Average Error: 58.4 → 0.3
Time: 19.6s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\left(2 \cdot x + \left(\frac{2}{5} \cdot {x}^{5} + x \cdot \left(\left(\frac{2}{3} \cdot x\right) \cdot x\right)\right)\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\left(2 \cdot x + \left(\frac{2}{5} \cdot {x}^{5} + x \cdot \left(\left(\frac{2}{3} \cdot x\right) \cdot x\right)\right)\right) \cdot \frac{1}{2}
double f(double x) {
        double r3031887 = 1.0;
        double r3031888 = 2.0;
        double r3031889 = r3031887 / r3031888;
        double r3031890 = x;
        double r3031891 = r3031887 + r3031890;
        double r3031892 = r3031887 - r3031890;
        double r3031893 = r3031891 / r3031892;
        double r3031894 = log(r3031893);
        double r3031895 = r3031889 * r3031894;
        return r3031895;
}
↓
double f(double x) {
        double r3031896 = 2.0;
        double r3031897 = x;
        double r3031898 = r3031896 * r3031897;
        double r3031899 = 0.4;
        double r3031900 = 5.0;
        double r3031901 = pow(r3031897, r3031900);
        double r3031902 = r3031899 * r3031901;
        double r3031903 = 0.6666666666666666;
        double r3031904 = r3031903 * r3031897;
        double r3031905 = r3031904 * r3031897;
        double r3031906 = r3031897 * r3031905;
        double r3031907 = r3031902 + r3031906;
        double r3031908 = r3031898 + r3031907;
        double r3031909 = 0.5;
        double r3031910 = r3031908 * r3031909;
        return r3031910;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.4

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

  2. Simplified58.4

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

  3. Taylor expanded around 0 0.3

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

  4. Simplified0.3

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

  6. Applied distribute-lft-in0.3

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

  7. Applied associate-+r+0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019163 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))