Average Error: 1.0 → 0.0
Time: 12.9s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(\cos \left(\frac{\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{\sqrt{3}}}{\sqrt{3}}\right) \cdot \cos \left(\frac{2}{3} \cdot \pi\right) - \sin \left(\frac{\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{\sqrt{3}}}{\sqrt{3}}\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{2}{3} \cdot \pi\right)\right)\right)\right)\]
2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)
↓
2 \cdot \left(\cos \left(\frac{\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{\sqrt{3}}}{\sqrt{3}}\right) \cdot \cos \left(\frac{2}{3} \cdot \pi\right) - \sin \left(\frac{\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{\sqrt{3}}}{\sqrt{3}}\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{2}{3} \cdot \pi\right)\right)\right)\right)
double f(double g, double h) {
        double r88693 = 2.0;
        double r88694 = atan2(1.0, 0.0);
        double r88695 = r88693 * r88694;
        double r88696 = 3.0;
        double r88697 = r88695 / r88696;
        double r88698 = g;
        double r88699 = -r88698;
        double r88700 = h;
        double r88701 = r88699 / r88700;
        double r88702 = acos(r88701);
        double r88703 = r88702 / r88696;
        double r88704 = r88697 + r88703;
        double r88705 = cos(r88704);
        double r88706 = r88693 * r88705;
        return r88706;
}
↓
double f(double g, double h) {
        double r88707 = 2.0;
        double r88708 = g;
        double r88709 = -r88708;
        double r88710 = h;
        double r88711 = r88709 / r88710;
        double r88712 = acos(r88711);
        double r88713 = 3.0;
        double r88714 = sqrt(r88713);
        double r88715 = r88712 / r88714;
        double r88716 = r88715 / r88714;
        double r88717 = cos(r88716);
        double r88718 = r88707 / r88713;
        double r88719 = atan2(1.0, 0.0);
        double r88720 = r88718 * r88719;
        double r88721 = cos(r88720);
        double r88722 = r88717 * r88721;
        double r88723 = sin(r88716);
        double r88724 = sin(r88720);
        double r88725 = expm1(r88724);
        double r88726 = log1p(r88725);
        double r88727 = r88723 * r88726;
        double r88728 = r88722 - r88727;
        double r88729 = r88707 * r88728;
        return r88729;
}

Error

Bits error versus g

Bits error versus h

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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 fma-udef1.0

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

  9. Applied cos-sum1.0

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

  10. Simplified1.0

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

  11. Simplified1.0

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

  13. Applied log1p-expm1-u0.0

    \[\leadsto 2 \cdot \left(\cos \left(\frac{\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{\sqrt{3}}}{\sqrt{3}}\right) \cdot \cos \left(\frac{2}{3} \cdot \pi\right) - \sin \left(\frac{\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{\sqrt{3}}}{\sqrt{3}}\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{2}{3} \cdot \pi\right)\right)\right)}\right)\]

  14. Final simplification0.0

    \[\leadsto 2 \cdot \left(\cos \left(\frac{\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{\sqrt{3}}}{\sqrt{3}}\right) \cdot \cos \left(\frac{2}{3} \cdot \pi\right) - \sin \left(\frac{\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{\sqrt{3}}}{\sqrt{3}}\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{2}{3} \cdot \pi\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2019305 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))