Average Error: 58.5 → 0.2
Time: 18.3s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\left(\frac{2}{5} \cdot {x}^{5} + \left(\left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right) + 2 \cdot x\right)\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\left(\frac{2}{5} \cdot {x}^{5} + \left(\left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right) + 2 \cdot x\right)\right) \cdot \frac{1}{2}
double f(double x) {
        double r2650382 = 1.0;
        double r2650383 = 2.0;
        double r2650384 = r2650382 / r2650383;
        double r2650385 = x;
        double r2650386 = r2650382 + r2650385;
        double r2650387 = r2650382 - r2650385;
        double r2650388 = r2650386 / r2650387;
        double r2650389 = log(r2650388);
        double r2650390 = r2650384 * r2650389;
        return r2650390;
}
↓
double f(double x) {
        double r2650391 = 0.4;
        double r2650392 = x;
        double r2650393 = 5.0;
        double r2650394 = pow(r2650392, r2650393);
        double r2650395 = r2650391 * r2650394;
        double r2650396 = r2650392 * r2650392;
        double r2650397 = 0.6666666666666666;
        double r2650398 = r2650392 * r2650397;
        double r2650399 = r2650396 * r2650398;
        double r2650400 = 2.0;
        double r2650401 = r2650400 * r2650392;
        double r2650402 = r2650399 + r2650401;
        double r2650403 = r2650395 + r2650402;
        double r2650404 = 0.5;
        double r2650405 = r2650403 * r2650404;
        return r2650405;
}

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. 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

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

  5. Taylor expanded around 0 0.2

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

  6. Simplified0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019158 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))