Average Error: 58.6 → 0.6
Time: 17.6s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\left(\log 1 + \left(\left(x + x \cdot x\right) - \frac{x \cdot x}{1 \cdot 1}\right) \cdot 2\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\left(\log 1 + \left(\left(x + x \cdot x\right) - \frac{x \cdot x}{1 \cdot 1}\right) \cdot 2\right) \cdot \frac{1}{2}
double f(double x) {
        double r3236499 = 1.0;
        double r3236500 = 2.0;
        double r3236501 = r3236499 / r3236500;
        double r3236502 = x;
        double r3236503 = r3236499 + r3236502;
        double r3236504 = r3236499 - r3236502;
        double r3236505 = r3236503 / r3236504;
        double r3236506 = log(r3236505);
        double r3236507 = r3236501 * r3236506;
        return r3236507;
}
↓
double f(double x) {
        double r3236508 = 1.0;
        double r3236509 = log(r3236508);
        double r3236510 = x;
        double r3236511 = r3236510 * r3236510;
        double r3236512 = r3236510 + r3236511;
        double r3236513 = r3236508 * r3236508;
        double r3236514 = r3236511 / r3236513;
        double r3236515 = r3236512 - r3236514;
        double r3236516 = 2.0;
        double r3236517 = r3236515 * r3236516;
        double r3236518 = r3236509 + r3236517;
        double r3236519 = r3236508 / r3236516;
        double r3236520 = r3236518 * r3236519;
        return r3236520;
}

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. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019172 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))