Average Error: 58.5 → 0.7
Time: 10.3s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \left(2 \cdot \left({x}^{2} + x\right) + \left(\log 1 - 2 \cdot \frac{{x}^{2}}{{1}^{2}}\right)\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \left(2 \cdot \left({x}^{2} + x\right) + \left(\log 1 - 2 \cdot \frac{{x}^{2}}{{1}^{2}}\right)\right)
double f(double x) {
        double r104187 = 1.0;
        double r104188 = 2.0;
        double r104189 = r104187 / r104188;
        double r104190 = x;
        double r104191 = r104187 + r104190;
        double r104192 = r104187 - r104190;
        double r104193 = r104191 / r104192;
        double r104194 = log(r104193);
        double r104195 = r104189 * r104194;
        return r104195;
}
↓
double f(double x) {
        double r104196 = 1.0;
        double r104197 = 2.0;
        double r104198 = r104196 / r104197;
        double r104199 = x;
        double r104200 = 2.0;
        double r104201 = pow(r104199, r104200);
        double r104202 = r104201 + r104199;
        double r104203 = r104197 * r104202;
        double r104204 = log(r104196);
        double r104205 = pow(r104196, r104200);
        double r104206 = r104201 / r104205;
        double r104207 = r104197 * r104206;
        double r104208 = r104204 - r104207;
        double r104209 = r104203 + r104208;
        double r104210 = r104198 * r104209;
        return r104210;
}

Error

Bits error versus x

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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. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019322 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))