Average Error: 1.0 → 0.1
Time: 13.2s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(\cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right) \cdot \cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\pi}{3}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\pi}{3}\right)\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)\]
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 r86059 = 2.0;
        double r86060 = atan2(1.0, 0.0);
        double r86061 = r86059 * r86060;
        double r86062 = 3.0;
        double r86063 = r86061 / r86062;
        double r86064 = g;
        double r86065 = -r86064;
        double r86066 = h;
        double r86067 = r86065 / r86066;
        double r86068 = acos(r86067);
        double r86069 = r86068 / r86062;
        double r86070 = r86063 + r86069;
        double r86071 = cos(r86070);
        double r86072 = r86059 * r86071;
        return r86072;
}
↓
double f(double g, double h) {
        double r86073 = 2.0;
        double r86074 = g;
        double r86075 = h;
        double r86076 = r86074 / r86075;
        double r86077 = acos(r86076);
        double r86078 = 3.0;
        double r86079 = r86077 / r86078;
        double r86080 = cos(r86079);
        double r86081 = r86073 / r86078;
        double r86082 = atan2(1.0, 0.0);
        double r86083 = r86082 / r86078;
        double r86084 = fma(r86081, r86082, r86083);
        double r86085 = cos(r86084);
        double r86086 = r86080 * r86085;
        double r86087 = sin(r86084);
        double r86088 = sin(r86079);
        double r86089 = r86087 * r86088;
        double r86090 = r86086 + r86089;
        double r86091 = r86073 * r86090;
        return r86091;
}

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-diff0.1

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

  8. Simplified0.1

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

  9. Simplified0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))