Average Error: 1.5 → 0.1
Time: 16.6s
Precision: 64
\[\frac{\cos^{-1} \left(\frac{x - 1}{2}\right)}{2 \cdot \sin \left(\cos^{-1} \left(\frac{x - 1}{2}\right)\right)}\]
\[\frac{\cos^{-1} \left(\frac{x - 1}{2}\right)}{\sqrt{1 - \left(\frac{x - 1}{\sqrt[3]{2}} \cdot \frac{x - 1}{\sqrt[3]{2}}\right) \cdot \left(\frac{1}{\sqrt[3]{2} \cdot \sqrt[3]{2}} \cdot \frac{1}{\sqrt[3]{2} \cdot \sqrt[3]{2}}\right)} \cdot 2}\]
\frac{\cos^{-1} \left(\frac{x - 1}{2}\right)}{2 \cdot \sin \left(\cos^{-1} \left(\frac{x - 1}{2}\right)\right)}
\frac{\cos^{-1} \left(\frac{x - 1}{2}\right)}{\sqrt{1 - \left(\frac{x - 1}{\sqrt[3]{2}} \cdot \frac{x - 1}{\sqrt[3]{2}}\right) \cdot \left(\frac{1}{\sqrt[3]{2} \cdot \sqrt[3]{2}} \cdot \frac{1}{\sqrt[3]{2} \cdot \sqrt[3]{2}}\right)} \cdot 2}
double f(double x) {
        double r14039730 = x;
        double r14039731 = 1.0;
        double r14039732 = r14039730 - r14039731;
        double r14039733 = 2.0;
        double r14039734 = r14039732 / r14039733;
        double r14039735 = acos(r14039734);
        double r14039736 = sin(r14039735);
        double r14039737 = r14039733 * r14039736;
        double r14039738 = r14039735 / r14039737;
        return r14039738;
}

double f(double x) {
        double r14039739 = x;
        double r14039740 = 1.0;
        double r14039741 = r14039739 - r14039740;
        double r14039742 = 2.0;
        double r14039743 = r14039741 / r14039742;
        double r14039744 = acos(r14039743);
        double r14039745 = cbrt(r14039742);
        double r14039746 = r14039741 / r14039745;
        double r14039747 = r14039746 * r14039746;
        double r14039748 = r14039745 * r14039745;
        double r14039749 = r14039740 / r14039748;
        double r14039750 = r14039749 * r14039749;
        double r14039751 = r14039747 * r14039750;
        double r14039752 = r14039740 - r14039751;
        double r14039753 = sqrt(r14039752);
        double r14039754 = r14039753 * r14039742;
        double r14039755 = r14039744 / r14039754;
        return r14039755;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 1.5

    \[\frac{\cos^{-1} \left(\frac{x - 1}{2}\right)}{2 \cdot \sin \left(\cos^{-1} \left(\frac{x - 1}{2}\right)\right)}\]
  2. Using strategy rm
  3. Applied sin-acos1.0

    \[\leadsto \frac{\cos^{-1} \left(\frac{x - 1}{2}\right)}{2 \cdot \color{blue}{\sqrt{1 - \frac{x - 1}{2} \cdot \frac{x - 1}{2}}}}\]
  4. Using strategy rm
  5. Applied add-cube-cbrt1.0

    \[\leadsto \frac{\cos^{-1} \left(\frac{x - 1}{2}\right)}{2 \cdot \sqrt{1 - \frac{x - 1}{2} \cdot \frac{x - 1}{\color{blue}{\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{2}}}}}\]
  6. Applied *-un-lft-identity1.0

    \[\leadsto \frac{\cos^{-1} \left(\frac{x - 1}{2}\right)}{2 \cdot \sqrt{1 - \frac{x - 1}{2} \cdot \frac{x - \color{blue}{1 \cdot 1}}{\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{2}}}}\]
  7. Applied *-un-lft-identity1.0

    \[\leadsto \frac{\cos^{-1} \left(\frac{x - 1}{2}\right)}{2 \cdot \sqrt{1 - \frac{x - 1}{2} \cdot \frac{\color{blue}{1 \cdot x} - 1 \cdot 1}{\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{2}}}}\]
  8. Applied distribute-lft-out--1.0

