Average Error: 58.6 → 0.6
Time: 13.1s
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 r3143022 = 1.0;
        double r3143023 = 2.0;
        double r3143024 = r3143022 / r3143023;
        double r3143025 = x;
        double r3143026 = r3143022 + r3143025;
        double r3143027 = r3143022 - r3143025;
        double r3143028 = r3143026 / r3143027;
        double r3143029 = log(r3143028);
        double r3143030 = r3143024 * r3143029;
        return r3143030;
}
↓
double f(double x) {
        double r3143031 = 1.0;
        double r3143032 = log(r3143031);
        double r3143033 = x;
        double r3143034 = r3143033 * r3143033;
        double r3143035 = r3143033 + r3143034;
        double r3143036 = r3143031 * r3143031;
        double r3143037 = r3143034 / r3143036;
        double r3143038 = r3143035 - r3143037;
        double r3143039 = 2.0;
        double r3143040 = r3143038 * r3143039;
        double r3143041 = r3143032 + r3143040;
        double r3143042 = r3143031 / r3143039;
        double r3143043 = r3143041 * r3143042;
        return r3143043;
}

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 2019192 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))