Average Error: 58.4 → 0.2
Time: 35.0s
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 r10426920 = 1.0;
        double r10426921 = 2.0;
        double r10426922 = r10426920 / r10426921;
        double r10426923 = x;
        double r10426924 = r10426920 + r10426923;
        double r10426925 = r10426920 - r10426923;
        double r10426926 = r10426924 / r10426925;
        double r10426927 = log(r10426926);
        double r10426928 = r10426922 * r10426927;
        return r10426928;
}
↓
double f(double x) {
        double r10426929 = 0.5;
        double r10426930 = x;
        double r10426931 = 5.0;
        double r10426932 = pow(r10426930, r10426931);
        double r10426933 = 0.4;
        double r10426934 = r10426932 * r10426933;
        double r10426935 = 2.0;
        double r10426936 = r10426935 * r10426930;
        double r10426937 = 0.6666666666666666;
        double r10426938 = r10426930 * r10426937;
        double r10426939 = r10426930 * r10426938;
        double r10426940 = r10426939 * r10426930;
        double r10426941 = r10426936 + r10426940;
        double r10426942 = r10426934 + r10426941;
        double r10426943 = r10426929 * r10426942;
        return r10426943;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.4

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

  2. Simplified58.4

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

  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 2019125 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))