Average Error: 1.0 → 0.0
Time: 10.3s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(\cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right) \cdot \cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\pi}{3}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\pi}{3}\right)\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)\]
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{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right) \cdot \cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\pi}{3}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\pi}{3}\right)\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)
double f(double g, double h) {
        double r38923 = 2.0;
        double r38924 = atan2(1.0, 0.0);
        double r38925 = r38923 * r38924;
        double r38926 = 3.0;
        double r38927 = r38925 / r38926;
        double r38928 = g;
        double r38929 = -r38928;
        double r38930 = h;
        double r38931 = r38929 / r38930;
        double r38932 = acos(r38931);
        double r38933 = r38932 / r38926;
        double r38934 = r38927 + r38933;
        double r38935 = cos(r38934);
        double r38936 = r38923 * r38935;
        return r38936;
}
↓
double f(double g, double h) {
        double r38937 = 2.0;
        double r38938 = g;
        double r38939 = h;
        double r38940 = r38938 / r38939;
        double r38941 = acos(r38940);
        double r38942 = 3.0;
        double r38943 = r38941 / r38942;
        double r38944 = cos(r38943);
        double r38945 = r38937 / r38942;
        double r38946 = atan2(1.0, 0.0);
        double r38947 = r38946 / r38942;
        double r38948 = fma(r38945, r38946, r38947);
        double r38949 = cos(r38948);
        double r38950 = r38944 * r38949;
        double r38951 = sin(r38948);
        double r38952 = sin(r38943);
        double r38953 = r38951 * r38952;
        double r38954 = r38950 + r38953;
        double r38955 = r38937 * r38954;
        return r38955;
}

Error

Bits error versus g

Bits error versus h

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-diff0.0

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

  8. Simplified0.0

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))