Average Error: 58.5 → 0.2
Time: 17.2s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\left({x}^{5} \cdot \frac{2}{5} + \left(x \cdot 2 + \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right)\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\left({x}^{5} \cdot \frac{2}{5} + \left(x \cdot 2 + \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right)\right) \cdot \frac{1}{2}
double f(double x) {
        double r3038318 = 1.0;
        double r3038319 = 2.0;
        double r3038320 = r3038318 / r3038319;
        double r3038321 = x;
        double r3038322 = r3038318 + r3038321;
        double r3038323 = r3038318 - r3038321;
        double r3038324 = r3038322 / r3038323;
        double r3038325 = log(r3038324);
        double r3038326 = r3038320 * r3038325;
        return r3038326;
}
↓
double f(double x) {
        double r3038327 = x;
        double r3038328 = 5.0;
        double r3038329 = pow(r3038327, r3038328);
        double r3038330 = 0.4;
        double r3038331 = r3038329 * r3038330;
        double r3038332 = 2.0;
        double r3038333 = r3038327 * r3038332;
        double r3038334 = 0.6666666666666666;
        double r3038335 = r3038327 * r3038327;
        double r3038336 = r3038334 * r3038335;
        double r3038337 = r3038336 * r3038327;
        double r3038338 = r3038333 + r3038337;
        double r3038339 = r3038331 + r3038338;
        double r3038340 = 0.5;
        double r3038341 = r3038339 * r3038340;
        return r3038341;
}

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

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

  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

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

  6. Applied +-commutative0.2

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

  7. Applied distribute-rgt-in0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019158 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))