Average Error: 58.5 → 0.2
Time: 21.4s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\left(\frac{2}{5} \cdot {x}^{5} + \left(x \cdot 2 + x \cdot \left(\left(\frac{2}{3} \cdot x\right) \cdot x\right)\right)\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\left(\frac{2}{5} \cdot {x}^{5} + \left(x \cdot 2 + x \cdot \left(\left(\frac{2}{3} \cdot x\right) \cdot x\right)\right)\right) \cdot \frac{1}{2}
double f(double x) {
        double r2412720 = 1.0;
        double r2412721 = 2.0;
        double r2412722 = r2412720 / r2412721;
        double r2412723 = x;
        double r2412724 = r2412720 + r2412723;
        double r2412725 = r2412720 - r2412723;
        double r2412726 = r2412724 / r2412725;
        double r2412727 = log(r2412726);
        double r2412728 = r2412722 * r2412727;
        return r2412728;
}
↓
double f(double x) {
        double r2412729 = 0.4;
        double r2412730 = x;
        double r2412731 = 5.0;
        double r2412732 = pow(r2412730, r2412731);
        double r2412733 = r2412729 * r2412732;
        double r2412734 = 2.0;
        double r2412735 = r2412730 * r2412734;
        double r2412736 = 0.6666666666666666;
        double r2412737 = r2412736 * r2412730;
        double r2412738 = r2412737 * r2412730;
        double r2412739 = r2412730 * r2412738;
        double r2412740 = r2412735 + r2412739;
        double r2412741 = r2412733 + r2412740;
        double r2412742 = 0.5;
        double r2412743 = r2412741 * r2412742;
        return r2412743;
}

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. Simplified58.5

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

  3. Taylor expanded around 0 0.2

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

  4. Simplified0.2

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

  6. Applied distribute-rgt-in0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019162 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))