Average Error: 58.7 → 0.2
Time: 46.7s
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 r7964313 = 1.0;
        double r7964314 = 2.0;
        double r7964315 = r7964313 / r7964314;
        double r7964316 = x;
        double r7964317 = r7964313 + r7964316;
        double r7964318 = r7964313 - r7964316;
        double r7964319 = r7964317 / r7964318;
        double r7964320 = log(r7964319);
        double r7964321 = r7964315 * r7964320;
        return r7964321;
}
↓
double f(double x) {
        double r7964322 = 0.5;
        double r7964323 = x;
        double r7964324 = 5.0;
        double r7964325 = pow(r7964323, r7964324);
        double r7964326 = 0.4;
        double r7964327 = r7964325 * r7964326;
        double r7964328 = 2.0;
        double r7964329 = r7964328 * r7964323;
        double r7964330 = 0.6666666666666666;
        double r7964331 = r7964323 * r7964330;
        double r7964332 = r7964323 * r7964331;
        double r7964333 = r7964332 * r7964323;
        double r7964334 = r7964329 + r7964333;
        double r7964335 = r7964327 + r7964334;
        double r7964336 = r7964322 * r7964335;
        return r7964336;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.7

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

  2. Simplified58.7

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

  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 2019128 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))