Average Error: 58.8 → 0.2
Time: 17.0s
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 r2701188 = 1.0;
        double r2701189 = 2.0;
        double r2701190 = r2701188 / r2701189;
        double r2701191 = x;
        double r2701192 = r2701188 + r2701191;
        double r2701193 = r2701188 - r2701191;
        double r2701194 = r2701192 / r2701193;
        double r2701195 = log(r2701194);
        double r2701196 = r2701190 * r2701195;
        return r2701196;
}
↓
double f(double x) {
        double r2701197 = 0.5;
        double r2701198 = x;
        double r2701199 = 5.0;
        double r2701200 = pow(r2701198, r2701199);
        double r2701201 = 0.4;
        double r2701202 = r2701200 * r2701201;
        double r2701203 = 2.0;
        double r2701204 = r2701198 * r2701203;
        double r2701205 = 0.6666666666666666;
        double r2701206 = r2701205 * r2701198;
        double r2701207 = r2701206 * r2701198;
        double r2701208 = r2701198 * r2701207;
        double r2701209 = r2701204 + r2701208;
        double r2701210 = r2701202 + r2701209;
        double r2701211 = r2701197 * r2701210;
        return r2701211;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.8

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

  2. Simplified58.8

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

  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

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

  6. Applied distribute-rgt-in0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019149 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))