Average Error: 58.5 → 0.2
Time: 8.4s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \left(2 \cdot x + \left(0.66666666666666663 \cdot {x}^{3} + 0.40000000000000002 \cdot {x}^{5}\right)\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \left(2 \cdot x + \left(0.66666666666666663 \cdot {x}^{3} + 0.40000000000000002 \cdot {x}^{5}\right)\right)
double f(double x) {
        double r59173 = 1.0;
        double r59174 = 2.0;
        double r59175 = r59173 / r59174;
        double r59176 = x;
        double r59177 = r59173 + r59176;
        double r59178 = r59173 - r59176;
        double r59179 = r59177 / r59178;
        double r59180 = log(r59179);
        double r59181 = r59175 * r59180;
        return r59181;
}
↓
double f(double x) {
        double r59182 = 1.0;
        double r59183 = 2.0;
        double r59184 = r59182 / r59183;
        double r59185 = x;
        double r59186 = r59183 * r59185;
        double r59187 = 0.6666666666666666;
        double r59188 = 3.0;
        double r59189 = pow(r59185, r59188);
        double r59190 = r59187 * r59189;
        double r59191 = 0.4;
        double r59192 = 5.0;
        double r59193 = pow(r59185, r59192);
        double r59194 = r59191 * r59193;
        double r59195 = r59190 + r59194;
        double r59196 = r59186 + r59195;
        double r59197 = r59184 * r59196;
        return r59197;
}

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. Using strategy rm

  3. Applied log-div58.5

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

  4. Taylor expanded around 0 0.2

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

  5. Simplified0.2

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

  6. Taylor expanded around 0 0.2

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(0.66666666666666663 \cdot {x}^{3} + \left(0.40000000000000002 \cdot {x}^{5} + 2 \cdot x\right)\right)}\]

  7. Simplified0.2

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(2 \cdot x + \left(0.66666666666666663 \cdot {x}^{3} + 0.40000000000000002 \cdot {x}^{5}\right)\right)}\]

  8. Final simplification0.2

    \[\leadsto \frac{1}{2} \cdot \left(2 \cdot x + \left(0.66666666666666663 \cdot {x}^{3} + 0.40000000000000002 \cdot {x}^{5}\right)\right)\]

Reproduce

herbie shell --seed 2020046 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))