    \[\leadsto \frac{\cos^{-1} \left(\frac{x - 1}{2}\right)}{2 \cdot \sqrt{1 - \frac{x - 1}{2} \cdot \frac{\color{blue}{1 \cdot \left(x - 1\right)}}{\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{2}}}}\]
  9. Applied times-frac1.0

    \[\leadsto \frac{\cos^{-1} \left(\frac{x - 1}{2}\right)}{2 \cdot \sqrt{1 - \frac{x - 1}{2} \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{2} \cdot \sqrt[3]{2}} \cdot \frac{x - 1}{\sqrt[3]{2}}\right)}}}\]
  10. Applied add-cube-cbrt0.0

    \[\leadsto \frac{\cos^{-1} \left(\frac{x - 1}{2}\right)}{2 \cdot \sqrt{1 - \frac{x - 1}{\color{blue}{\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{2}}} \cdot \left(\frac{1}{\sqrt[3]{2} \cdot \sqrt[3]{2}} \cdot \frac{x - 1}{\sqrt[3]{2}}\right)}}\]
  11. Applied *-un-lft-identity0.0

    \[\leadsto \frac{\cos^{-1} \left(\frac{x - 1}{2}\right)}{2 \cdot \sqrt{1 - \frac{x - \color{blue}{1 \cdot 1}}{\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{2}} \cdot \left(\frac{1}{\sqrt[3]{2} \cdot \sqrt[3]{2}} \cdot \frac{x - 1}{\sqrt[3]{2}}\right)}}\]
  12. Applied *-un-lft-identity0.0

    \[\leadsto \frac{\cos^{-1} \left(\frac{x - 1}{2}\right)}{2 \cdot \sqrt{1 - \frac{\color{blue}{1 \cdot x} - 1 \cdot 1}{\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{2}} \cdot \left(\frac{1}{\sqrt[3]{2} \cdot \sqrt[3]{2}} \cdot \frac{x - 1}{\sqrt[3]{2}}\right)}}\]
  13. Applied distribute-lft-out--0.0

    \[\leadsto \frac{\cos^{-1} \left(\frac{x - 1}{2}\right)}{2 \cdot \sqrt{1 - \frac{\color{blue}{1 \cdot \left(x - 1\right)}}{\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{2}} \cdot \left(\frac{1}{\sqrt[3]{2} \cdot \sqrt[3]{2}} \cdot \frac{x - 1}{\sqrt[3]{2}}\right)}}\]
  14. Applied times-frac0.1

    \[\leadsto \frac{\cos^{-1} \left(\frac{x - 1}{2}\right)}{2 \cdot \sqrt{1 - \color{blue}{\left(\frac{1}{\sqrt[3]{2} \cdot \sqrt[3]{2}} \cdot \frac{x - 1}{\sqrt[3]{2}}\right)} \cdot \left(\frac{1}{\sqrt[3]{2} \cdot \sqrt[3]{2}} \cdot \frac{x - 1}{\sqrt[3]{2}}\right)}}\]
  15. Applied swap-sqr0.1

    \[\leadsto \frac{\cos^{-1} \left(\frac{x - 1}{2}\right)}{2 \cdot \sqrt{1 - \color{blue}{\left(\frac{1}{\sqrt[3]{2} \cdot \sqrt[3]{2}} \cdot \frac{1}{\sqrt[3]{2} \cdot \sqrt[3]{2}}\right) \cdot \left(\frac{x - 1}{\sqrt[3]{2}} \cdot \frac{x - 1}{\sqrt[3]{2}}\right)}}}\]
  16. Final simplification0.1

    \[\leadsto \frac{\cos^{-1} \left(\frac{x - 1}{2}\right)}{\sqrt{1 - \left(\frac{x - 1}{\sqrt[3]{2}} \cdot \frac{x - 1}{\sqrt[3]{2}}\right) \cdot \left(\frac{1}{\sqrt[3]{2} \cdot \sqrt[3]{2}} \cdot \frac{1}{\sqrt[3]{2} \cdot \sqrt[3]{2}}\right)} \cdot 2}\]

Reproduce

herbie shell --seed 1 
(FPCore (x)
  :name "acos((x-1)/2)/(2*sin(acos((x-1)/2)))"
  (/ (acos (/ (- x 1) 2)) (* 2 (sin (acos (/ (- x 1) 2))))))