Average Error: 58.6 → 0.2
Time: 25.9s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \left({x}^{5} \cdot \frac{2}{5} + \left(2 \cdot x + \left(x \cdot \left(x \cdot \frac{2}{3}\right)\right) \cdot x\right)\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \left({x}^{5} \cdot \frac{2}{5} + \left(2 \cdot x + \left(x \cdot \left(x \cdot \frac{2}{3}\right)\right) \cdot x\right)\right)
double f(double x) {
        double r7923366 = 1.0;
        double r7923367 = 2.0;
        double r7923368 = r7923366 / r7923367;
        double r7923369 = x;
        double r7923370 = r7923366 + r7923369;
        double r7923371 = r7923366 - r7923369;
        double r7923372 = r7923370 / r7923371;
        double r7923373 = log(r7923372);
        double r7923374 = r7923368 * r7923373;
        return r7923374;
}
↓
double f(double x) {
        double r7923375 = 0.5;
        double r7923376 = x;
        double r7923377 = 5.0;
        double r7923378 = pow(r7923376, r7923377);
        double r7923379 = 0.4;
        double r7923380 = r7923378 * r7923379;
        double r7923381 = 2.0;
        double r7923382 = r7923381 * r7923376;
        double r7923383 = 0.6666666666666666;
        double r7923384 = r7923376 * r7923383;
        double r7923385 = r7923376 * r7923384;
        double r7923386 = r7923385 * r7923376;
        double r7923387 = r7923382 + r7923386;
        double r7923388 = r7923380 + r7923387;
        double r7923389 = r7923375 * r7923388;
        return r7923389;
}

Error

Bits error versus x

Try it out

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

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

  3. Taylor expanded around 0 0.2

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

  4. Simplified0.2

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

  6. Applied distribute-rgt-in0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019119 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))