Average Error: 58.5 → 0.0
Time: 19.0s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2}
double f(double x) {
        double r3558860 = 1.0;
        double r3558861 = 2.0;
        double r3558862 = r3558860 / r3558861;
        double r3558863 = x;
        double r3558864 = r3558860 + r3558863;
        double r3558865 = r3558860 - r3558863;
        double r3558866 = r3558864 / r3558865;
        double r3558867 = log(r3558866);
        double r3558868 = r3558862 * r3558867;
        return r3558868;
}
↓
double f(double x) {
        double r3558869 = x;
        double r3558870 = log1p(r3558869);
        double r3558871 = -r3558869;
        double r3558872 = log1p(r3558871);
        double r3558873 = r3558870 - r3558872;
        double r3558874 = 0.5;
        double r3558875 = r3558873 * r3558874;
        return r3558875;
}

Error

Bits error versus x

Try it out

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. Using strategy rm

  4. Applied log-div58.5

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

  5. Simplified50.4

    \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\mathsf{log1p}\left(x\right)} - \log \left(1 - x\right)\right)\]
  6. Using strategy rm

  7. Applied log1p-expm1-u50.4

    \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(x\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(1 - x\right)\right)\right)}\right)\]

  8. Simplified0.0

    \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(\color{blue}{-x}\right)\right)\]

  9. Final simplification0.0

    \[\leadsto \left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2}\]

Reproduce

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