Average Error: 1.0 → 0.1
Time: 5.5s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(\cos \left(\frac{2 \cdot \pi}{3} + \frac{\pi}{3}\right) \cdot \cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right) + \sin \left(\frac{2 \cdot \pi}{3} + \frac{\pi}{3}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)\]
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{2 \cdot \pi}{3} + \frac{\pi}{3}\right) \cdot \cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right) + \sin \left(\frac{2 \cdot \pi}{3} + \frac{\pi}{3}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)
double f(double g, double h) {
        double r146002 = 2.0;
        double r146003 = atan2(1.0, 0.0);
        double r146004 = r146002 * r146003;
        double r146005 = 3.0;
        double r146006 = r146004 / r146005;
        double r146007 = g;
        double r146008 = -r146007;
        double r146009 = h;
        double r146010 = r146008 / r146009;
        double r146011 = acos(r146010);
        double r146012 = r146011 / r146005;
        double r146013 = r146006 + r146012;
        double r146014 = cos(r146013);
        double r146015 = r146002 * r146014;
        return r146015;
}
↓
double f(double g, double h) {
        double r146016 = 2.0;
        double r146017 = atan2(1.0, 0.0);
        double r146018 = r146016 * r146017;
        double r146019 = 3.0;
        double r146020 = r146018 / r146019;
        double r146021 = r146017 / r146019;
        double r146022 = r146020 + r146021;
        double r146023 = cos(r146022);
        double r146024 = g;
        double r146025 = h;
        double r146026 = r146024 / r146025;
        double r146027 = acos(r146026);
        double r146028 = r146027 / r146019;
        double r146029 = cos(r146028);
        double r146030 = r146023 * r146029;
        double r146031 = sin(r146022);
        double r146032 = sin(r146028);
        double r146033 = r146031 * r146032;
        double r146034 = r146030 + r146033;
        double r146035 = r146016 * r146034;
        return r146035;
}

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. Using strategy rm

  3. Applied distribute-frac-neg1.0

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

  4. Applied acos-neg1.0

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

  5. Applied div-sub1.0

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

  6. Applied associate-+r-1.0

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

  7. Applied cos-diff0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2020056 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))