2-ancestry mixing, positive discriminant

Average Error: 36.4 → 31.5

Time: 8.9s

Precision: binary64

Math

\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;g \leq -3.322449351468917 \cdot 10^{-214}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{\frac{h \cdot h}{g - \sqrt{g \cdot g - h \cdot h}} \cdot -0.5}}{\sqrt[3]{a}}\\ \mathbf{elif}\;g \leq 7.692507907003283 \cdot 10^{+92}:\\ \;\;\;\;\sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{2 \cdot a}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + g\right)}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{-0.5 \cdot \left(\sqrt{g + \sqrt{g \cdot g - h \cdot h}} \cdot \sqrt{g + \sqrt{g \cdot g - h \cdot h}}\right)}}{\sqrt[3]{a}}\\ \end{array}\]

FPCore

(FPCore (g h a)
 :precision binary64
 (+
  (cbrt (* (/ 1.0 (* 2.0 a)) (+ (- g) (sqrt (- (* g g) (* h h))))))
  (cbrt (* (/ 1.0 (* 2.0 a)) (- (- g) (sqrt (- (* g g) (* h h))))))))

↓

(FPCore (g h a)
 :precision binary64
 (if (<= g -3.322449351468917e-214)
   (+
    (/ (cbrt (- (sqrt (- (* g g) (* h h))) g)) (cbrt (* 2.0 a)))
    (/ (cbrt (* (/ (* h h) (- g (sqrt (- (* g g) (* h h))))) -0.5)) (cbrt a)))
   (if (<= g 7.692507907003283e+92)
     (+
      (cbrt (/ (- (sqrt (- (* g g) (* h h))) g) (* 2.0 a)))
      (/ (cbrt (* -0.5 (+ g g))) (cbrt a)))
     (+
      (/ (cbrt (- (sqrt (- (* g g) (* h h))) g)) (cbrt (* 2.0 a)))
      (/
       (cbrt
        (*
         -0.5
         (*
          (sqrt (+ g (sqrt (- (* g g) (* h h)))))
          (sqrt (+ g (sqrt (- (* g g) (* h h))))))))
       (cbrt a))))))

C

double code(double g, double h, double a) {
	return cbrt((1.0 / (2.0 * a)) * (-g + sqrt((g * g) - (h * h)))) + cbrt((1.0 / (2.0 * a)) * (-g - sqrt((g * g) - (h * h))));
}

↓

double code(double g, double h, double a) {
	double tmp;
	if (g <= -3.322449351468917e-214) {
		tmp = (cbrt(sqrt((g * g) - (h * h)) - g) / cbrt(2.0 * a)) + (cbrt(((h * h) / (g - sqrt((g * g) - (h * h)))) * -0.5) / cbrt(a));
	} else if (g <= 7.692507907003283e+92) {
		tmp = cbrt((sqrt((g * g) - (h * h)) - g) / (2.0 * a)) + (cbrt(-0.5 * (g + g)) / cbrt(a));
	} else {
		tmp = (cbrt(sqrt((g * g) - (h * h)) - g) / cbrt(2.0 * a)) + (cbrt(-0.5 * (sqrt(g + sqrt((g * g) - (h * h))) * sqrt(g + sqrt((g * g) - (h * h))))) / cbrt(a));
	}
	return tmp;
}

Error

Bits error versus g

Bits error versus h

Bits error versus a

Derivation

1. Split input into 3 regimes

2. if g < -3.3224493514689172e-214

   1. Initial program 36.4

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]

   2. Simplified 36.4

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{2 \cdot a}} + \sqrt[3]{\frac{g + \sqrt{g \cdot g - h \cdot h}}{a} \cdot -0.5}}\]
   3. Using strategy rm

   4. Applied associate-*l/_binary64_2408 36.4

      \[\leadsto \sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{2 \cdot a}} + \sqrt[3]{\color{blue}{\frac{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot -0.5}{a}}}\]

   5. Applied cbrt-div_binary64_2497 36.3

      \[\leadsto \sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{2 \cdot a}} + \color{blue}{\frac{\sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot -0.5}}{\sqrt[3]{a}}}\]
   6. Using strategy rm

   7. Applied difference-of-squares_binary64_2434 36.3

      \[\leadsto \sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{2 \cdot a}} + \frac{\sqrt[3]{\left(g + \sqrt{\color{blue}{\left(g + h\right) \cdot \left(g - h\right)}}\right) \cdot -0.5}}{\sqrt[3]{a}}\]
   8. Using strategy rm

   9. Applied cbrt-div_binary64_2497 32.3

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{2 \cdot a}}} + \frac{\sqrt[3]{\left(g + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot -0.5}}{\sqrt[3]{a}}\]
   10. Using strategy rm

