Average Error: 58.5 → 0.0
Time: 15.1s
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 r1343350 = 1.0;
        double r1343351 = 2.0;
        double r1343352 = r1343350 / r1343351;
        double r1343353 = x;
        double r1343354 = r1343350 + r1343353;
        double r1343355 = r1343350 - r1343353;
        double r1343356 = r1343354 / r1343355;
        double r1343357 = log(r1343356);
        double r1343358 = r1343352 * r1343357;
        return r1343358;
}
↓
double f(double x) {
        double r1343359 = x;
        double r1343360 = log1p(r1343359);
        double r1343361 = -r1343359;
        double r1343362 = log1p(r1343361);
        double r1343363 = r1343360 - r1343362;
        double r1343364 = 0.5;
        double r1343365 = r1343363 * r1343364;
        return r1343365;
}

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

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

  4. Applied log-div58.5

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

  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 log1p-expm1-u50.5

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

  8. Simplified0.0

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

  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 2019129 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))