2-ancestry mixing, positive discriminant

Average Error: 36.2 → 31.3

Time: 13.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ t_1 := t_0 - g\\ \mathbf{if}\;g \leq -1 \cdot 10^{-170}:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot t_1}}{\sqrt[3]{a}} + \frac{\sqrt[3]{\left(\left(h \cdot h\right) \cdot \frac{0.5}{g}\right) \cdot -0.5}}{\sqrt[3]{a}}\\ \mathbf{elif}\;g \leq 10^{-153}:\\ \;\;\;\;\sqrt[3]{t_1 \cdot \frac{0.5}{a}} + \frac{\sqrt[3]{-0.5 \cdot \left(g \cdot 2\right)}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot \left(-0.5 \cdot \frac{{h}^{2}}{g}\right)}}{\sqrt[3]{a}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + t_0\right)}}{\sqrt[3]{a}}\\ \end{array} \]

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
 (let* ((t_0 (sqrt (- (* g g) (* h h)))) (t_1 (- t_0 g)))
   (if (<= g -1e-170)
     (+
      (/ (cbrt (* 0.5 t_1)) (cbrt a))
      (/ (cbrt (* (* (* h h) (/ 0.5 g)) -0.5)) (cbrt a)))
     (if (<= g 1e-153)
       (+ (cbrt (* t_1 (/ 0.5 a))) (/ (cbrt (* -0.5 (* g 2.0))) (cbrt a)))
       (+
        (/ (cbrt (* 0.5 (* -0.5 (/ (pow h 2.0) g)))) (cbrt a))
        (/ (cbrt (* -0.5 (+ g t_0))) (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 = sqrt(((g * g) - (h * h)));
	double t_1 = t_0 - g;
	double tmp;
	if (g <= -1e-170) {
		tmp = (cbrt((0.5 * t_1)) / cbrt(a)) + (cbrt((((h * h) * (0.5 / g)) * -0.5)) / cbrt(a));
	} else if (g <= 1e-153) {
		tmp = cbrt((t_1 * (0.5 / a))) + (cbrt((-0.5 * (g * 2.0))) / cbrt(a));
	} else {
		tmp = (cbrt((0.5 * (-0.5 * (pow(h, 2.0) / g)))) / cbrt(a)) + (cbrt((-0.5 * (g + t_0))) / cbrt(a));
	}
	return tmp;
}

Java

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

↓

public static double code(double g, double h, double a) {
	double t_0 = Math.sqrt(((g * g) - (h * h)));
	double t_1 = t_0 - g;
	double tmp;
	if (g <= -1e-170) {
		tmp = (Math.cbrt((0.5 * t_1)) / Math.cbrt(a)) + (Math.cbrt((((h * h) * (0.5 / g)) * -0.5)) / Math.cbrt(a));
	} else if (g <= 1e-153) {
		tmp = Math.cbrt((t_1 * (0.5 / a))) + (Math.cbrt((-0.5 * (g * 2.0))) / Math.cbrt(a));
	} else {
		tmp = (Math.cbrt((0.5 * (-0.5 * (Math.pow(h, 2.0) / g)))) / Math.cbrt(a)) + (Math.cbrt((-0.5 * (g + t_0))) / Math.cbrt(a));
	}
	return tmp;
}

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(1.0 / Float64(2.0 * a)) * Float64(Float64(-g) + sqrt(Float64(Float64(g * g) - Float64(h * h)))))) + cbrt(Float64(Float64(1.0 / Float64(2.0 * a)) * Float64(Float64(-g) - sqrt(Float64(Float64(g * g) - Float64(h * h)))))))
end

↓

function code(g, h, a)
	t_0 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	t_1 = Float64(t_0 - g)
	tmp = 0.0
	if (g <= -1e-170)
		tmp = Float64(Float64(cbrt(Float64(0.5 * t_1)) / cbrt(a)) + Float64(cbrt(Float64(Float64(Float64(h * h) * Float64(0.5 / g)) * -0.5)) / cbrt(a)));
	elseif (g <= 1e-153)
		tmp = Float64(cbrt(Float64(t_1 * Float64(0.5 / a))) + Float64(cbrt(Float64(-0.5 * Float64(g * 2.0))) / cbrt(a)));
	else
		tmp = Float64(Float64(cbrt(Float64(0.5 * Float64(-0.5 * Float64((h ^ 2.0) / g)))) / cbrt(a)) + Float64(cbrt(Float64(-0.5 * Float64(g + t_0))) / cbrt(a)));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * N[((-g) + N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * N[((-g) - N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

↓

code[g_, h_, a_] := Block[{t$95$0 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - g), $MachinePrecision]}, If[LessEqual[g, -1e-170], N[(N[(N[Power[N[(0.5 * t$95$1), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(N[(N[(h * h), $MachinePrecision] * N[(0.5 / g), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[g, 1e-153], N[(N[Power[N[(t$95$1 * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(-0.5 * N[(g * 2.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(0.5 * N[(-0.5 * N[(N[Power[h, 2.0], $MachinePrecision] / g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(-0.5 * N[(g + t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if g < -9.99999999999999983e-171

   1. Initial program 35.1

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

   2. Simplified 35.1

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

   3. Applied egg-rr 35.0

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

   4. Applied egg-rr 31.3

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

   5. Taylor expanded in g around -inf 30.7

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

   6. Simplified 30.7

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

   if -9.99999999999999983e-171 < g < 1.00000000000000004e-153

   1. Initial program 53.7

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

   2. Simplified 53.7

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

   3. Applied egg-rr 50.3

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

   4. Taylor expanded in g around inf 37.9

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

   if 1.00000000000000004e-153 < g

   1. Initial program 35.4

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

   2. Simplified 35.4

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

   3. Applied egg-rr 31.7

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

   4. Applied egg-rr 31.6

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

   5. Taylor expanded in g around inf 31.2

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

4. Final simplification 31.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -1 \cdot 10^{-170}:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)}}{\sqrt[3]{a}} + \frac{\sqrt[3]{\left(\left(h \cdot h\right) \cdot \frac{0.5}{g}\right) \cdot -0.5}}{\sqrt[3]{a}}\\ \mathbf{elif}\;g \leq 10^{-153}:\\ \;\;\;\;\sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} - g\right) \cdot \frac{0.5}{a}} + \frac{\sqrt[3]{-0.5 \cdot \left(g \cdot 2\right)}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot \left(-0.5 \cdot \frac{{h}^{2}}{g}\right)}}{\sqrt[3]{a}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)}}{\sqrt[3]{a}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022197 
(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))))))))