   11. Applied flip-+_binary64_2439 32.2

       \[\leadsto \frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{\color{blue}{\frac{g \cdot g - \sqrt{\left(g + h\right) \cdot \left(g - h\right)} \cdot \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}{g - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}} \cdot -0.5}}{\sqrt[3]{a}}\]

   12. Simplified 31.5

       \[\leadsto \frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{\frac{\color{blue}{h \cdot h}}{g - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}} \cdot -0.5}}{\sqrt[3]{a}}\]

   13. Simplified 31.5

       \[\leadsto \frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{\frac{h \cdot h}{\color{blue}{g - \sqrt{g \cdot g - h \cdot h}}} \cdot -0.5}}{\sqrt[3]{a}}\]

   if -3.3224493514689172e-214 < g < 7.69250790700328266e92

   1. Initial program 16.4

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]

   2. Simplified 16.4

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{2 \cdot a}} + \sqrt[3]{\frac{g + \sqrt{g \cdot g - h \cdot h}}{a} \cdot -0.5}}\]
   3. Using strategy rm

   4. Applied associate-*l/_binary64_2408 16.4

      \[\leadsto \sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{2 \cdot a}} + \sqrt[3]{\color{blue}{\frac{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot -0.5}{a}}}\]

   5. Applied cbrt-div_binary64_2497 11.2

      \[\leadsto \sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{2 \cdot a}} + \color{blue}{\frac{\sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot -0.5}}{\sqrt[3]{a}}}\]

   6. Taylor expanded around inf 8.9

      \[\leadsto \sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{2 \cdot a}} + \frac{\sqrt[3]{\left(g + \color{blue}{g}\right) \cdot -0.5}}{\sqrt[3]{a}}\]

   if 7.69250790700328266e92 < g

   1. Initial program 52.6

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]

   2. Simplified 52.6

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{2 \cdot a}} + \sqrt[3]{\frac{g + \sqrt{g \cdot g - h \cdot h}}{a} \cdot -0.5}}\]
   3. Using strategy rm

   4. Applied associate-*l/_binary64_2408 52.6

      \[\leadsto \sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{2 \cdot a}} + \sqrt[3]{\color{blue}{\frac{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot -0.5}{a}}}\]

   5. Applied cbrt-div_binary64_2497 49.6

      \[\leadsto \sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{2 \cdot a}} + \color{blue}{\frac{\sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot -0.5}}{\sqrt[3]{a}}}\]
   6. Using strategy rm

   7. Applied difference-of-squares_binary64_2434 49.6

      \[\leadsto \sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{2 \cdot a}} + \frac{\sqrt[3]{\left(g + \sqrt{\color{blue}{\left(g + h\right) \cdot \left(g - h\right)}}\right) \cdot -0.5}}{\sqrt[3]{a}}\]
   8. Using strategy rm

   9. Applied cbrt-div_binary64_2497 49.6

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{2 \cdot a}}} + \frac{\sqrt[3]{\left(g + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot -0.5}}{\sqrt[3]{a}}\]
   10. Using strategy rm

   11. Applied add-sqr-sqrt_binary64_2487 49.6

       \[\leadsto \frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{\color{blue}{\left(\sqrt{g + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}} \cdot \sqrt{g + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}\right)} \cdot -0.5}}{\sqrt[3]{a}}\]

   12. Simplified 49.6

       \[\leadsto \frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{\left(\color{blue}{\sqrt{g + \sqrt{g \cdot g - h \cdot h}}} \cdot \sqrt{g + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}\right) \cdot -0.5}}{\sqrt[3]{a}}\]

   13. Simplified 49.6

       \[\leadsto \frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{\left(\sqrt{g + \sqrt{g \cdot g - h \cdot h}} \cdot \color{blue}{\sqrt{g + \sqrt{g \cdot g - h \cdot h}}}\right) \cdot -0.5}}{\sqrt[3]{a}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 31.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -3.322449351468917 \cdot 10^{-214}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{\frac{h \cdot h}{g - \sqrt{g \cdot g - h \cdot h}} \cdot -0.5}}{\sqrt[3]{a}}\\ \mathbf{elif}\;g \leq 7.692507907003283 \cdot 10^{+92}:\\ \;\;\;\;\sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{2 \cdot a}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + g\right)}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{-0.5 \cdot \left(\sqrt{g + \sqrt{g \cdot g - h \cdot h}} \cdot \sqrt{g + \sqrt{g \cdot g - h \cdot h}}\right)}}{\sqrt[3]{a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020298 
(FPCore (g h a)
  :name "2-ancestry mixing, positive discriminant"
  :precision binary64
  (+ (cbrt (* (/ 1.0 (* 2.0 a)) (+ (- g) (sqrt (- (* g g) (* h h)))))) (cbrt (* (/ 1.0 (* 2.0 a)) (- (- g) (sqrt (- (* g g) (* h h))))))))