2-ancestry mixing, positive discriminant

Percentage Accurate: 44.8% → 47.4%

Time: 8.8s

Alternatives: 7

Speedup: 1.3×

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: 44.8% 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: 47.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} h = |h|\\ \\ \begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ t_1 := \sqrt[3]{\frac{0.5}{a} \cdot \left(t_0 - g\right)}\\ \mathbf{if}\;h \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;t_1 + \frac{\sqrt[3]{\left(g + \mathsf{hypot}\left(g, \sqrt{-h \cdot h}\right)\right) \cdot -0.5}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;t_1 + \sqrt[3]{\left(g + t_0\right) \cdot \frac{-0.5}{a}}\\ \end{array} \end{array} \]

FPCore

NOTE: h should be positive before calling this function
(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (sqrt (- (* g g) (* h h)))) (t_1 (cbrt (* (/ 0.5 a) (- t_0 g)))))
   (if (<= h 1.55e-162)
     (+ t_1 (/ (cbrt (* (+ g (hypot g (sqrt (- (* h h))))) -0.5)) (cbrt a)))
     (+ t_1 (cbrt (* (+ g t_0) (/ -0.5 a)))))))

C

h = abs(h);
double code(double g, double h, double a) {
	double t_0 = sqrt(((g * g) - (h * h)));
	double t_1 = cbrt(((0.5 / a) * (t_0 - g)));
	double tmp;
	if (h <= 1.55e-162) {
		tmp = t_1 + (cbrt(((g + hypot(g, sqrt(-(h * h)))) * -0.5)) / cbrt(a));
	} else {
		tmp = t_1 + cbrt(((g + t_0) * (-0.5 / a)));
	}
	return tmp;
}

Java

h = Math.abs(h);
public static double code(double g, double h, double a) {
	double t_0 = Math.sqrt(((g * g) - (h * h)));
	double t_1 = Math.cbrt(((0.5 / a) * (t_0 - g)));
	double tmp;
	if (h <= 1.55e-162) {
		tmp = t_1 + (Math.cbrt(((g + Math.hypot(g, Math.sqrt(-(h * h)))) * -0.5)) / Math.cbrt(a));
	} else {
		tmp = t_1 + Math.cbrt(((g + t_0) * (-0.5 / a)));
	}
	return tmp;
}

Julia

h = abs(h)
function code(g, h, a)
	t_0 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	t_1 = cbrt(Float64(Float64(0.5 / a) * Float64(t_0 - g)))
	tmp = 0.0
	if (h <= 1.55e-162)
		tmp = Float64(t_1 + Float64(cbrt(Float64(Float64(g + hypot(g, sqrt(Float64(-Float64(h * h))))) * -0.5)) / cbrt(a)));
	else
		tmp = Float64(t_1 + cbrt(Float64(Float64(g + t_0) * Float64(-0.5 / a))));
	end
	return tmp
end

Wolfram

NOTE: h should be positive before calling this function
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[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(t$95$0 - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[h, 1.55e-162], N[(t$95$1 + N[(N[Power[N[(N[(g + N[Sqrt[g ^ 2 + N[Sqrt[(-N[(h * h), $MachinePrecision])], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[Power[N[(N[(g + t$95$0), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if h < 1.5499999999999999e-162

   1. Initial program 44.0%

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

   3. Simplified 44.0%

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

      1. associate-*r/ 44.0%

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

      2. cbrt-div 47.5%

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

      3. sub-neg 47.5%

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

      4. add-sqr-sqrt 34.8%

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

      5. hypot-def 36.5%

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

      6. distribute-rgt-neg-in 36.5%

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

   5. Applied egg-rr 36.5%

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

   if 1.5499999999999999e-162 < h

   1. Initial program 39.9%

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

   3. Simplified 39.9%

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

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \frac{\sqrt[3]{\left(g + \mathsf{hypot}\left(g, \sqrt{-h \cdot h}\right)\right) \cdot -0.5}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}\\ \end{array} \]

Alternative 2: 47.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} h = |h|\\ \\ \begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ t_1 := \sqrt[3]{\frac{0.5}{a} \cdot \left(t_0 - g\right)}\\ \mathbf{if}\;h \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;t_1 + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + \mathsf{hypot}\left(g, \sqrt{-h \cdot h}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 + \sqrt[3]{\left(g + t_0\right) \cdot \frac{-0.5}{a}}\\ \end{array} \end{array} \]

