2-ancestry mixing, negative discriminant

Average Error: 1.0 → 0.1

Time: 4.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \log \left(\sqrt[3]{e^{\cos \left(\mathsf{fma}\left(0.6666666666666666, \pi, \frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right)\right)}}\right)\\ 2 \cdot \left(t_0 + 2 \cdot t_0\right) \end{array} \]

FPCore

(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
         (log
          (cbrt
           (exp
            (cos (fma 0.6666666666666666 PI (/ (acos (- (/ g h))) 3.0))))))))
   (* 2.0 (+ t_0 (* 2.0 t_0)))))

C

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 = log(cbrt(exp(cos(fma(0.6666666666666666, ((double) M_PI), (acos(-(g / h)) / 3.0))))));
	return 2.0 * (t_0 + (2.0 * 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. Simplified 1.0

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

3. Applied add-log-exp_binary64 1.0

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

4. Applied add-cube-cbrt_binary64 0.1

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

5. Applied log-prod_binary64 0.1

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

6. Simplified 0.1

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

7. Simplified 0.1

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

8. Final simplification 0.1

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

Reproduce

herbie shell --seed 2022055 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))