Average Error: 58.7 → 0.2
Time: 27.9s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\left(\frac{2}{5} \cdot {x}^{5} + \left(x \cdot \left(x \cdot \frac{2}{3}\right) + 2\right) \cdot x\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\left(\frac{2}{5} \cdot {x}^{5} + \left(x \cdot \left(x \cdot \frac{2}{3}\right) + 2\right) \cdot x\right) \cdot \frac{1}{2}
double f(double x) {
        double r7546212 = 1.0;
        double r7546213 = 2.0;
        double r7546214 = r7546212 / r7546213;
        double r7546215 = x;
        double r7546216 = r7546212 + r7546215;
        double r7546217 = r7546212 - r7546215;
        double r7546218 = r7546216 / r7546217;
        double r7546219 = log(r7546218);
        double r7546220 = r7546214 * r7546219;
        return r7546220;
}
↓
double f(double x) {
        double r7546221 = 0.4;
        double r7546222 = x;
        double r7546223 = 5.0;
        double r7546224 = pow(r7546222, r7546223);
        double r7546225 = r7546221 * r7546224;
        double r7546226 = 0.6666666666666666;
        double r7546227 = r7546222 * r7546226;
        double r7546228 = r7546222 * r7546227;
        double r7546229 = 2.0;
        double r7546230 = r7546228 + r7546229;
        double r7546231 = r7546230 * r7546222;
        double r7546232 = r7546225 + r7546231;
        double r7546233 = 0.5;
        double r7546234 = r7546232 * r7546233;
        return r7546234;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.7

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

  2. Simplified58.7

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

  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

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019112 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))