Average Error: 58.5 → 0.2
Time: 18.0s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\left({x}^{5} \cdot \frac{2}{5} + \left(\left(x \cdot x\right) \cdot \frac{2}{3} + 2\right) \cdot x\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\left({x}^{5} \cdot \frac{2}{5} + \left(\left(x \cdot x\right) \cdot \frac{2}{3} + 2\right) \cdot x\right) \cdot \frac{1}{2}
double f(double x) {
        double r3704960 = 1.0;
        double r3704961 = 2.0;
        double r3704962 = r3704960 / r3704961;
        double r3704963 = x;
        double r3704964 = r3704960 + r3704963;
        double r3704965 = r3704960 - r3704963;
        double r3704966 = r3704964 / r3704965;
        double r3704967 = log(r3704966);
        double r3704968 = r3704962 * r3704967;
        return r3704968;
}
↓
double f(double x) {
        double r3704969 = x;
        double r3704970 = 5.0;
        double r3704971 = pow(r3704969, r3704970);
        double r3704972 = 0.4;
        double r3704973 = r3704971 * r3704972;
        double r3704974 = r3704969 * r3704969;
        double r3704975 = 0.6666666666666666;
        double r3704976 = r3704974 * r3704975;
        double r3704977 = 2.0;
        double r3704978 = r3704976 + r3704977;
        double r3704979 = r3704978 * r3704969;
        double r3704980 = r3704973 + r3704979;
        double r3704981 = 0.5;
        double r3704982 = r3704980 * r3704981;
        return r3704982;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.5

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

  2. Simplified58.5

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

  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

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

  5. Taylor expanded around 0 0.2

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

  6. Simplified0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019165 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))