Average Error: 58.6 → 0.2
Time: 23.7s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \left({x}^{5} \cdot \frac{2}{5} + \left(2 \cdot x + \left(x \cdot \left(x \cdot \frac{2}{3}\right)\right) \cdot x\right)\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \left({x}^{5} \cdot \frac{2}{5} + \left(2 \cdot x + \left(x \cdot \left(x \cdot \frac{2}{3}\right)\right) \cdot x\right)\right)
double f(double x) {
        double r7674862 = 1.0;
        double r7674863 = 2.0;
        double r7674864 = r7674862 / r7674863;
        double r7674865 = x;
        double r7674866 = r7674862 + r7674865;
        double r7674867 = r7674862 - r7674865;
        double r7674868 = r7674866 / r7674867;
        double r7674869 = log(r7674868);
        double r7674870 = r7674864 * r7674869;
        return r7674870;
}
↓
double f(double x) {
        double r7674871 = 0.5;
        double r7674872 = x;
        double r7674873 = 5.0;
        double r7674874 = pow(r7674872, r7674873);
        double r7674875 = 0.4;
        double r7674876 = r7674874 * r7674875;
        double r7674877 = 2.0;
        double r7674878 = r7674877 * r7674872;
        double r7674879 = 0.6666666666666666;
        double r7674880 = r7674872 * r7674879;
        double r7674881 = r7674872 * r7674880;
        double r7674882 = r7674881 * r7674872;
        double r7674883 = r7674878 + r7674882;
        double r7674884 = r7674876 + r7674883;
        double r7674885 = r7674871 * r7674884;
        return r7674885;
}

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

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

  6. Applied distribute-lft-in0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019104 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))