2-ancestry mixing, positive discriminant

Percentage Accurate: 44.3% → 77.1%

Time: 11.4s

Alternatives: 3

Speedup: 2.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{2 \cdot a}\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ \sqrt[3]{t_0 \cdot \left(\left(-g\right) + t_1\right)} + \sqrt[3]{t_0 \cdot \left(\left(-g\right) - t_1\right)} \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* 2.0 a))) (t_1 (sqrt (- (* g g) (* h h)))))
   (+ (cbrt (* t_0 (+ (- g) t_1))) (cbrt (* t_0 (- (- g) t_1))))))

C

double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double t_1 = sqrt(((g * g) - (h * h)));
	return cbrt((t_0 * (-g + t_1))) + cbrt((t_0 * (-g - t_1)));
}

Java

public static double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double t_1 = Math.sqrt(((g * g) - (h * h)));
	return Math.cbrt((t_0 * (-g + t_1))) + Math.cbrt((t_0 * (-g - t_1)));
}

Julia

function code(g, h, a)
	t_0 = Float64(1.0 / Float64(2.0 * a))
	t_1 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	return Float64(cbrt(Float64(t_0 * Float64(Float64(-g) + t_1))) + cbrt(Float64(t_0 * Float64(Float64(-g) - t_1))))
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[Power[N[(t$95$0 * N[((-g) + t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(t$95$0 * N[((-g) - t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]

Initial Program: 44.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{2 \cdot a}\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ \sqrt[3]{t_0 \cdot \left(\left(-g\right) + t_1\right)} + \sqrt[3]{t_0 \cdot \left(\left(-g\right) - t_1\right)} \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* 2.0 a))) (t_1 (sqrt (- (* g g) (* h h)))))
   (+ (cbrt (* t_0 (+ (- g) t_1))) (cbrt (* t_0 (- (- g) t_1))))))

C

double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double t_1 = sqrt(((g * g) - (h * h)));
	return cbrt((t_0 * (-g + t_1))) + cbrt((t_0 * (-g - t_1)));
}

Java

public static double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double t_1 = Math.sqrt(((g * g) - (h * h)));
	return Math.cbrt((t_0 * (-g + t_1))) + Math.cbrt((t_0 * (-g - t_1)));
}

Julia

function code(g, h, a)
	t_0 = Float64(1.0 / Float64(2.0 * a))
	t_1 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	return Float64(cbrt(Float64(t_0 * Float64(Float64(-g) + t_1))) + cbrt(Float64(t_0 * Float64(Float64(-g) - t_1))))
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[Power[N[(t$95$0 * N[((-g) + t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(t$95$0 * N[((-g) - t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]

Alternative 1: 77.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}}\\ \mathbf{if}\;g \leq 1.5 \cdot 10^{-168}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot \left(h \cdot \frac{h}{g}\right)\right)} + t_0\\ \mathbf{elif}\;g \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{g + g} \cdot \sqrt[3]{\frac{-0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (* (+ g g) (/ -0.5 a)))))
   (if (<= g 1.5e-168)
     (+ (cbrt (* (/ 0.5 a) (* -0.5 (* h (/ h g))))) t_0)
     (if (<= g 1.35e+154)
       (+
        (cbrt (* (/ 0.5 a) (- (sqrt (- (* g g) (* h h))) g)))
        (* (cbrt (+ g g)) (cbrt (/ -0.5 a))))
       (+ t_0 (cbrt (* (/ 0.5 a) (- g g))))))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt(((g + g) * (-0.5 / a)));
	double tmp;
	if (g <= 1.5e-168) {
		tmp = cbrt(((0.5 / a) * (-0.5 * (h * (h / g))))) + t_0;
	} else if (g <= 1.35e+154) {
		tmp = cbrt(((0.5 / a) * (sqrt(((g * g) - (h * h))) - g))) + (cbrt((g + g)) * cbrt((-0.5 / a)));
	} else {
		tmp = t_0 + cbrt(((0.5 / a) * (g - g)));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.cbrt(((g + g) * (-0.5 / a)));
	double tmp;
	if (g <= 1.5e-168) {
		tmp = Math.cbrt(((0.5 / a) * (-0.5 * (h * (h / g))))) + t_0;
	} else if (g <= 1.35e+154) {
		tmp = Math.cbrt(((0.5 / a) * (Math.sqrt(((g * g) - (h * h))) - g))) + (Math.cbrt((g + g)) * Math.cbrt((-0.5 / a)));
	} else {
		tmp = t_0 + Math.cbrt(((0.5 / a) * (g - g)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(Float64(g + g) * Float64(-0.5 / a)))
	tmp = 0.0
	if (g <= 1.5e-168)
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(-0.5 * Float64(h * Float64(h / g))))) + t_0);
	elseif (g <= 1.35e+154)
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(sqrt(Float64(Float64(g * g) - Float64(h * h))) - g))) + Float64(cbrt(Float64(g + g)) * cbrt(Float64(-0.5 / a))));
	else
		tmp = Float64(t_0 + cbrt(Float64(Float64(0.5 / a) * Float64(g - g))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[(N[(g + g), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[g, 1.5e-168], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(-0.5 * N[(h * N[(h / g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[g, 1.35e+154], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(g + g), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(-0.5 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(g - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if g < 1.49999999999999996e-168

