2-ancestry mixing, positive discriminant

Average Error: 36.5 → 31.8

Time: 13.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := g + \sqrt{g \cdot g - h \cdot h}\\ t_1 := \frac{h \cdot h}{g}\\ \mathbf{if}\;g \leq 1.4710116353411875 \cdot 10^{-158}:\\ \;\;\;\;\frac{\sqrt[3]{\left(0.5 \cdot t_1 - g\right) - g}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{t_0}{a} \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{t_1 \cdot -0.5}{2}}}{\sqrt[3]{a}} + \frac{\sqrt[3]{t_0 \cdot -0.5}}{\sqrt[3]{a}}\\ \end{array} \]

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 (+ g (sqrt (- (* g g) (* h h))))) (t_1 (/ (* h h) g)))
   (if (<= g 1.4710116353411875e-158)
     (+
      (/ (cbrt (- (- (* 0.5 t_1) g) g)) (cbrt (* 2.0 a)))
      (cbrt (* (/ t_0 a) -0.5)))
     (+
      (/ (cbrt (/ (* t_1 -0.5) 2.0)) (cbrt a))
      (/ (cbrt (* t_0 -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 = g + sqrt(((g * g) - (h * h)));
	double t_1 = (h * h) / g;
	double tmp;
	if (g <= 1.4710116353411875e-158) {
		tmp = (cbrt((((0.5 * t_1) - g) - g)) / cbrt((2.0 * a))) + cbrt(((t_0 / a) * -0.5));
	} else {
		tmp = (cbrt(((t_1 * -0.5) / 2.0)) / cbrt(a)) + (cbrt((t_0 * -0.5)) / 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 = g + Math.sqrt(((g * g) - (h * h)));
	double t_1 = (h * h) / g;
	double tmp;
	if (g <= 1.4710116353411875e-158) {
		tmp = (Math.cbrt((((0.5 * t_1) - g) - g)) / Math.cbrt((2.0 * a))) + Math.cbrt(((t_0 / a) * -0.5));
	} else {
		tmp = (Math.cbrt(((t_1 * -0.5) / 2.0)) / Math.cbrt(a)) + (Math.cbrt((t_0 * -0.5)) / 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 = Float64(g + sqrt(Float64(Float64(g * g) - Float64(h * h))))
	t_1 = Float64(Float64(h * h) / g)
	tmp = 0.0
	if (g <= 1.4710116353411875e-158)
		tmp = Float64(Float64(cbrt(Float64(Float64(Float64(0.5 * t_1) - g) - g)) / cbrt(Float64(2.0 * a))) + cbrt(Float64(Float64(t_0 / a) * -0.5)));
	else
		tmp = Float64(Float64(cbrt(Float64(Float64(t_1 * -0.5) / 2.0)) / cbrt(a)) + Float64(cbrt(Float64(t_0 * -0.5)) / 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[(g + N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(h * h), $MachinePrecision] / g), $MachinePrecision]}, If[LessEqual[g, 1.4710116353411875e-158], N[(N[(N[Power[N[(N[(N[(0.5 * t$95$1), $MachinePrecision] - g), $MachinePrecision] - g), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[N[(2.0 * a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(t$95$0 / a), $MachinePrecision] * -0.5), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(N[(t$95$1 * -0.5), $MachinePrecision] / 2.0), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(t$95$0 * -0.5), $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 2 regimes

2. if g < 1.4710116353411875e-158

   1. Initial program 38.0

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

   2. Simplified 38.0

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

   3. Applied egg-rr 34.4

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

   4. Taylor expanded in g around -inf 33.1

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

   5. Simplified 33.1

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

   if 1.4710116353411875e-158 < g

   1. Initial program 34.9

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

   2. Simplified 34.8

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

   3. Applied egg-rr 34.8

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

   4. Applied egg-rr 30.9

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

   5. Taylor expanded in g around inf 30.4

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

   6. Simplified 30.4

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

4. Final simplification 31.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq 1.4710116353411875 \cdot 10^{-158}:\\ \;\;\;\;\frac{\sqrt[3]{\left(0.5 \cdot \frac{h \cdot h}{g} - g\right) - g}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{g + \sqrt{g \cdot g - h \cdot h}}{a} \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\frac{h \cdot h}{g} \cdot -0.5}{2}}}{\sqrt[3]{a}} + \frac{\sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot -0.5}}{\sqrt[3]{a}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022144 
(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))))))))