Average Error: 1.0 → 0.0
Time: 24.1s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(\cos \left(\frac{1}{\sqrt{3}} \cdot \frac{\cos^{-1} \left(-\frac{g}{h}\right)}{\sqrt{3}}\right) \cdot \cos \left(\pi \cdot \frac{2}{3}\right) - \left(\sqrt{\sin \left(\pi \cdot \frac{2}{3}\right)} \cdot \sqrt{\sin \left(\pi \cdot \frac{2}{3}\right)}\right) \cdot \sin \left(\frac{1}{\sqrt{3}} \cdot \frac{\cos^{-1} \left(-\frac{g}{h}\right)}{\sqrt{3}}\right)\right)\]
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{1}{\sqrt{3}} \cdot \frac{\cos^{-1} \left(-\frac{g}{h}\right)}{\sqrt{3}}\right) \cdot \cos \left(\pi \cdot \frac{2}{3}\right) - \left(\sqrt{\sin \left(\pi \cdot \frac{2}{3}\right)} \cdot \sqrt{\sin \left(\pi \cdot \frac{2}{3}\right)}\right) \cdot \sin \left(\frac{1}{\sqrt{3}} \cdot \frac{\cos^{-1} \left(-\frac{g}{h}\right)}{\sqrt{3}}\right)\right)
double f(double g, double h) {
        double r6826666 = 2.0;
        double r6826667 = atan2(1.0, 0.0);
        double r6826668 = r6826666 * r6826667;
        double r6826669 = 3.0;
        double r6826670 = r6826668 / r6826669;
        double r6826671 = g;
        double r6826672 = -r6826671;
        double r6826673 = h;
        double r6826674 = r6826672 / r6826673;
        double r6826675 = acos(r6826674);
        double r6826676 = r6826675 / r6826669;
        double r6826677 = r6826670 + r6826676;
        double r6826678 = cos(r6826677);
        double r6826679 = r6826666 * r6826678;
        return r6826679;
}
↓
double f(double g, double h) {
        double r6826680 = 2.0;
        double r6826681 = 1.0;
        double r6826682 = 3.0;
        double r6826683 = sqrt(r6826682);
        double r6826684 = r6826681 / r6826683;
        double r6826685 = g;
        double r6826686 = h;
        double r6826687 = r6826685 / r6826686;
        double r6826688 = -r6826687;
        double r6826689 = acos(r6826688);
        double r6826690 = r6826689 / r6826683;
        double r6826691 = r6826684 * r6826690;
        double r6826692 = cos(r6826691);
        double r6826693 = atan2(1.0, 0.0);
        double r6826694 = 0.6666666666666666;
        double r6826695 = r6826693 * r6826694;
        double r6826696 = cos(r6826695);
        double r6826697 = r6826692 * r6826696;
        double r6826698 = sin(r6826695);
        double r6826699 = sqrt(r6826698);
        double r6826700 = r6826699 * r6826699;
        double r6826701 = sin(r6826691);
        double r6826702 = r6826700 * r6826701;
        double r6826703 = r6826697 - r6826702;
        double r6826704 = r6826680 * r6826703;
        return r6826704;
}

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

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

  4. Applied add-sqr-sqrt1.0

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

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

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

  6. Applied times-frac1.0

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

  8. Applied fma-udef1.0

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

  9. Applied cos-sum1.0

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

  11. Applied add-sqr-sqrt0.0

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

  12. Final simplification0.0

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))