Average Error: 58.6 → 0.3
Time: 12.8s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\left(\frac{2}{5} \cdot {x}^{5} + x \cdot \left(\left(\frac{2}{3} \cdot x\right) \cdot x + 2\right)\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\left(\frac{2}{5} \cdot {x}^{5} + x \cdot \left(\left(\frac{2}{3} \cdot x\right) \cdot x + 2\right)\right) \cdot \frac{1}{2}
double f(double x) {
        double r3526195 = 1.0;
        double r3526196 = 2.0;
        double r3526197 = r3526195 / r3526196;
        double r3526198 = x;
        double r3526199 = r3526195 + r3526198;
        double r3526200 = r3526195 - r3526198;
        double r3526201 = r3526199 / r3526200;
        double r3526202 = log(r3526201);
        double r3526203 = r3526197 * r3526202;
        return r3526203;
}
↓
double f(double x) {
        double r3526204 = 0.4;
        double r3526205 = x;
        double r3526206 = 5.0;
        double r3526207 = pow(r3526205, r3526206);
        double r3526208 = r3526204 * r3526207;
        double r3526209 = 0.6666666666666666;
        double r3526210 = r3526209 * r3526205;
        double r3526211 = r3526210 * r3526205;
        double r3526212 = 2.0;
        double r3526213 = r3526211 + r3526212;
        double r3526214 = r3526205 * r3526213;
        double r3526215 = r3526208 + r3526214;
        double r3526216 = 0.5;
        double r3526217 = r3526215 * r3526216;
        return r3526217;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.6

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

  2. Simplified58.6

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

  3. Taylor expanded around 0 0.3

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

  4. Simplified0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019168 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))