2-ancestry mixing, positive discriminant

Average Error: 35.8 → 31.1

Time: 10.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ t_1 := t_0 - g\\ \mathbf{if}\;g \leq -1.65 \cdot 10^{-191}:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot t_1}}{\sqrt[3]{a}} + \frac{\sqrt[3]{\left(0.5 \cdot \left(h \cdot \frac{h}{g}\right)\right) \cdot -0.5}}{\sqrt[3]{a}}\\ \mathbf{elif}\;g \leq 2.35 \cdot 10^{-136}:\\ \;\;\;\;\sqrt[3]{t_1 \cdot \frac{0.5}{a}} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + g}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot \left(-0.5 \cdot \frac{{h}^{2}}{g}\right)}}{\sqrt[3]{a}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + t_0\right)}}{\sqrt[3]{a}}\\ \end{array} \]

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 -1.65e-191)
     (+
      (/ (cbrt (* 0.5 t_1)) (cbrt a))
      (/ (cbrt (* (* 0.5 (* h (/ h g))) -0.5)) (cbrt a)))
     (if (<= g 2.35e-136)
       (+ (cbrt (* t_1 (/ 0.5 a))) (* (cbrt (/ -0.5 a)) (cbrt (+ g g))))
       (+
        (/ (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 <= -1.65e-191) {
		tmp = (cbrt((0.5 * t_1)) / cbrt(a)) + (cbrt(((0.5 * (h * (h / g))) * -0.5)) / cbrt(a));
	} else if (g <= 2.35e-136) {
		tmp = cbrt((t_1 * (0.5 / a))) + (cbrt((-0.5 / a)) * cbrt((g + g)));
	} 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 <= -1.65e-191) {
		tmp = (Math.cbrt((0.5 * t_1)) / Math.cbrt(a)) + (Math.cbrt(((0.5 * (h * (h / g))) * -0.5)) / Math.cbrt(a));
	} else if (g <= 2.35e-136) {
		tmp = Math.cbrt((t_1 * (0.5 / a))) + (Math.cbrt((-0.5 / a)) * Math.cbrt((g + g)));
	} 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 <= -1.65e-191)
		tmp = Float64(Float64(cbrt(Float64(0.5 * t_1)) / cbrt(a)) + Float64(cbrt(Float64(Float64(0.5 * Float64(h * Float64(h / g))) * -0.5)) / cbrt(a)));
	elseif (g <= 2.35e-136)
		tmp = Float64(cbrt(Float64(t_1 * Float64(0.5 / a))) + Float64(cbrt(Float64(-0.5 / a)) * cbrt(Float64(g + g))));
	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, -1.65e-191], 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[(0.5 * N[(h * N[(h / g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[g, 2.35e-136], N[(N[Power[N[(t$95$1 * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(-0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(g + g), $MachinePrecision], 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]]]]]

Error

Bits error versus g

Bits error versus h

Bits error versus a

Derivation

1. Split input into 3 regimes

2. if g < -1.64999999999999991e-191

   1. Initial program 35.2

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

   2. Simplified 35.2

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

   3. Applied egg-rr 35.1

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

   4. Applied egg-rr 31.4

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

   5. Taylor expanded in g around -inf 30.8

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

   6. Simplified 30.8

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

   if -1.64999999999999991e-191 < g < 2.35000000000000011e-136

   1. Initial program 48.0

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

   2. Simplified 48.0

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

   3. Applied egg-rr 43.1

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

   4. Taylor expanded in g around inf 33.8

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

   if 2.35000000000000011e-136 < g

   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 31.6

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

   4. Applied egg-rr 31.5

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

   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 31.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -1.65 \cdot 10^{-191}:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)}}{\sqrt[3]{a}} + \frac{\sqrt[3]{\left(0.5 \cdot \left(h \cdot \frac{h}{g}\right)\right) \cdot -0.5}}{\sqrt[3]{a}}\\ \mathbf{elif}\;g \leq 2.35 \cdot 10^{-136}:\\ \;\;\;\;\sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} - g\right) \cdot \frac{0.5}{a}} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + g}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot \left(-0.5 \cdot \frac{{h}^{2}}{g}\right)}}{\sqrt[3]{a}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)}}{\sqrt[3]{a}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(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))))))))