Average Error: 58.4 → 0.2
Time: 23.8s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\left(\frac{{x}^{5}}{{1}^{5}} \cdot \frac{2}{5} + \left(x \cdot 2 + \left(\frac{2}{3} \cdot \frac{x}{1}\right) \cdot \left(\frac{x}{1} \cdot \frac{x}{1}\right)\right)\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\left(\frac{{x}^{5}}{{1}^{5}} \cdot \frac{2}{5} + \left(x \cdot 2 + \left(\frac{2}{3} \cdot \frac{x}{1}\right) \cdot \left(\frac{x}{1} \cdot \frac{x}{1}\right)\right)\right) \cdot \frac{1}{2}
double f(double x) {
        double r3810367 = 1.0;
        double r3810368 = 2.0;
        double r3810369 = r3810367 / r3810368;
        double r3810370 = x;
        double r3810371 = r3810367 + r3810370;
        double r3810372 = r3810367 - r3810370;
        double r3810373 = r3810371 / r3810372;
        double r3810374 = log(r3810373);
        double r3810375 = r3810369 * r3810374;
        return r3810375;
}
↓
double f(double x) {
        double r3810376 = x;
        double r3810377 = 5.0;
        double r3810378 = pow(r3810376, r3810377);
        double r3810379 = 1.0;
        double r3810380 = pow(r3810379, r3810377);
        double r3810381 = r3810378 / r3810380;
        double r3810382 = 0.4;
        double r3810383 = r3810381 * r3810382;
        double r3810384 = 2.0;
        double r3810385 = r3810376 * r3810384;
        double r3810386 = 0.6666666666666666;
        double r3810387 = r3810376 / r3810379;
        double r3810388 = r3810386 * r3810387;
        double r3810389 = r3810387 * r3810387;
        double r3810390 = r3810388 * r3810389;
        double r3810391 = r3810385 + r3810390;
        double r3810392 = r3810383 + r3810391;
        double r3810393 = r3810379 / r3810384;
        double r3810394 = r3810392 * r3810393;
        return r3810394;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.4

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

  3. Applied log-div58.4

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

  4. Taylor expanded around 0 0.2

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

  5. Simplified0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019200 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))