Average Error: 1.0 → 0.0
Time: 19.1s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(\cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right) \cdot \frac{3}{2} - 3 \cdot \pi}{\frac{9}{2}}\right) \cdot \frac{1}{2} + \frac{\sqrt{3}}{2} \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3} - \frac{\pi}{\frac{3}{2}}\right)\right)\]
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{\cos^{-1} \left(\frac{g}{h}\right) \cdot \frac{3}{2} - 3 \cdot \pi}{\frac{9}{2}}\right) \cdot \frac{1}{2} + \frac{\sqrt{3}}{2} \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3} - \frac{\pi}{\frac{3}{2}}\right)\right)
double f(double g, double h) {
        double r7556926 = 2.0;
        double r7556927 = atan2(1.0, 0.0);
        double r7556928 = r7556926 * r7556927;
        double r7556929 = 3.0;
        double r7556930 = r7556928 / r7556929;
        double r7556931 = g;
        double r7556932 = -r7556931;
        double r7556933 = h;
        double r7556934 = r7556932 / r7556933;
        double r7556935 = acos(r7556934);
        double r7556936 = r7556935 / r7556929;
        double r7556937 = r7556930 + r7556936;
        double r7556938 = cos(r7556937);
        double r7556939 = r7556926 * r7556938;
        return r7556939;
}
↓
double f(double g, double h) {
        double r7556940 = 2.0;
        double r7556941 = g;
        double r7556942 = h;
        double r7556943 = r7556941 / r7556942;
        double r7556944 = acos(r7556943);
        double r7556945 = 1.5;
        double r7556946 = r7556944 * r7556945;
        double r7556947 = 3.0;
        double r7556948 = atan2(1.0, 0.0);
        double r7556949 = r7556947 * r7556948;
        double r7556950 = r7556946 - r7556949;
        double r7556951 = 4.5;
        double r7556952 = r7556950 / r7556951;
        double r7556953 = cos(r7556952);
        double r7556954 = 0.5;
        double r7556955 = r7556953 * r7556954;
        double r7556956 = sqrt(r7556947);
        double r7556957 = r7556956 / r7556940;
        double r7556958 = r7556944 / r7556947;
        double r7556959 = r7556948 / r7556945;
        double r7556960 = r7556958 - r7556959;
        double r7556961 = sin(r7556960);
        double r7556962 = r7556957 * r7556961;
        double r7556963 = r7556955 + r7556962;
        double r7556964 = r7556940 * r7556963;
        return r7556964;
}

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

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

  4. Applied distribute-frac-neg1.0

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

  5. Applied acos-neg1.0

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

  6. Applied div-sub1.0

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

  7. Applied associate-+l-1.0

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

  8. Applied cos-diff0.1

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

  9. Simplified0.1

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

  10. Simplified0.1

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

  12. Applied frac-sub0.0

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

  13. Simplified0.0

    \[\leadsto \left(\frac{1}{2} \cdot \cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right) \cdot \frac{3}{2} - 3 \cdot \pi}{\color{blue}{\frac{9}{2}}}\right) + \frac{\sqrt{3}}{2} \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3} - \frac{\pi}{\frac{3}{2}}\right)\right) \cdot 2\]

  14. Final simplification0.0

    \[\leadsto 2 \cdot \left(\cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right) \cdot \frac{3}{2} - 3 \cdot \pi}{\frac{9}{2}}\right) \cdot \frac{1}{2} + \frac{\sqrt{3}}{2} \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3} - \frac{\pi}{\frac{3}{2}}\right)\right)\]

Reproduce

herbie shell --seed 2019164 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))