Average Error: 58.6 → 0.2
Time: 17.3s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\left(2 \cdot x + \left(\frac{2}{5} \cdot {x}^{5} + x \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{3}\right)\right)\right) \cdot \frac{1}{2}\]
\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(x \cdot x\right) \cdot \frac{2}{3}\right)\right)\right) \cdot \frac{1}{2}
double f(double x) {
        double r1148502 = 1.0;
        double r1148503 = 2.0;
        double r1148504 = r1148502 / r1148503;
        double r1148505 = x;
        double r1148506 = r1148502 + r1148505;
        double r1148507 = r1148502 - r1148505;
        double r1148508 = r1148506 / r1148507;
        double r1148509 = log(r1148508);
        double r1148510 = r1148504 * r1148509;
        return r1148510;
}
↓
double f(double x) {
        double r1148511 = 2.0;
        double r1148512 = x;
        double r1148513 = r1148511 * r1148512;
        double r1148514 = 0.4;
        double r1148515 = 5.0;
        double r1148516 = pow(r1148512, r1148515);
        double r1148517 = r1148514 * r1148516;
        double r1148518 = r1148512 * r1148512;
        double r1148519 = 0.6666666666666666;
        double r1148520 = r1148518 * r1148519;
        double r1148521 = r1148512 * r1148520;
        double r1148522 = r1148517 + r1148521;
        double r1148523 = r1148513 + r1148522;
        double r1148524 = 0.5;
        double r1148525 = r1148523 * r1148524;
        return r1148525;
}

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. Simplified58.6

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

  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

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

  6. Applied distribute-lft-in0.2

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

  7. Applied associate-+r+0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019155 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))