Average Error: 58.6 → 0.0
Time: 19.1s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\left(\mathsf{log1p}\left(x\right) + \left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x \cdot x\right)\right)\right) \cdot \frac{1}{2}\]
\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(-x \cdot x\right)\right)\right) \cdot \frac{1}{2}
double f(double x) {
        double r1284566 = 1.0;
        double r1284567 = 2.0;
        double r1284568 = r1284566 / r1284567;
        double r1284569 = x;
        double r1284570 = r1284566 + r1284569;
        double r1284571 = r1284566 - r1284569;
        double r1284572 = r1284570 / r1284571;
        double r1284573 = log(r1284572);
        double r1284574 = r1284568 * r1284573;
        return r1284574;
}
↓
double f(double x) {
        double r1284575 = x;
        double r1284576 = log1p(r1284575);
        double r1284577 = r1284575 * r1284575;
        double r1284578 = -r1284577;
        double r1284579 = log1p(r1284578);
        double r1284580 = r1284576 - r1284579;
        double r1284581 = r1284576 + r1284580;
        double r1284582 = 0.5;
        double r1284583 = r1284581 * r1284582;
        return r1284583;
}

Error

Bits error versus x

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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 sub-neg0.5

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

  14. Applied log1p-def0.0

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

  15. Final simplification0.0

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

Reproduce

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