2-ancestry mixing, positive discriminant

Average Error: 36.1 → 32.7

Time: 13.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt[3]{2 \cdot a}\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ t_2 := g + t_1\\ t_3 := t_1 - g\\ t_4 := \sqrt[3]{\frac{1}{2 \cdot a} \cdot t_3} + \sqrt[3]{t_2 \cdot \frac{-1}{2 \cdot a}}\\ t_5 := \frac{\sqrt[3]{t_3}}{t_0}\\ t_6 := \sqrt[3]{\frac{t_2}{a} \cdot -0.5}\\ \mathbf{if}\;t_4 \leq -2.394584009627847 \cdot 10^{-97}:\\ \;\;\;\;t_5 + t_6\\ \mathbf{elif}\;t_4 \leq 2.0188474623397205 \cdot 10^{-106}:\\ \;\;\;\;t_6 + \frac{\sqrt[3]{g \cdot -2}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_5 + \frac{\sqrt[3]{t_2 \cdot -0.5}}{\sqrt[3]{a}}\\ \end{array} \]

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
 (let* ((t_0 (cbrt (* 2.0 a)))
        (t_1 (sqrt (- (* g g) (* h h))))
        (t_2 (+ g t_1))
        (t_3 (- t_1 g))
        (t_4
         (+
          (cbrt (* (/ 1.0 (* 2.0 a)) t_3))
          (cbrt (* t_2 (/ -1.0 (* 2.0 a))))))
        (t_5 (/ (cbrt t_3) t_0))
        (t_6 (cbrt (* (/ t_2 a) -0.5))))
   (if (<= t_4 -2.394584009627847e-97)
     (+ t_5 t_6)
     (if (<= t_4 2.0188474623397205e-106)
       (+ t_6 (/ (cbrt (* g -2.0)) t_0))
       (+ t_5 (/ (cbrt (* t_2 -0.5)) (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 t_0 = cbrt((2.0 * a));
	double t_1 = sqrt(((g * g) - (h * h)));
	double t_2 = g + t_1;
	double t_3 = t_1 - g;
	double t_4 = cbrt(((1.0 / (2.0 * a)) * t_3)) + cbrt((t_2 * (-1.0 / (2.0 * a))));
	double t_5 = cbrt(t_3) / t_0;
	double t_6 = cbrt(((t_2 / a) * -0.5));
	double tmp;
	if (t_4 <= -2.394584009627847e-97) {
		tmp = t_5 + t_6;
	} else if (t_4 <= 2.0188474623397205e-106) {
		tmp = t_6 + (cbrt((g * -2.0)) / t_0);
	} else {
		tmp = t_5 + (cbrt((t_2 * -0.5)) / 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 (+.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))))))) < -2.3945840096278469e-97

   1. Initial program 10.3

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

   2. Simplified 10.3

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

   3. Applied cbrt-div_binary64 8.4

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

   if -2.3945840096278469e-97 < (+.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))))))) < 2.01884746233972053e-106

   1. Initial program 53.6

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

   2. Simplified 53.6

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

   3. Applied cbrt-div_binary64 37.4

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

   4. Taylor expanded in g around -inf 11.8

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

   if 2.01884746233972053e-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)))))))

   1. Initial program 44.2

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

   2. Simplified 44.2

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

   3. Applied cbrt-div_binary64 43.3

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

   4. Applied associate-*l/_binary64 43.3

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

   5. Applied cbrt-div_binary64 42.4

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

   6. Simplified 42.4

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

   7. Applied *-un-lft-identity_binary64 42.4

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

4. Final simplification 32.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} \leq -2.394584009627847 \cdot 10^{-97}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{g + \sqrt{g \cdot g - h \cdot h}}{a} \cdot -0.5}\\ \mathbf{elif}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} \leq 2.0188474623397205 \cdot 10^{-106}:\\ \;\;\;\;\sqrt[3]{\frac{g + \sqrt{g \cdot g - h \cdot h}}{a} \cdot -0.5} + \frac{\sqrt[3]{g \cdot -2}}{\sqrt[3]{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot -0.5}}{\sqrt[3]{a}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022160 
(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))))))))