Average Error: 58.6 → 0.3
Time: 27.1s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\left(\left(\left(\frac{x}{1} \cdot \frac{x}{1}\right) \cdot \left(\frac{2}{3} \cdot \frac{x}{1}\right) + \frac{{x}^{5}}{{1}^{5}} \cdot \frac{2}{5}\right) + x \cdot 2\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\left(\left(\left(\frac{x}{1} \cdot \frac{x}{1}\right) \cdot \left(\frac{2}{3} \cdot \frac{x}{1}\right) + \frac{{x}^{5}}{{1}^{5}} \cdot \frac{2}{5}\right) + x \cdot 2\right) \cdot \frac{1}{2}
double f(double x) {
        double r2054562 = 1.0;
        double r2054563 = 2.0;
        double r2054564 = r2054562 / r2054563;
        double r2054565 = x;
        double r2054566 = r2054562 + r2054565;
        double r2054567 = r2054562 - r2054565;
        double r2054568 = r2054566 / r2054567;
        double r2054569 = log(r2054568);
        double r2054570 = r2054564 * r2054569;
        return r2054570;
}
↓
double f(double x) {
        double r2054571 = x;
        double r2054572 = 1.0;
        double r2054573 = r2054571 / r2054572;
        double r2054574 = r2054573 * r2054573;
        double r2054575 = 0.6666666666666666;
        double r2054576 = r2054575 * r2054573;
        double r2054577 = r2054574 * r2054576;
        double r2054578 = 5.0;
        double r2054579 = pow(r2054571, r2054578);
        double r2054580 = pow(r2054572, r2054578);
        double r2054581 = r2054579 / r2054580;
        double r2054582 = 0.4;
        double r2054583 = r2054581 * r2054582;
        double r2054584 = r2054577 + r2054583;
        double r2054585 = 2.0;
        double r2054586 = r2054571 * r2054585;
        double r2054587 = r2054584 + r2054586;
        double r2054588 = r2054572 / r2054585;
        double r2054589 = r2054587 * r2054588;
        return r2054589;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.6

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

  3. Applied log-div58.6

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

  4. Taylor expanded around 0 0.3

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

  5. Simplified0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019168 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))