2-ancestry mixing, positive discriminant

Percentage Accurate: 43.4% → 74.8%

Time: 16.8s

Alternatives: 4

Speedup: 1.5×

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: 43.4% 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: 74.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ t_1 := \frac{1}{2 \cdot a}\\ \mathbf{if}\;\sqrt[3]{t\_1 \cdot \left(\left(-g\right) + t\_0\right)} + \sqrt[3]{t\_1 \cdot \left(\left(-g\right) - t\_0\right)} \leq -2 \cdot 10^{+82}:\\ \;\;\;\;g \cdot \mathsf{fma}\left(\sqrt[3]{\frac{1}{a \cdot {g}^{2}}}, \sqrt[3]{-0.5} \cdot \sqrt[3]{2}, \sqrt[3]{\frac{{h}^{2}}{a \cdot {g}^{4}}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{t\_1 \cdot \left(\left(-g\right) + g\right)} + \sqrt[3]{-1 \cdot \frac{g}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (sqrt (- (* g g) (* h h)))) (t_1 (/ 1.0 (* 2.0 a))))
   (if (<=
        (+ (cbrt (* t_1 (+ (- g) t_0))) (cbrt (* t_1 (- (- g) t_0))))
        -2e+82)
     (*
      g
      (fma
       (cbrt (/ 1.0 (* a (pow g 2.0))))
       (* (cbrt -0.5) (cbrt 2.0))
       (*
        (cbrt (/ (pow h 2.0) (* a (pow g 4.0))))
        (* (cbrt -0.5) (cbrt 0.5)))))
     (+ (cbrt (* t_1 (+ (- g) g))) (cbrt (* -1.0 (/ g a)))))))

C

double code(double g, double h, double a) {
	double t_0 = sqrt(((g * g) - (h * h)));
	double t_1 = 1.0 / (2.0 * a);
	double tmp;
	if ((cbrt((t_1 * (-g + t_0))) + cbrt((t_1 * (-g - t_0)))) <= -2e+82) {
		tmp = g * fma(cbrt((1.0 / (a * pow(g, 2.0)))), (cbrt(-0.5) * cbrt(2.0)), (cbrt((pow(h, 2.0) / (a * pow(g, 4.0)))) * (cbrt(-0.5) * cbrt(0.5))));
	} else {
		tmp = cbrt((t_1 * (-g + g))) + cbrt((-1.0 * (g / a)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	t_1 = Float64(1.0 / Float64(2.0 * a))
	tmp = 0.0
	if (Float64(cbrt(Float64(t_1 * Float64(Float64(-g) + t_0))) + cbrt(Float64(t_1 * Float64(Float64(-g) - t_0)))) <= -2e+82)
		tmp = Float64(g * fma(cbrt(Float64(1.0 / Float64(a * (g ^ 2.0)))), Float64(cbrt(-0.5) * cbrt(2.0)), Float64(cbrt(Float64((h ^ 2.0) / Float64(a * (g ^ 4.0)))) * Float64(cbrt(-0.5) * cbrt(0.5)))));
	else
		tmp = Float64(cbrt(Float64(t_1 * Float64(Float64(-g) + g))) + cbrt(Float64(-1.0 * Float64(g / a))));
	end
	return tmp
end

Wolfram

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[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(t$95$1 * N[((-g) + t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(t$95$1 * N[((-g) - t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], -2e+82], N[(g * N[(N[Power[N[(1.0 / N[(a * N[Power[g, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(N[Power[h, 2.0], $MachinePrecision] / N[(a * N[Power[g, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[0.5, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(t$95$1 * N[((-g) + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(-1.0 * N[(g / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < -1.9999999999999999e82

   1. Initial program 47.2%

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

   2. Taylor expanded in g around inf

      \[\leadsto \color{blue}{g \cdot \left(\sqrt[3]{\frac{1}{a \cdot {g}^{2}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{2}\right) + \sqrt[3]{\frac{{h}^{2}}{a \cdot {g}^{4}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)\right)} \]

