Average Error: 58.6 → 0.3
Time: 11.2s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{\left(\left(\left(2 \cdot \left({x}^{2} + \left({x}^{3} + x\right)\right) + 2.666666666666666518636930049979127943516 \cdot \frac{{x}^{3}}{{1}^{3}}\right) - 2 \cdot \frac{{x}^{2}}{{1}^{2}}\right) - 4 \cdot \frac{{x}^{3}}{{1}^{2}}\right) \cdot 1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{\left(\left(\left(2 \cdot \left({x}^{2} + \left({x}^{3} + x\right)\right) + 2.666666666666666518636930049979127943516 \cdot \frac{{x}^{3}}{{1}^{3}}\right) - 2 \cdot \frac{{x}^{2}}{{1}^{2}}\right) - 4 \cdot \frac{{x}^{3}}{{1}^{2}}\right) \cdot 1}{2}
double f(double x) {
        double r73356 = 1.0;
        double r73357 = 2.0;
        double r73358 = r73356 / r73357;
        double r73359 = x;
        double r73360 = r73356 + r73359;
        double r73361 = r73356 - r73359;
        double r73362 = r73360 / r73361;
        double r73363 = log(r73362);
        double r73364 = r73358 * r73363;
        return r73364;
}
↓
double f(double x) {
        double r73365 = 2.0;
        double r73366 = x;
        double r73367 = 2.0;
        double r73368 = pow(r73366, r73367);
        double r73369 = 3.0;
        double r73370 = pow(r73366, r73369);
        double r73371 = r73370 + r73366;
        double r73372 = r73368 + r73371;
        double r73373 = r73365 * r73372;
        double r73374 = 2.6666666666666665;
        double r73375 = 1.0;
        double r73376 = pow(r73375, r73369);
        double r73377 = r73370 / r73376;
        double r73378 = r73374 * r73377;
        double r73379 = r73373 + r73378;
        double r73380 = pow(r73375, r73367);
        double r73381 = r73368 / r73380;
        double r73382 = r73365 * r73381;
        double r73383 = r73379 - r73382;
        double r73384 = 4.0;
        double r73385 = r73370 / r73380;
        double r73386 = r73384 * r73385;
        double r73387 = r73383 - r73386;
        double r73388 = r73387 * r73375;
        double r73389 = r73388 / r73365;
        return r73389;
}

Error

Bits error versus x

Try it out

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. Using strategy rm

  3. Applied flip3--58.6

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

  4. Applied associate-/r/58.6

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

  6. Applied associate-*l/58.6

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

  7. Applied log-div58.6

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

  8. Taylor expanded around 0 0.3

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(2 \cdot {x}^{2} + \left(2.666666666666666518636930049979127943516 \cdot \frac{{x}^{3}}{{1}^{3}} + \left(2 \cdot {x}^{3} + 2 \cdot x\right)\right)\right) - \left(4 \cdot \frac{{x}^{3}}{{1}^{2}} + 2 \cdot \frac{{x}^{2}}{{1}^{2}}\right)\right)}\]

  9. Simplified0.3

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(\left(\left(2 \cdot {x}^{2} + 2.666666666666666518636930049979127943516 \cdot \frac{{x}^{3}}{{1}^{3}}\right) + 2 \cdot \left({x}^{3} + x\right)\right) - 4 \cdot \frac{{x}^{3}}{{1}^{2}}\right) - 2 \cdot \frac{{x}^{2}}{{1}^{2}}\right)}\]

  10. Final simplification0.3

    \[\leadsto \frac{\left(\left(\left(2 \cdot \left({x}^{2} + \left({x}^{3} + x\right)\right) + 2.666666666666666518636930049979127943516 \cdot \frac{{x}^{3}}{{1}^{3}}\right) - 2 \cdot \frac{{x}^{2}}{{1}^{2}}\right) - 4 \cdot \frac{{x}^{3}}{{1}^{2}}\right) \cdot 1}{2}\]

Reproduce

herbie shell --seed 2019308 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))