Average Error: 1.0 → 1.0
Time: 5.7s
Precision: binary64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
↓
\[\begin{array}{l} t_0 := \frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}}\\ 2 \cdot \cos \left(\mathsf{fma}\left(\pi, 0.6666666666666666, t_0 \cdot t_0\right)\right) \end{array} \]
2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)
↓
\begin{array}{l}
t_0 := \frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}}\\
2 \cdot \cos \left(\mathsf{fma}\left(\pi, 0.6666666666666666, t_0 \cdot t_0\right)\right)
\end{array}

(FPCore (g h)
 :precision binary64
 (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))
↓
(FPCore (g h)
 :precision binary64
 (let* ((t_0 (/ (sqrt (acos (/ (- g) h))) (sqrt 3.0))))
   (* 2.0 (cos (fma PI 0.6666666666666666 (* t_0 t_0))))))
double code(double g, double h) {
	return 2.0 * cos(((2.0 * ((double) M_PI)) / 3.0) + (acos(-g / h) / 3.0));
}
↓
double code(double g, double h) {
	double t_0 = sqrt(acos(-g / h)) / sqrt(3.0);
	return 2.0 * cos(fma(((double) M_PI), 0.6666666666666666, (t_0 * t_0)));
}

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. Simplified1.0

    \[\leadsto \color{blue}{2 \cdot \cos \left(\mathsf{fma}\left(\pi, 0.6666666666666666, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)} \]

  3. Applied add-sqr-sqrt_binary641.0

    \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\pi, 0.6666666666666666, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{\color{blue}{\sqrt{3} \cdot \sqrt{3}}}\right)\right) \]

  4. Applied add-sqr-sqrt_binary641.0

    \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\pi, 0.6666666666666666, \frac{\color{blue}{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)} \cdot \sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}}{\sqrt{3} \cdot \sqrt{3}}\right)\right) \]

  5. Applied times-frac_binary641.0

    \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\pi, 0.6666666666666666, \color{blue}{\frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}} \cdot \frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}}}\right)\right) \]

  6. Final simplification1.0

    \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\pi, 0.6666666666666666, \frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}} \cdot \frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}}\right)\right) \]

Reproduce

herbie shell --seed 2022077 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))