Average Error: 58.5 → 0.2
Time: 22.0s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \left(x \cdot 2 + \left(\left(x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{2}{5} + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right)\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \left(x \cdot 2 + \left(\left(x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{2}{5} + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right)\right)
double f(double x) {
        double r2892190 = 1.0;
        double r2892191 = 2.0;
        double r2892192 = r2892190 / r2892191;
        double r2892193 = x;
        double r2892194 = r2892190 + r2892193;
        double r2892195 = r2892190 - r2892193;
        double r2892196 = r2892194 / r2892195;
        double r2892197 = log(r2892196);
        double r2892198 = r2892192 * r2892197;
        return r2892198;
}
↓
double f(double x) {
        double r2892199 = 0.5;
        double r2892200 = x;
        double r2892201 = 2.0;
        double r2892202 = r2892200 * r2892201;
        double r2892203 = r2892200 * r2892200;
        double r2892204 = r2892203 * r2892203;
        double r2892205 = r2892200 * r2892204;
        double r2892206 = 0.4;
        double r2892207 = r2892205 * r2892206;
        double r2892208 = 0.6666666666666666;
        double r2892209 = r2892200 * r2892208;
        double r2892210 = r2892203 * r2892209;
        double r2892211 = r2892207 + r2892210;
        double r2892212 = r2892202 + r2892211;
        double r2892213 = r2892199 * r2892212;
        return r2892213;
}

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. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{\left(2 \cdot x + \left(\frac{2}{3} \cdot {x}^{3} + \frac{2}{5} \cdot {x}^{5}\right)\right)} \cdot \frac{1}{2}\]

  4. Simplified0.2

    \[\leadsto \color{blue}{\left(\frac{2}{5} \cdot {x}^{5} + x \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{3} + 2\right)\right)} \cdot \frac{1}{2}\]
  5. Using strategy rm

  6. Applied distribute-lft-in0.2

    \[\leadsto \left(\frac{2}{5} \cdot {x}^{5} + \color{blue}{\left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{3}\right) + x \cdot 2\right)}\right) \cdot \frac{1}{2}\]

  7. Applied associate-+r+0.2

    \[\leadsto \color{blue}{\left(\left(\frac{2}{5} \cdot {x}^{5} + x \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{3}\right)\right) + x \cdot 2\right)} \cdot \frac{1}{2}\]

  8. Simplified0.2

    \[\leadsto \left(\color{blue}{\left(\left(x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{2}{5} + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right)} + x \cdot 2\right) \cdot \frac{1}{2}\]

  9. Final simplification0.2

    \[\leadsto \frac{1}{2} \cdot \left(x \cdot 2 + \left(\left(x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{2}{5} + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right)\right)\]

Reproduce

herbie shell --seed 2019130 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))