Average Error: 58.4 → 0.3
Time: 21.0s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \left({x}^{5} \cdot \frac{2}{5} + \left(x \cdot 2 + x \cdot \left(\left(\frac{2}{3} \cdot x\right) \cdot x\right)\right)\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \left({x}^{5} \cdot \frac{2}{5} + \left(x \cdot 2 + x \cdot \left(\left(\frac{2}{3} \cdot x\right) \cdot x\right)\right)\right)
double f(double x) {
        double r2239322 = 1.0;
        double r2239323 = 2.0;
        double r2239324 = r2239322 / r2239323;
        double r2239325 = x;
        double r2239326 = r2239322 + r2239325;
        double r2239327 = r2239322 - r2239325;
        double r2239328 = r2239326 / r2239327;
        double r2239329 = log(r2239328);
        double r2239330 = r2239324 * r2239329;
        return r2239330;
}
↓
double f(double x) {
        double r2239331 = 0.5;
        double r2239332 = x;
        double r2239333 = 5.0;
        double r2239334 = pow(r2239332, r2239333);
        double r2239335 = 0.4;
        double r2239336 = r2239334 * r2239335;
        double r2239337 = 2.0;
        double r2239338 = r2239332 * r2239337;
        double r2239339 = 0.6666666666666666;
        double r2239340 = r2239339 * r2239332;
        double r2239341 = r2239340 * r2239332;
        double r2239342 = r2239332 * r2239341;
        double r2239343 = r2239338 + r2239342;
        double r2239344 = r2239336 + r2239343;
        double r2239345 = r2239331 * r2239344;
        return r2239345;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.4

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

  2. Simplified58.4

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

  3. Taylor expanded around 0 0.3

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

  4. Simplified0.3

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

  6. Applied distribute-rgt-in0.3

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

  7. Final simplification0.3

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

Reproduce

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