Average Error: 1.0 → 0.1
Time: 19.2s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(\log \left(e^{\cos \left(\frac{1}{3} \cdot \cos^{-1} \left(\frac{g}{h}\right) - \pi \cdot \frac{2}{3}\right)}\right) \cdot \cos \left(\frac{\pi}{3}\right) + \sin \left(\frac{\pi}{3}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3} - \pi \cdot \frac{2}{3}\right)\right)\]
2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)
↓
2 \cdot \left(\log \left(e^{\cos \left(\frac{1}{3} \cdot \cos^{-1} \left(\frac{g}{h}\right) - \pi \cdot \frac{2}{3}\right)}\right) \cdot \cos \left(\frac{\pi}{3}\right) + \sin \left(\frac{\pi}{3}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3} - \pi \cdot \frac{2}{3}\right)\right)
double f(double g, double h) {
        double r5686777 = 2.0;
        double r5686778 = atan2(1.0, 0.0);
        double r5686779 = r5686777 * r5686778;
        double r5686780 = 3.0;
        double r5686781 = r5686779 / r5686780;
        double r5686782 = g;
        double r5686783 = -r5686782;
        double r5686784 = h;
        double r5686785 = r5686783 / r5686784;
        double r5686786 = acos(r5686785);
        double r5686787 = r5686786 / r5686780;
        double r5686788 = r5686781 + r5686787;
        double r5686789 = cos(r5686788);
        double r5686790 = r5686777 * r5686789;
        return r5686790;
}
↓
double f(double g, double h) {
        double r5686791 = 2.0;
        double r5686792 = 0.3333333333333333;
        double r5686793 = g;
        double r5686794 = h;
        double r5686795 = r5686793 / r5686794;
        double r5686796 = acos(r5686795);
        double r5686797 = r5686792 * r5686796;
        double r5686798 = atan2(1.0, 0.0);
        double r5686799 = 0.6666666666666666;
        double r5686800 = r5686798 * r5686799;
        double r5686801 = r5686797 - r5686800;
        double r5686802 = cos(r5686801);
        double r5686803 = exp(r5686802);
        double r5686804 = log(r5686803);
        double r5686805 = 3.0;
        double r5686806 = r5686798 / r5686805;
        double r5686807 = cos(r5686806);
        double r5686808 = r5686804 * r5686807;
        double r5686809 = sin(r5686806);
        double r5686810 = r5686796 / r5686805;
        double r5686811 = r5686810 - r5686800;
        double r5686812 = sin(r5686811);
        double r5686813 = r5686809 * r5686812;
        double r5686814 = r5686808 + r5686813;
        double r5686815 = r5686791 * r5686814;
        return r5686815;
}

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

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

  4. Applied distribute-frac-neg1.0

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

  5. Applied acos-neg1.0

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

  6. Applied div-sub1.0

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

  7. Applied associate-+l-1.0

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

  8. Applied cos-diff0.1

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

  9. Taylor expanded around 0 0.1

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

  11. Applied add-log-exp0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019165 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))