   4. Applied rewrites 93.8%

      \[\leadsto \color{blue}{g \cdot \mathsf{fma}\left(\sqrt[3]{\frac{1}{a \cdot {g}^{2}}}, \sqrt[3]{-0.5} \cdot \sqrt[3]{2}, \sqrt[3]{\frac{{h}^{2}}{a \cdot {g}^{4}}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right)\right)} \]

   if -1.9999999999999999e82 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))))

   1. Initial program 43.2%

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

   2. Taylor expanded in g around inf

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

   4. Applied rewrites 23.5%

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

   5. Taylor expanded in g around inf

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

   7. Applied rewrites 73.9%

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

   8. Taylor expanded in g around inf

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

   10. Applied rewrites 74.1%

       \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + g\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 73.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 43.4%

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

2. Taylor expanded in g around inf

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

4. Applied rewrites 23.5%

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

5. Taylor expanded in g around inf

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

7. Applied rewrites 72.9%

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

8. Taylor expanded in g around inf

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

10. Applied rewrites 73.0%

    \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + g\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
11. Add Preprocessing

Alternative 3: 72.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{0.5}{a} \cdot \left(\left(-g\right) + g\right)} + \sqrt[3]{\frac{0.5}{a} \cdot \left(-2 \cdot g\right)} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (+ (cbrt (* (/ 0.5 a) (+ (- g) g))) (cbrt (* (/ 0.5 a) (* -2.0 g)))))

C

double code(double g, double h, double a) {
	return cbrt(((0.5 / a) * (-g + g))) + cbrt(((0.5 / a) * (-2.0 * g)));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt(((0.5 / a) * (-g + g))) + Math.cbrt(((0.5 / a) * (-2.0 * g)));
}

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(0.5 / a) * Float64(Float64(-g) + g))) + cbrt(Float64(Float64(0.5 / a) * Float64(-2.0 * g))))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[((-g) + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(-2.0 * g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.4%

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

2. Taylor expanded in g around inf

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

4. Applied rewrites 23.5%

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

5. Taylor expanded in g around inf

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

7. Applied rewrites 72.9%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 72.9%

    \[\leadsto \sqrt[3]{\color{blue}{\frac{0.5}{a}} \cdot \left(\left(-g\right) + g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - g\right)} \]

11. Taylor expanded in a around 0

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

13. Applied rewrites 72.9%

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

14. Taylor expanded in g around inf

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

16. Applied rewrites 72.9%

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

Alternative 4: 15.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{-1 \cdot \frac{g}{a}} + \sqrt[3]{\frac{0.5}{a} \cdot \left(\left(-g\right) - g\right)} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (+ (cbrt (* -1.0 (/ g a))) (cbrt (* (/ 0.5 a) (- (- g) g)))))

C

double code(double g, double h, double a) {
	return cbrt((-1.0 * (g / a))) + cbrt(((0.5 / a) * (-g - g)));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt((-1.0 * (g / a))) + Math.cbrt(((0.5 / a) * (-g - g)));
}

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(-1.0 * Float64(g / a))) + cbrt(Float64(Float64(0.5 / a) * Float64(Float64(-g) - g))))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(-1.0 * N[(g / 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 43.4%

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

2. Taylor expanded in g around inf

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

4. Applied rewrites 23.5%

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

5. Taylor expanded in g around inf

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

7. Applied rewrites 72.9%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 72.9%

    \[\leadsto \sqrt[3]{\color{blue}{\frac{0.5}{a}} \cdot \left(\left(-g\right) + g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - g\right)} \]

11. Taylor expanded in a around 0

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

13. Applied rewrites 72.9%

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

14. Taylor expanded in g around -inf

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

16. Applied rewrites 15.2%

    \[\leadsto \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} + \sqrt[3]{\frac{0.5}{a} \cdot \left(\left(-g\right) - g\right)} \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2025121 
(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))))))))