FPCore

NOTE: h should be positive before calling this function
(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (sqrt (- (* g g) (* h h)))) (t_1 (cbrt (* (/ 0.5 a) (- t_0 g)))))
   (if (<= h 1.55e-162)
     (+ t_1 (* (cbrt (/ -0.5 a)) (cbrt (+ g (hypot g (sqrt (- (* h h))))))))
     (+ t_1 (cbrt (* (+ g t_0) (/ -0.5 a)))))))

C

h = abs(h);
double code(double g, double h, double a) {
	double t_0 = sqrt(((g * g) - (h * h)));
	double t_1 = cbrt(((0.5 / a) * (t_0 - g)));
	double tmp;
	if (h <= 1.55e-162) {
		tmp = t_1 + (cbrt((-0.5 / a)) * cbrt((g + hypot(g, sqrt(-(h * h))))));
	} else {
		tmp = t_1 + cbrt(((g + t_0) * (-0.5 / a)));
	}
	return tmp;
}

Java

h = Math.abs(h);
public static double code(double g, double h, double a) {
	double t_0 = Math.sqrt(((g * g) - (h * h)));
	double t_1 = Math.cbrt(((0.5 / a) * (t_0 - g)));
	double tmp;
	if (h <= 1.55e-162) {
		tmp = t_1 + (Math.cbrt((-0.5 / a)) * Math.cbrt((g + Math.hypot(g, Math.sqrt(-(h * h))))));
	} else {
		tmp = t_1 + Math.cbrt(((g + t_0) * (-0.5 / a)));
	}
	return tmp;
}

Julia

h = abs(h)
function code(g, h, a)
	t_0 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	t_1 = cbrt(Float64(Float64(0.5 / a) * Float64(t_0 - g)))
	tmp = 0.0
	if (h <= 1.55e-162)
		tmp = Float64(t_1 + Float64(cbrt(Float64(-0.5 / a)) * cbrt(Float64(g + hypot(g, sqrt(Float64(-Float64(h * h))))))));
	else
		tmp = Float64(t_1 + cbrt(Float64(Float64(g + t_0) * Float64(-0.5 / a))));
	end
	return tmp
end

Wolfram

NOTE: h should be positive before calling this function
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[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(t$95$0 - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[h, 1.55e-162], N[(t$95$1 + N[(N[Power[N[(-0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(g + N[Sqrt[g ^ 2 + N[Sqrt[(-N[(h * h), $MachinePrecision])], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[Power[N[(N[(g + t$95$0), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if h < 1.5499999999999999e-162

   1. Initial program 44.0%

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

   3. Simplified 44.0%

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

      1. cbrt-prod 47.5%

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

      2. *-commutative 47.5%

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

      3. sub-neg 47.5%

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

      4. add-sqr-sqrt 34.8%

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

      5. hypot-def 36.4%

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

      6. distribute-rgt-neg-in 36.4%

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

   5. Applied egg-rr 36.4%

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

   if 1.5499999999999999e-162 < h

   1. Initial program 39.9%

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

   3. Simplified 39.9%

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

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + \mathsf{hypot}\left(g, \sqrt{-h \cdot h}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}\\ \end{array} \]

Alternative 3: 44.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} h = |h|\\ \\ \begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ \sqrt[3]{\frac{0.5}{a} \cdot \left(t_0 - g\right)} + \sqrt[3]{\left(g + t_0\right) \cdot \frac{-0.5}{a}} \end{array} \end{array} \]

FPCore

NOTE: h should be positive before calling this function
(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (sqrt (- (* g g) (* h h)))))
   (+ (cbrt (* (/ 0.5 a) (- t_0 g))) (cbrt (* (+ g t_0) (/ -0.5 a))))))

C

h = abs(h);
double code(double g, double h, double a) {
	double t_0 = sqrt(((g * g) - (h * h)));
	return cbrt(((0.5 / a) * (t_0 - g))) + cbrt(((g + t_0) * (-0.5 / a)));
}

