Average Error: 1.0 → 0.1
Time: 11.8s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(\log \left(e^{\sqrt[3]{{\left(\cos \left(\mathsf{fma}\left(0.3333333333333333148296162562473909929395, \cos^{-1} \left(-\frac{g}{h}\right), 0.6666666666666666296592325124947819858789 \cdot \pi\right)\right)\right)}^{2}}}\right) \cdot \sqrt[3]{\cos \left(\mathsf{fma}\left(\pi, \frac{2}{3}, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)}\right)\]
2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)
↓
2 \cdot \left(\log \left(e^{\sqrt[3]{{\left(\cos \left(\mathsf{fma}\left(0.3333333333333333148296162562473909929395, \cos^{-1} \left(-\frac{g}{h}\right), 0.6666666666666666296592325124947819858789 \cdot \pi\right)\right)\right)}^{2}}}\right) \cdot \sqrt[3]{\cos \left(\mathsf{fma}\left(\pi, \frac{2}{3}, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)}\right)
double f(double g, double h) {
        double r154729 = 2.0;
        double r154730 = atan2(1.0, 0.0);
        double r154731 = r154729 * r154730;
        double r154732 = 3.0;
        double r154733 = r154731 / r154732;
        double r154734 = g;
        double r154735 = -r154734;
        double r154736 = h;
        double r154737 = r154735 / r154736;
        double r154738 = acos(r154737);
        double r154739 = r154738 / r154732;
        double r154740 = r154733 + r154739;
        double r154741 = cos(r154740);
        double r154742 = r154729 * r154741;
        return r154742;
}
↓
double f(double g, double h) {
        double r154743 = 2.0;
        double r154744 = 0.3333333333333333;
        double r154745 = g;
        double r154746 = h;
        double r154747 = r154745 / r154746;
        double r154748 = -r154747;
        double r154749 = acos(r154748);
        double r154750 = 0.6666666666666666;
        double r154751 = atan2(1.0, 0.0);
        double r154752 = r154750 * r154751;
        double r154753 = fma(r154744, r154749, r154752);
        double r154754 = cos(r154753);
        double r154755 = 2.0;
        double r154756 = pow(r154754, r154755);
        double r154757 = cbrt(r154756);
        double r154758 = exp(r154757);
        double r154759 = log(r154758);
        double r154760 = 3.0;
        double r154761 = r154743 / r154760;
        double r154762 = -r154745;
        double r154763 = r154762 / r154746;
        double r154764 = acos(r154763);
        double r154765 = r154764 / r154760;
        double r154766 = fma(r154751, r154761, r154765);
        double r154767 = cos(r154766);
        double r154768 = cbrt(r154767);
        double r154769 = r154759 * r154768;
        double r154770 = r154743 * r154769;
        return r154770;
}

Error

Bits error versus g

Bits error versus h

Derivation

  1. Initial program 1.0

    \[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]

  2. Simplified1.0

    \[\leadsto \color{blue}{2 \cdot \cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt1.0

    \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{\color{blue}{\sqrt{3} \cdot \sqrt{3}}}\right)\right)\]

  5. Applied *-un-lft-identity1.0

    \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\color{blue}{1 \cdot \cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3} \cdot \sqrt{3}}\right)\right)\]

  6. Applied times-frac1.0

    \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \color{blue}{\frac{1}{\sqrt{3}} \cdot \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{\sqrt{3}}}\right)\right)\]
  7. Using strategy rm

  8. Applied add-cube-cbrt1.0

    \[\leadsto 2 \cdot \color{blue}{\left(\left(\sqrt[3]{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{1}{\sqrt{3}} \cdot \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{\sqrt{3}}\right)\right)} \cdot \sqrt[3]{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{1}{\sqrt{3}} \cdot \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{\sqrt{3}}\right)\right)}\right) \cdot \sqrt[3]{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{1}{\sqrt{3}} \cdot \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{\sqrt{3}}\right)\right)}\right)}\]

  9. Simplified1.0

    \[\leadsto 2 \cdot \left(\color{blue}{\left(\sqrt[3]{\cos \left(\mathsf{fma}\left(\pi, \frac{2}{3}, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)} \cdot \sqrt[3]{\cos \left(\mathsf{fma}\left(\pi, \frac{2}{3}, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)}\right)} \cdot \sqrt[3]{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{1}{\sqrt{3}} \cdot \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{\sqrt{3}}\right)\right)}\right)\]

  10. Simplified1.0

    \[\leadsto 2 \cdot \left(\left(\sqrt[3]{\cos \left(\mathsf{fma}\left(\pi, \frac{2}{3}, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)} \cdot \sqrt[3]{\cos \left(\mathsf{fma}\left(\pi, \frac{2}{3}, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)}\right) \cdot \color{blue}{\sqrt[3]{\cos \left(\mathsf{fma}\left(\pi, \frac{2}{3}, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)}}\right)\]

  11. Taylor expanded around 0 1.0

    \[\leadsto 2 \cdot \left(\color{blue}{{\left({\left(\cos \left(0.3333333333333333148296162562473909929395 \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right) + 0.6666666666666666296592325124947819858789 \cdot \pi\right)\right)}^{2}\right)}^{\frac{1}{3}}} \cdot \sqrt[3]{\cos \left(\mathsf{fma}\left(\pi, \frac{2}{3}, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)}\right)\]

  12. Simplified0.1

    \[\leadsto 2 \cdot \left(\color{blue}{\sqrt[3]{{\left(\cos \left(\mathsf{fma}\left(0.3333333333333333148296162562473909929395, \cos^{-1} \left(-\frac{g}{h}\right), 0.6666666666666666296592325124947819858789 \cdot \pi\right)\right)\right)}^{2}}} \cdot \sqrt[3]{\cos \left(\mathsf{fma}\left(\pi, \frac{2}{3}, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)}\right)\]
  13. Using strategy rm

  14. Applied add-log-exp0.1

    \[\leadsto 2 \cdot \left(\color{blue}{\log \left(e^{\sqrt[3]{{\left(\cos \left(\mathsf{fma}\left(0.3333333333333333148296162562473909929395, \cos^{-1} \left(-\frac{g}{h}\right), 0.6666666666666666296592325124947819858789 \cdot \pi\right)\right)\right)}^{2}}}\right)} \cdot \sqrt[3]{\cos \left(\mathsf{fma}\left(\pi, \frac{2}{3}, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)}\right)\]

  15. Final simplification0.1

    \[\leadsto 2 \cdot \left(\log \left(e^{\sqrt[3]{{\left(\cos \left(\mathsf{fma}\left(0.3333333333333333148296162562473909929395, \cos^{-1} \left(-\frac{g}{h}\right), 0.6666666666666666296592325124947819858789 \cdot \pi\right)\right)\right)}^{2}}}\right) \cdot \sqrt[3]{\cos \left(\mathsf{fma}\left(\pi, \frac{2}{3}, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)}\right)\]

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))