Average Error: 58.5 → 0.2
Time: 10.2s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \left(\frac{2}{3} \cdot \frac{{x}^{3}}{{1}^{3}} + \left(2 \cdot x + \frac{2}{5} \cdot \frac{{x}^{5}}{{1}^{5}}\right)\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \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)
double f(double x) {
        double r71555 = 1.0;
        double r71556 = 2.0;
        double r71557 = r71555 / r71556;
        double r71558 = x;
        double r71559 = r71555 + r71558;
        double r71560 = r71555 - r71558;
        double r71561 = r71559 / r71560;
        double r71562 = log(r71561);
        double r71563 = r71557 * r71562;
        return r71563;
}
↓
double f(double x) {
        double r71564 = 1.0;
        double r71565 = 2.0;
        double r71566 = r71564 / r71565;
        double r71567 = 0.6666666666666666;
        double r71568 = x;
        double r71569 = 3.0;
        double r71570 = pow(r71568, r71569);
        double r71571 = pow(r71564, r71569);
        double r71572 = r71570 / r71571;
        double r71573 = r71567 * r71572;
        double r71574 = r71565 * r71568;
        double r71575 = 0.4;
        double r71576 = 5.0;
        double r71577 = pow(r71568, r71576);
        double r71578 = pow(r71564, r71576);
        double r71579 = r71577 / r71578;
        double r71580 = r71575 * r71579;
        double r71581 = r71574 + r71580;
        double r71582 = r71573 + r71581;
        double r71583 = r71566 * r71582;
        return r71583;
}

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. Using strategy rm

  3. Applied log-div58.5

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

  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. Final simplification0.2

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

Reproduce

herbie shell --seed 2019304 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))