Average Error: 58.5 → 0.3
Time: 18.0s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \left(\left(\left(2 \cdot \left({x}^{3} + \left(x + \frac{{x}^{2}}{{1}^{2}}\right)\right) - 4 \cdot \frac{{x}^{3}}{{1}^{2}}\right) - 2 \cdot {x}^{2}\right) + 2.666666666666666518636930049979127943516 \cdot {\left(\frac{x}{1}\right)}^{3}\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \left(\left(\left(2 \cdot \left({x}^{3} + \left(x + \frac{{x}^{2}}{{1}^{2}}\right)\right) - 4 \cdot \frac{{x}^{3}}{{1}^{2}}\right) - 2 \cdot {x}^{2}\right) + 2.666666666666666518636930049979127943516 \cdot {\left(\frac{x}{1}\right)}^{3}\right)
double f(double x) {
        double r77875 = 1.0;
        double r77876 = 2.0;
        double r77877 = r77875 / r77876;
        double r77878 = x;
        double r77879 = r77875 + r77878;
        double r77880 = r77875 - r77878;
        double r77881 = r77879 / r77880;
        double r77882 = log(r77881);
        double r77883 = r77877 * r77882;
        return r77883;
}
↓
double f(double x) {
        double r77884 = 1.0;
        double r77885 = 2.0;
        double r77886 = r77884 / r77885;
        double r77887 = x;
        double r77888 = 3.0;
        double r77889 = pow(r77887, r77888);
        double r77890 = 2.0;
        double r77891 = pow(r77887, r77890);
        double r77892 = pow(r77884, r77890);
        double r77893 = r77891 / r77892;
        double r77894 = r77887 + r77893;
        double r77895 = r77889 + r77894;
        double r77896 = r77885 * r77895;
        double r77897 = 4.0;
        double r77898 = r77889 / r77892;
        double r77899 = r77897 * r77898;
        double r77900 = r77896 - r77899;
        double r77901 = r77885 * r77891;
        double r77902 = r77900 - r77901;
        double r77903 = 2.6666666666666665;
        double r77904 = r77887 / r77884;
        double r77905 = pow(r77904, r77888);
        double r77906 = r77903 * r77905;
        double r77907 = r77902 + r77906;
        double r77908 = r77886 * r77907;
        return r77908;
}

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. Using strategy rm

  3. Applied div-inv58.5

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

  5. Applied flip3-+58.6

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

  6. Applied frac-times58.6

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

  7. Applied log-div58.6

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

  8. Simplified58.6

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

  9. Simplified58.6

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

  10. Taylor expanded around 0 0.3

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

  11. Simplified0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019325 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))