Average Error: 58.6 → 0.2
Time: 17.1s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\left(\frac{2}{5} \cdot {x}^{5} + x \cdot \left(\left(\frac{2}{3} \cdot x\right) \cdot x + 2\right)\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\left(\frac{2}{5} \cdot {x}^{5} + x \cdot \left(\left(\frac{2}{3} \cdot x\right) \cdot x + 2\right)\right) \cdot \frac{1}{2}
double f(double x) {
        double r3165973 = 1.0;
        double r3165974 = 2.0;
        double r3165975 = r3165973 / r3165974;
        double r3165976 = x;
        double r3165977 = r3165973 + r3165976;
        double r3165978 = r3165973 - r3165976;
        double r3165979 = r3165977 / r3165978;
        double r3165980 = log(r3165979);
        double r3165981 = r3165975 * r3165980;
        return r3165981;
}
↓
double f(double x) {
        double r3165982 = 0.4;
        double r3165983 = x;
        double r3165984 = 5.0;
        double r3165985 = pow(r3165983, r3165984);
        double r3165986 = r3165982 * r3165985;
        double r3165987 = 0.6666666666666666;
        double r3165988 = r3165987 * r3165983;
        double r3165989 = r3165988 * r3165983;
        double r3165990 = 2.0;
        double r3165991 = r3165989 + r3165990;
        double r3165992 = r3165983 * r3165991;
        double r3165993 = r3165986 + r3165992;
        double r3165994 = 0.5;
        double r3165995 = r3165993 * r3165994;
        return r3165995;
}

Error

Bits error versus x

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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

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

  3. Taylor expanded around 0 0.2

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019164 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))