   1. Initial program 44.6%

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

   3. Simplified 45.3%

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

   4. Taylor expanded in g around inf 11.2%

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

   5. Taylor expanded in g around inf 75.2%

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

      1. unpow2 75.2%

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

      2. associate-*l/ 81.6%

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

      3. *-commutative 81.6%

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

   7. Simplified 81.6%

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

   if 1.49999999999999996e-168 < g < 1.35000000000000003e154

   1. Initial program 82.5%

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

   3. Simplified 82.5%

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

   4. Taylor expanded in g around inf 81.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \color{blue}{g}\right) \cdot \frac{-0.5}{a}} \]
   5. Step-by-step derivation

      1. cbrt-prod 96.7%

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

   6. Applied egg-rr 96.7%

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

   if 1.35000000000000003e154 < g

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in g around inf 0.3%

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

   5. Taylor expanded in g around inf 61.5%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq 1.5 \cdot 10^{-168}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot \left(h \cdot \frac{h}{g}\right)\right)} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}}\\ \mathbf{elif}\;g \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{g + g} \cdot \sqrt[3]{\frac{-0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)}\\ \end{array} \]

Alternative 2: 75.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot \left(h \cdot \frac{h}{g}\right)\right)} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (+ (cbrt (* (/ 0.5 a) (* -0.5 (* h (/ h g))))) (cbrt (* (+ g g) (/ -0.5 a)))))

C

double code(double g, double h, double a) {
	return cbrt(((0.5 / a) * (-0.5 * (h * (h / g))))) + cbrt(((g + g) * (-0.5 / a)));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt(((0.5 / a) * (-0.5 * (h * (h / g))))) + Math.cbrt(((g + g) * (-0.5 / a)));
}

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(0.5 / a) * Float64(-0.5 * Float64(h * Float64(h / g))))) + cbrt(Float64(Float64(g + g) * Float64(-0.5 / a))))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(-0.5 * N[(h * N[(h / g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(g + g), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.2%

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

3. Simplified 46.5%

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

4. Taylor expanded in g around inf 27.6%

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

5. Taylor expanded in g around inf 73.6%

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

   1. unpow2 73.6%

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

   2. associate-*l/ 78.2%

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

   3. *-commutative 78.2%

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

7. Simplified 78.2%

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

8. Final simplification 78.2%

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

Alternative 3: 73.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (+ (cbrt (* (+ g g) (/ -0.5 a))) (cbrt (* (/ 0.5 a) (- g g)))))

C

double code(double g, double h, double a) {
	return cbrt(((g + g) * (-0.5 / a))) + cbrt(((0.5 / a) * (g - g)));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt(((g + g) * (-0.5 / a))) + Math.cbrt(((0.5 / a) * (g - g)));
}

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(g + g) * Float64(-0.5 / a))) + cbrt(Float64(Float64(0.5 / a) * Float64(g - g))))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(N[(g + g), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(g - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.2%

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

3. Simplified 46.5%

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

4. Taylor expanded in g around inf 27.6%

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

5. Taylor expanded in g around inf 77.1%

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

6. Final simplification 77.1%

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

Reproduce

herbie shell --seed 2023273 
(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))))))))