Java

h = Math.abs(h);
public static double code(double g, double h, double a) {
	double t_0 = Math.sqrt(((g * g) - (h * h)));
	return Math.cbrt(((0.5 / a) * (t_0 - g))) + Math.cbrt(((g + t_0) * (-0.5 / a)));
}

Julia

h = abs(h)
function code(g, h, a)
	t_0 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	return Float64(cbrt(Float64(Float64(0.5 / a) * Float64(t_0 - g))) + cbrt(Float64(Float64(g + t_0) * Float64(-0.5 / a))))
end

Wolfram

NOTE: h should be positive before calling this function
code[g_, h_, a_] := Block[{t$95$0 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(t$95$0 - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(g + t$95$0), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 42.7%

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

3. Simplified 42.7%

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

4. Final simplification 42.7%

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

Alternative 4: 46.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} h = |h|\\ \\ \begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ \mathbf{if}\;g \leq 10^{-173}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(t_0 - g\right)} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(0.5 \cdot \frac{h}{\frac{g}{h}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(g + t_0\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: h should be positive before calling this function
(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (sqrt (- (* g g) (* h h)))))
   (if (<= g 1e-173)
     (+
      (cbrt (* (/ 0.5 a) (- t_0 g)))
      (cbrt (* (/ -0.5 a) (* 0.5 (/ h (/ g h))))))
     (+ (cbrt (* (+ g t_0) (/ -0.5 a))) (cbrt (* (/ 0.5 a) (- g g)))))))

C

h = abs(h);
double code(double g, double h, double a) {
	double t_0 = sqrt(((g * g) - (h * h)));
	double tmp;
	if (g <= 1e-173) {
		tmp = cbrt(((0.5 / a) * (t_0 - g))) + cbrt(((-0.5 / a) * (0.5 * (h / (g / h)))));
	} else {
		tmp = cbrt(((g + t_0) * (-0.5 / a))) + cbrt(((0.5 / a) * (g - g)));
	}
	return tmp;
}

Java

h = Math.abs(h);
public static double code(double g, double h, double a) {
	double t_0 = Math.sqrt(((g * g) - (h * h)));
	double tmp;
	if (g <= 1e-173) {
		tmp = Math.cbrt(((0.5 / a) * (t_0 - g))) + Math.cbrt(((-0.5 / a) * (0.5 * (h / (g / h)))));
	} else {
		tmp = Math.cbrt(((g + t_0) * (-0.5 / a))) + Math.cbrt(((0.5 / a) * (g - g)));
	}
	return tmp;
}

Julia

h = abs(h)
function code(g, h, a)
	t_0 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	tmp = 0.0
	if (g <= 1e-173)
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(t_0 - g))) + cbrt(Float64(Float64(-0.5 / a) * Float64(0.5 * Float64(h / Float64(g / h))))));
	else
		tmp = Float64(cbrt(Float64(Float64(g + t_0) * Float64(-0.5 / a))) + cbrt(Float64(Float64(0.5 / a) * Float64(g - g))));
	end
	return tmp
end

Wolfram

NOTE: h should be positive before calling this function
code[g_, h_, a_] := Block[{t$95$0 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[g, 1e-173], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(t$95$0 - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(-0.5 / a), $MachinePrecision] * N[(0.5 * N[(h / N[(g / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(g + t$95$0), $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. Split input into 2 regimes

2. if g < 1e-173

   1. Initial program 42.8%

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

   3. Simplified 42.8%

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

   4. Taylor expanded in g around -inf 44.3%

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

      1. unpow2 44.3%

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

      2. associate-/l* 44.3%

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

   6. Simplified 44.3%

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

   if 1e-173 < g

   1. Initial program 42.7%

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

   3. Simplified 42.7%

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

   4. Taylor expanded in g around inf 43.7%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq 10^{-173}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(0.5 \cdot \frac{h}{\frac{g}{h}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)}\\ \end{array} \]

