Average Error: 58.6 → 0.2
Time: 19.7s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \left({x}^{5} \cdot \frac{2}{5} + \left(x \cdot 2 + x \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{3}\right)\right)\right)\]
\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(x \cdot x\right) \cdot \frac{2}{3}\right)\right)\right)
double f(double x) {
        double r3125125 = 1.0;
        double r3125126 = 2.0;
        double r3125127 = r3125125 / r3125126;
        double r3125128 = x;
        double r3125129 = r3125125 + r3125128;
        double r3125130 = r3125125 - r3125128;
        double r3125131 = r3125129 / r3125130;
        double r3125132 = log(r3125131);
        double r3125133 = r3125127 * r3125132;
        return r3125133;
}
↓
double f(double x) {
        double r3125134 = 0.5;
        double r3125135 = x;
        double r3125136 = 5.0;
        double r3125137 = pow(r3125135, r3125136);
        double r3125138 = 0.4;
        double r3125139 = r3125137 * r3125138;
        double r3125140 = 2.0;
        double r3125141 = r3125135 * r3125140;
        double r3125142 = r3125135 * r3125135;
        double r3125143 = 0.6666666666666666;
        double r3125144 = r3125142 * r3125143;
        double r3125145 = r3125135 * r3125144;
        double r3125146 = r3125141 + r3125145;
        double r3125147 = r3125139 + r3125146;
        double r3125148 = r3125134 * r3125147;
        return r3125148;
}

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

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

  6. Applied distribute-lft-in0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019133 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))