Average Error: 58.6 → 0.0
Time: 16.0s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\left(\mathsf{log1p}\left(x\right) + \left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(\left(-x\right) \cdot x\right)\right)\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\left(\mathsf{log1p}\left(x\right) + \left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(\left(-x\right) \cdot x\right)\right)\right) \cdot \frac{1}{2}
double f(double x) {
        double r2049590 = 1.0;
        double r2049591 = 2.0;
        double r2049592 = r2049590 / r2049591;
        double r2049593 = x;
        double r2049594 = r2049590 + r2049593;
        double r2049595 = r2049590 - r2049593;
        double r2049596 = r2049594 / r2049595;
        double r2049597 = log(r2049596);
        double r2049598 = r2049592 * r2049597;
        return r2049598;
}
↓
double f(double x) {
        double r2049599 = x;
        double r2049600 = log1p(r2049599);
        double r2049601 = -r2049599;
        double r2049602 = r2049601 * r2049599;
        double r2049603 = log1p(r2049602);
        double r2049604 = r2049600 - r2049603;
        double r2049605 = r2049600 + r2049604;
        double r2049606 = 0.5;
        double r2049607 = r2049605 * r2049606;
        return r2049607;
}

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

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

  4. Applied flip--58.6

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

  5. Applied associate-/r/58.6

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

  6. Applied log-prod58.6

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

  7. Simplified58.6

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

  8. Simplified50.5

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

  10. Applied log-div50.5

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

  11. Simplified0.5

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

  13. Applied log1p-expm1-u0.5

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

  14. Simplified0.0

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

  15. Final simplification0.0

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

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))