Average Error: 58.6 → 0.0
Time: 27.8s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(\left(-x\right)\right)\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(\left(-x\right)\right)\right) \cdot \frac{1}{2}
double f(double x) {
        double r12955024 = 1.0;
        double r12955025 = 2.0;
        double r12955026 = r12955024 / r12955025;
        double r12955027 = x;
        double r12955028 = r12955024 + r12955027;
        double r12955029 = r12955024 - r12955027;
        double r12955030 = r12955028 / r12955029;
        double r12955031 = log(r12955030);
        double r12955032 = r12955026 * r12955031;
        return r12955032;
}
↓
double f(double x) {
        double r12955033 = x;
        double r12955034 = log1p(r12955033);
        double r12955035 = -r12955033;
        double r12955036 = log1p(r12955035);
        double r12955037 = r12955034 - r12955036;
        double r12955038 = 0.5;
        double r12955039 = r12955037 * r12955038;
        return r12955039;
}

Error

Bits error versus x

Try it out

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. Simplified58.6

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

  4. Applied log-div58.6

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

  5. Simplified50.5

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

  7. Applied sub-neg50.5

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

  8. Applied log1p-def0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))