Average Error: 58.6 → 0.2
Time: 18.6s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{{x}^{5} \cdot \frac{2}{5} + \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{3}\right) + 2 \cdot x\right)}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{{x}^{5} \cdot \frac{2}{5} + \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{3}\right) + 2 \cdot x\right)}{2}
double f(double x) {
        double r2515059 = 1.0;
        double r2515060 = 2.0;
        double r2515061 = r2515059 / r2515060;
        double r2515062 = x;
        double r2515063 = r2515059 + r2515062;
        double r2515064 = r2515059 - r2515062;
        double r2515065 = r2515063 / r2515064;
        double r2515066 = log(r2515065);
        double r2515067 = r2515061 * r2515066;
        return r2515067;
}
↓
double f(double x) {
        double r2515068 = x;
        double r2515069 = 5.0;
        double r2515070 = pow(r2515068, r2515069);
        double r2515071 = 0.4;
        double r2515072 = r2515070 * r2515071;
        double r2515073 = r2515068 * r2515068;
        double r2515074 = 0.6666666666666666;
        double r2515075 = r2515073 * r2515074;
        double r2515076 = r2515068 * r2515075;
        double r2515077 = 2.0;
        double r2515078 = r2515077 * r2515068;
        double r2515079 = r2515076 + r2515078;
        double r2515080 = r2515072 + r2515079;
        double r2515081 = r2515080 / r2515077;
        return r2515081;
}

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

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

  3. Taylor expanded around 0 0.2

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

  4. Simplified0.2

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

  6. Applied distribute-rgt-in0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019134 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))