2-ancestry mixing, positive discriminant

Average Error: 35.5 → 31.2

Time: 8.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} \leq -7.878564664183389 \cdot 10^{-106}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{0.5} \cdot \left(\sqrt[3]{g + \sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\frac{-1}{a}}\right)\\ \mathbf{elif}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} \leq 0:\\ \;\;\;\;\sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} + \frac{\sqrt[3]{\left(0.5 \cdot \frac{h \cdot h}{g} - g\right) - g}}{\sqrt[3]{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}{\sqrt[3]{2 \cdot a}}\\ \end{array}\]

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 (<=
      (+
       (cbrt (* (/ 1.0 (* 2.0 a)) (- (sqrt (- (* g g) (* h h))) g)))
       (cbrt (* (+ g (sqrt (- (* g g) (* h h)))) (/ -1.0 (* 2.0 a)))))
      -7.878564664183389e-106)
   (+
    (/ (cbrt (- (sqrt (- (* g g) (* h h))) g)) (cbrt (* 2.0 a)))
    (*
     (cbrt 0.5)
     (* (cbrt (+ g (sqrt (- (* g g) (* h h))))) (cbrt (/ -1.0 a)))))
   (if (<=
        (+
         (cbrt (* (/ 1.0 (* 2.0 a)) (- (sqrt (- (* g g) (* h h))) g)))
         (cbrt (* (+ g (sqrt (- (* g g) (* h h)))) (/ -1.0 (* 2.0 a)))))
        0.0)
     (+
      (cbrt (* (+ g (sqrt (- (* g g) (* h h)))) (/ -1.0 (* 2.0 a))))
      (/ (cbrt (- (- (* 0.5 (/ (* h h) g)) g) g)) (cbrt (* 2.0 a))))
     (+
      (/ (cbrt (- (sqrt (- (* g g) (* h h))) g)) (cbrt (* 2.0 a)))
      (/ (cbrt (- (- g) (sqrt (- (* g g) (* h h))))) (cbrt (* 2.0 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 ((cbrt((1.0 / (2.0 * a)) * (sqrt((g * g) - (h * h)) - g)) + cbrt((g + sqrt((g * g) - (h * h))) * (-1.0 / (2.0 * a)))) <= -7.878564664183389e-106) {
		tmp = (cbrt(sqrt((g * g) - (h * h)) - g) / cbrt(2.0 * a)) + (cbrt(0.5) * (cbrt(g + sqrt((g * g) - (h * h))) * cbrt(-1.0 / a)));
	} else if ((cbrt((1.0 / (2.0 * a)) * (sqrt((g * g) - (h * h)) - g)) + cbrt((g + sqrt((g * g) - (h * h))) * (-1.0 / (2.0 * a)))) <= 0.0) {
		tmp = cbrt((g + sqrt((g * g) - (h * h))) * (-1.0 / (2.0 * a))) + (cbrt(((0.5 * ((h * h) / g)) - g) - g) / cbrt(2.0 * a));
	} else {
		tmp = (cbrt(sqrt((g * g) - (h * h)) - g) / cbrt(2.0 * a)) + (cbrt(-g - sqrt((g * g) - (h * h))) / cbrt(2.0 * 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 (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < -7.87856466418338913e-106

   1. Initial program 9.8

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

   3. Applied associate-*l/_binary64_1385 9.8

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

   4. Applied cbrt-div_binary64_1474 7.4

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

   5. Simplified 7.4

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

   6. Taylor expanded around -inf 35.4

      \[\leadsto \frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{2 \cdot a}} + \color{blue}{\sqrt[3]{0.5} \cdot e^{0.3333333333333333 \cdot \left(\log \left(\sqrt{{g}^{2} - {h}^{2}} + g\right) + \log \left(\frac{-1}{a}\right)\right)}}\]

   7. Simplified 5.2

      \[\leadsto \frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{2 \cdot a}} + \color{blue}{\sqrt[3]{0.5} \cdot \left(\sqrt[3]{g + \sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\frac{-1}{a}}\right)}\]

   if -7.87856466418338913e-106 < (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < -0.0

   1. Initial program 60.1

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

   3. Applied associate-*l/_binary64_1385 60.1

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

   4. Applied cbrt-div_binary64_1474 41.1

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

   5. Simplified 41.1

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

   6. Taylor expanded around -inf 8.5

      \[\leadsto \frac{\sqrt[3]{\color{blue}{\left(0.5 \cdot \frac{{h}^{2}}{g} - g\right)} - g}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]

   7. Simplified 8.5

      \[\leadsto \frac{\sqrt[3]{\color{blue}{\left(0.5 \cdot \frac{h \cdot h}{g} - g\right)} - g}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]

   if -0.0 < (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))))

   1. Initial program 44.0

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

   3. Applied associate-*l/_binary64_1385 44.0

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

   4. Applied cbrt-div_binary64_1474 43.1

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

   5. Simplified 43.1

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
   6. Using strategy rm

   7. Applied associate-*l/_binary64_1385 43.1

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

   8. Applied cbrt-div_binary64_1474 42.3

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

   9. Simplified 42.3

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

4. Final simplification 31.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} \leq -7.878564664183389 \cdot 10^{-106}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{0.5} \cdot \left(\sqrt[3]{g + \sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\frac{-1}{a}}\right)\\ \mathbf{elif}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} \leq 0:\\ \;\;\;\;\sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} + \frac{\sqrt[3]{\left(0.5 \cdot \frac{h \cdot h}{g} - g\right) - g}}{\sqrt[3]{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}{\sqrt[3]{2 \cdot a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020339 
(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))))))))