Average Error: 58.6 → 0.2
Time: 16.3s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \left({x}^{5} \cdot \frac{2}{5} + \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{3}\right) + 2 \cdot x\right)\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \left({x}^{5} \cdot \frac{2}{5} + \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{3}\right) + 2 \cdot x\right)\right)
double f(double x) {
        double r1481171 = 1.0;
        double r1481172 = 2.0;
        double r1481173 = r1481171 / r1481172;
        double r1481174 = x;
        double r1481175 = r1481171 + r1481174;
        double r1481176 = r1481171 - r1481174;
        double r1481177 = r1481175 / r1481176;
        double r1481178 = log(r1481177);
        double r1481179 = r1481173 * r1481178;
        return r1481179;
}
↓
double f(double x) {
        double r1481180 = 0.5;
        double r1481181 = x;
        double r1481182 = 5.0;
        double r1481183 = pow(r1481181, r1481182);
        double r1481184 = 0.4;
        double r1481185 = r1481183 * r1481184;
        double r1481186 = r1481181 * r1481181;
        double r1481187 = 0.6666666666666666;
        double r1481188 = r1481186 * r1481187;
        double r1481189 = r1481181 * r1481188;
        double r1481190 = 2.0;
        double r1481191 = r1481190 * r1481181;
        double r1481192 = r1481189 + r1481191;
        double r1481193 = r1481185 + r1481192;
        double r1481194 = r1481180 * r1481193;
        return r1481194;
}

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

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

  6. Applied distribute-rgt-in0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019135 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))