Average Error: 58.5 → 0.2
Time: 33.7s
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 r5987986 = 1.0;
        double r5987987 = 2.0;
        double r5987988 = r5987986 / r5987987;
        double r5987989 = x;
        double r5987990 = r5987986 + r5987989;
        double r5987991 = r5987986 - r5987989;
        double r5987992 = r5987990 / r5987991;
        double r5987993 = log(r5987992);
        double r5987994 = r5987988 * r5987993;
        return r5987994;
}
↓
double f(double x) {
        double r5987995 = 0.5;
        double r5987996 = x;
        double r5987997 = 5.0;
        double r5987998 = pow(r5987996, r5987997);
        double r5987999 = 0.4;
        double r5988000 = r5987998 * r5987999;
        double r5988001 = 2.0;
        double r5988002 = r5988001 * r5987996;
        double r5988003 = 0.6666666666666666;
        double r5988004 = r5987996 * r5988003;
        double r5988005 = r5987996 * r5988004;
        double r5988006 = r5988005 * r5987996;
        double r5988007 = r5988002 + r5988006;
        double r5988008 = r5988000 + r5988007;
        double r5988009 = r5987995 * r5988008;
        return r5988009;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.5

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

  2. Simplified58.5

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

  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. Simplified0.3

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

  6. Applied distribute-rgt-in0.2

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

  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 2019107 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))