Average Error: 58.7 → 0.2
Time: 15.1s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\left(x \cdot \left(\left(\frac{2}{3} \cdot x\right) \cdot x + 2\right) + {x}^{5} \cdot \frac{2}{5}\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\left(x \cdot \left(\left(\frac{2}{3} \cdot x\right) \cdot x + 2\right) + {x}^{5} \cdot \frac{2}{5}\right) \cdot \frac{1}{2}
double f(double x) {
        double r1378877 = 1.0;
        double r1378878 = 2.0;
        double r1378879 = r1378877 / r1378878;
        double r1378880 = x;
        double r1378881 = r1378877 + r1378880;
        double r1378882 = r1378877 - r1378880;
        double r1378883 = r1378881 / r1378882;
        double r1378884 = log(r1378883);
        double r1378885 = r1378879 * r1378884;
        return r1378885;
}
↓
double f(double x) {
        double r1378886 = x;
        double r1378887 = 0.6666666666666666;
        double r1378888 = r1378887 * r1378886;
        double r1378889 = r1378888 * r1378886;
        double r1378890 = 2.0;
        double r1378891 = r1378889 + r1378890;
        double r1378892 = r1378886 * r1378891;
        double r1378893 = 5.0;
        double r1378894 = pow(r1378886, r1378893);
        double r1378895 = 0.4;
        double r1378896 = r1378894 * r1378895;
        double r1378897 = r1378892 + r1378896;
        double r1378898 = 0.5;
        double r1378899 = r1378897 * r1378898;
        return r1378899;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.7

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

  2. Simplified58.7

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

  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

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019152 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))