Average Error: 58.4 → 0.3
Time: 18.3s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\left(\frac{2}{5} \cdot {x}^{5} + \left(x \cdot 2 + x \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{3}\right)\right)\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\left(\frac{2}{5} \cdot {x}^{5} + \left(x \cdot 2 + x \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{3}\right)\right)\right) \cdot \frac{1}{2}
double f(double x) {
        double r1334451 = 1.0;
        double r1334452 = 2.0;
        double r1334453 = r1334451 / r1334452;
        double r1334454 = x;
        double r1334455 = r1334451 + r1334454;
        double r1334456 = r1334451 - r1334454;
        double r1334457 = r1334455 / r1334456;
        double r1334458 = log(r1334457);
        double r1334459 = r1334453 * r1334458;
        return r1334459;
}
↓
double f(double x) {
        double r1334460 = 0.4;
        double r1334461 = x;
        double r1334462 = 5.0;
        double r1334463 = pow(r1334461, r1334462);
        double r1334464 = r1334460 * r1334463;
        double r1334465 = 2.0;
        double r1334466 = r1334461 * r1334465;
        double r1334467 = r1334461 * r1334461;
        double r1334468 = 0.6666666666666666;
        double r1334469 = r1334467 * r1334468;
        double r1334470 = r1334461 * r1334469;
        double r1334471 = r1334466 + r1334470;
        double r1334472 = r1334464 + r1334471;
        double r1334473 = 0.5;
        double r1334474 = r1334472 * r1334473;
        return r1334474;
}

Error

Bits error versus x

Try it out

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

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

  6. Applied distribute-lft-in0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2019151 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))