Alternative 5: 45.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} h = |h|\\ \\ \begin{array}{l} t_0 := \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}\\ \mathbf{if}\;g \leq -1.55 \cdot 10^{-162}:\\ \;\;\;\;t_0 + \sqrt[3]{\frac{0.5}{a} \cdot \left(\left(-g\right) - g\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: h should be positive before calling this function
(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (* (+ g (sqrt (- (* g g) (* h h)))) (/ -0.5 a)))))
   (if (<= g -1.55e-162)
     (+ t_0 (cbrt (* (/ 0.5 a) (- (- g) g))))
     (+ t_0 (cbrt (* (/ 0.5 a) (- g g)))))))

C

h = abs(h);
double code(double g, double h, double a) {
	double t_0 = cbrt(((g + sqrt(((g * g) - (h * h)))) * (-0.5 / a)));
	double tmp;
	if (g <= -1.55e-162) {
		tmp = t_0 + cbrt(((0.5 / a) * (-g - g)));
	} else {
		tmp = t_0 + cbrt(((0.5 / a) * (g - g)));
	}
	return tmp;
}

Java

h = Math.abs(h);
public static double code(double g, double h, double a) {
	double t_0 = Math.cbrt(((g + Math.sqrt(((g * g) - (h * h)))) * (-0.5 / a)));
	double tmp;
	if (g <= -1.55e-162) {
		tmp = t_0 + Math.cbrt(((0.5 / a) * (-g - g)));
	} else {
		tmp = t_0 + Math.cbrt(((0.5 / a) * (g - g)));
	}
	return tmp;
}

Julia

h = abs(h)
function code(g, h, a)
	t_0 = cbrt(Float64(Float64(g + sqrt(Float64(Float64(g * g) - Float64(h * h)))) * Float64(-0.5 / a)))
	tmp = 0.0
	if (g <= -1.55e-162)
		tmp = Float64(t_0 + cbrt(Float64(Float64(0.5 / a) * Float64(Float64(-g) - g))));
	else
		tmp = Float64(t_0 + cbrt(Float64(Float64(0.5 / a) * Float64(g - g))));
	end
	return tmp
end

Wolfram

NOTE: h should be positive before calling this function
code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[(N[(g + N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[g, -1.55e-162], N[(t$95$0 + N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[((-g) - g), $MachinePrecision]), $MachinePrecision], 1/3], $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 2 regimes

2. if g < -1.5499999999999999e-162

   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 44.6%

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

   4. Taylor expanded in g around -inf 44.0%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\color{blue}{-1 \cdot g} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
   5. Step-by-step derivation

      1. mul-1-neg 44.0%

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

   6. Simplified 44.0%

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

   if -1.5499999999999999e-162 < g

   1. Initial program 40.9%

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

   3. Simplified 40.9%

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

   4. Taylor expanded in g around inf 42.0%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -1.55 \cdot 10^{-162}:\\ \;\;\;\;\sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{0.5}{a} \cdot \left(\left(-g\right) - g\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)}\\ \end{array} \]

Alternative 6: 23.8% accurate, 1.3× speedup

Math

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

FPCore

NOTE: h should be positive before calling this function
(FPCore (g h a)
 :precision binary64
 (+
  (cbrt (* (+ g (sqrt (- (* g g) (* h h)))) (/ -0.5 a)))
  (cbrt (* (/ 0.5 a) (- g g)))))

C

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

Java

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

Julia

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

Wolfram

NOTE: h should be positive before calling this function
code[g_, h_, a_] := N[(N[Power[N[(N[(g + N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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 42.7%

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

3. Simplified 42.7%

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

4. Taylor expanded in g around inf 22.1%

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

5. Final simplification 22.1%

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

Alternative 7: 27.3% accurate, 1.3× speedup

Math

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

FPCore

NOTE: h should be positive before calling this function
(FPCore (g h a)
 :precision binary64
 (+
  (cbrt (* (/ 0.5 a) (- (sqrt (- (* g g) (* h h))) g)))
  (cbrt (* (/ -0.5 a) (+ g g)))))

C

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

Java

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

Julia

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

Wolfram

NOTE: h should be positive before calling this function
code[g_, h_, a_] := 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[Power[N[(N[(-0.5 / a), $MachinePrecision] * N[(g + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 42.7%

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

3. Simplified 42.7%

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

4. Taylor expanded in g around inf 25.6%

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

5. Final simplification 25.6%

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

Reproduce

herbie shell --seed 2023287 
(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))))))))