Average Error: 58.6 → 0.2
Time: 15.6s
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 r1395438 = 1.0;
        double r1395439 = 2.0;
        double r1395440 = r1395438 / r1395439;
        double r1395441 = x;
        double r1395442 = r1395438 + r1395441;
        double r1395443 = r1395438 - r1395441;
        double r1395444 = r1395442 / r1395443;
        double r1395445 = log(r1395444);
        double r1395446 = r1395440 * r1395445;
        return r1395446;
}
↓
double f(double x) {
        double r1395447 = 2.0;
        double r1395448 = x;
        double r1395449 = r1395447 * r1395448;
        double r1395450 = 0.4;
        double r1395451 = 5.0;
        double r1395452 = pow(r1395448, r1395451);
        double r1395453 = r1395450 * r1395452;
        double r1395454 = r1395448 * r1395448;
        double r1395455 = 0.6666666666666666;
        double r1395456 = r1395454 * r1395455;
        double r1395457 = r1395448 * r1395456;
        double r1395458 = r1395453 + r1395457;
        double r1395459 = r1395449 + r1395458;
        double r1395460 = 0.5;
        double r1395461 = r1395459 * r1395460;
        return r1395461;
}

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-rgt-in0.2

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

  7. Applied associate-+r+0.2

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

  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 2019153 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))