \cdot \left(2 \cdot 2 - \left(\left(x \cdot \frac{2}{3}\right) \cdot x\right) \cdot \left(\left(x \cdot \frac{2}{3}\right) \cdot x\right)\right)}{2 - \left(x \cdot \frac{2}{3}\right) \cdot x}}\right)\]

  8. Final simplification0.2

    \[\leadsto \frac{1}{2} \cdot \left(\frac{x \cdot \left(4 - \left(x \cdot \left(x \cdot \frac{2}{3}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{2}{3}\right)\right)\right)}{2 - x \cdot \left(x \cdot \frac{2}{3}\right)} + \frac{2}{5} \cdot {x}^{5}\right)\]

Reproduce

herbie shell --seed 2019138 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))