Average Error: 1.0 → 0.0
Time: 17.4s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(\cos \left(\frac{\sqrt{\cos^{-1} \left(-\frac{g}{h}\right)}}{\sqrt{3}} \cdot \frac{\sqrt{\cos^{-1} \left(-\frac{g}{h}\right)}}{\sqrt{3}}\right) \cdot \cos \left(\frac{2}{3} \cdot \pi\right) - \left(\sqrt{\sin \left(\frac{2}{3} \cdot \pi\right)} \cdot \sqrt{\sin \left(\frac{2}{3} \cdot \pi\right)}\right) \cdot \sin \left(\frac{\sqrt{\cos^{-1} \left(-\frac{g}{h}\right)}}{\sqrt{3}} \cdot \frac{\sqrt{\cos^{-1} \left(-\frac{g}{h}\right)}}{\sqrt{3}}\right)\right)\]
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{\sqrt{\cos^{-1} \left(-\frac{g}{h}\right)}}{\sqrt{3}} \cdot \frac{\sqrt{\cos^{-1} \left(-\frac{g}{h}\right)}}{\sqrt{3}}\right) \cdot \cos \left(\frac{2}{3} \cdot \pi\right) - \left(\sqrt{\sin \left(\frac{2}{3} \cdot \pi\right)} \cdot \sqrt{\sin \left(\frac{2}{3} \cdot \pi\right)}\right) \cdot \sin \left(\frac{\sqrt{\cos^{-1} \left(-\frac{g}{h}\right)}}{\sqrt{3}} \cdot \frac{\sqrt{\cos^{-1} \left(-\frac{g}{h}\right)}}{\sqrt{3}}\right)\right)
double f(double g, double h) {
        double r5278701 = 2.0;
        double r5278702 = atan2(1.0, 0.0);
        double r5278703 = r5278701 * r5278702;
        double r5278704 = 3.0;
        double r5278705 = r5278703 / r5278704;
        double r5278706 = g;
        double r5278707 = -r5278706;
        double r5278708 = h;
        double r5278709 = r5278707 / r5278708;
        double r5278710 = acos(r5278709);
        double r5278711 = r5278710 / r5278704;
        double r5278712 = r5278705 + r5278711;
        double r5278713 = cos(r5278712);
        double r5278714 = r5278701 * r5278713;
        return r5278714;
}
↓
double f(double g, double h) {
        double r5278715 = 2.0;
        double r5278716 = g;
        double r5278717 = h;
        double r5278718 = r5278716 / r5278717;
        double r5278719 = -r5278718;
        double r5278720 = acos(r5278719);
        double r5278721 = sqrt(r5278720);
        double r5278722 = 3.0;
        double r5278723 = sqrt(r5278722);
        double r5278724 = r5278721 / r5278723;
        double r5278725 = r5278724 * r5278724;
        double r5278726 = cos(r5278725);
        double r5278727 = r5278715 / r5278722;
        double r5278728 = atan2(1.0, 0.0);
        double r5278729 = r5278727 * r5278728;
        double r5278730 = cos(r5278729);
        double r5278731 = r5278726 * r5278730;
        double r5278732 = sin(r5278729);
        double r5278733 = sqrt(r5278732);
        double r5278734 = r5278733 * r5278733;
        double r5278735 = sin(r5278725);
        double r5278736 = r5278734 * r5278735;
        double r5278737 = r5278731 - r5278736;
        double r5278738 = r5278715 * r5278737;
        return r5278738;
}

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

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

  4. Applied add-sqr-sqrt1.0

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

  5. Applied add-sqr-sqrt1.0

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

  6. Applied times-frac1.0

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

  8. Applied fma-udef1.0

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

  9. Applied cos-sum1.0

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

  11. Applied add-sqr-sqrt0.0

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

  12. Final simplification0.0

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))