Average Error: 58.4 → 0.3
Time: 18.1s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\left(2 \cdot x + \left(\frac{2}{5} \cdot {x}^{5} + x \cdot \left(\left(\frac{2}{3} \cdot x\right) \cdot x\right)\right)\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\left(2 \cdot x + \left(\frac{2}{5} \cdot {x}^{5} + x \cdot \left(\left(\frac{2}{3} \cdot x\right) \cdot x\right)\right)\right) \cdot \frac{1}{2}
double f(double x) {
        double r2568629 = 1.0;
        double r2568630 = 2.0;
        double r2568631 = r2568629 / r2568630;
        double r2568632 = x;
        double r2568633 = r2568629 + r2568632;
        double r2568634 = r2568629 - r2568632;
        double r2568635 = r2568633 / r2568634;
        double r2568636 = log(r2568635);
        double r2568637 = r2568631 * r2568636;
        return r2568637;
}
↓
double f(double x) {
        double r2568638 = 2.0;
        double r2568639 = x;
        double r2568640 = r2568638 * r2568639;
        double r2568641 = 0.4;
        double r2568642 = 5.0;
        double r2568643 = pow(r2568639, r2568642);
        double r2568644 = r2568641 * r2568643;
        double r2568645 = 0.6666666666666666;
        double r2568646 = r2568645 * r2568639;
        double r2568647 = r2568646 * r2568639;
        double r2568648 = r2568639 * r2568647;
        double r2568649 = r2568644 + r2568648;
        double r2568650 = r2568640 + r2568649;
        double r2568651 = 0.5;
        double r2568652 = r2568650 * r2568651;
        return r2568652;
}

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. Simplified58.4

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

  3. Taylor expanded around 0 0.3

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

  4. Simplified0.3

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

  6. Applied distribute-rgt-in0.3

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

  7. Applied associate-+r+0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019163 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))