2-ancestry mixing, positive discriminant

Percentage Accurate: 43.9% → 95.9%

Time: 11.5s

Alternatives: 4

Speedup: 1.4×

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: 43.9% 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: 95.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{0.5}{a}} \cdot \left(\sqrt[3]{g} \cdot \left(\sqrt[3]{-1} \cdot {4}^{0.16666666666666666}\right)\right) \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (*
  (cbrt (/ 0.5 a))
  (* (cbrt g) (* (cbrt -1.0) (pow 4.0 0.16666666666666666)))))

C

double code(double g, double h, double a) {
	return cbrt((0.5 / a)) * (cbrt(g) * (cbrt(-1.0) * pow(4.0, 0.16666666666666666)));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt((0.5 / a)) * (Math.cbrt(g) * (Math.cbrt(-1.0) * Math.pow(4.0, 0.16666666666666666)));
}

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(0.5 / a)) * Float64(cbrt(g) * Float64(cbrt(-1.0) * (4.0 ^ 0.16666666666666666))))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[(N[Power[g, 1/3], $MachinePrecision] * N[(N[Power[-1.0, 1/3], $MachinePrecision] * N[Power[4.0, 0.16666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.7%

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

4. Applied rewrites 13.5%

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

5. Taylor expanded in g around inf

   \[\leadsto \sqrt[3]{\frac{\frac{1}{2}}{a}} \cdot \color{blue}{\left(\sqrt[3]{g} \cdot \left(\sqrt[3]{-1} \cdot \sqrt[3]{2}\right)\right)} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-cbrt.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-cbrt.f64 96.1

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

7. Applied rewrites 96.1%

   \[\leadsto \sqrt[3]{\frac{0.5}{a}} \cdot \color{blue}{\left(\sqrt[3]{g} \cdot \left(\sqrt[3]{-1} \cdot \sqrt[3]{2}\right)\right)} \]
8. Step-by-step derivation

9. Applied rewrites 96.6%

   \[\leadsto \sqrt[3]{\frac{0.5}{a}} \cdot \left(\sqrt[3]{g} \cdot \left(\sqrt[3]{-1} \cdot {4}^{\color{blue}{0.16666666666666666}}\right)\right) \]
10. Add Preprocessing

Alternative 2: 80.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(2 \cdot a\right)}^{-1} \leq 2 \cdot 10^{+76}:\\ \;\;\;\;\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot -2}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= (pow (* 2.0 a) -1.0) 2e+76)
   (* (cbrt (/ g a)) (cbrt -1.0))
   (* (pow (* 2.0 a) -0.3333333333333333) (cbrt (* g -2.0)))))

C

double code(double g, double h, double a) {
	double tmp;
	if (pow((2.0 * a), -1.0) <= 2e+76) {
		tmp = cbrt((g / a)) * cbrt(-1.0);
	} else {
		tmp = pow((2.0 * a), -0.3333333333333333) * cbrt((g * -2.0));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if (Math.pow((2.0 * a), -1.0) <= 2e+76) {
		tmp = Math.cbrt((g / a)) * Math.cbrt(-1.0);
	} else {
		tmp = Math.pow((2.0 * a), -0.3333333333333333) * Math.cbrt((g * -2.0));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if ((Float64(2.0 * a) ^ -1.0) <= 2e+76)
		tmp = Float64(cbrt(Float64(g / a)) * cbrt(-1.0));
	else
		tmp = Float64((Float64(2.0 * a) ^ -0.3333333333333333) * cbrt(Float64(g * -2.0)));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[N[Power[N[(2.0 * a), $MachinePrecision], -1.0], $MachinePrecision], 2e+76], N[(N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[-1.0, 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(2.0 * a), $MachinePrecision], -0.3333333333333333], $MachinePrecision] * N[Power[N[(g * -2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) < 2.0000000000000001e76

   1. Initial program 51.5%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. sqr-pow N/A

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

      4. fabs-sqr N/A

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

      5. sqr-pow N/A

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

      6. pow1/3 N/A

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

      7. lift-cbrt.f64 N/A

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

      8. lower-fabs.f64 37.5

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

   4. Applied rewrites 14.1%

      \[\leadsto \color{blue}{\left|\sqrt[3]{\mathsf{fma}\left(\sqrt{g - h}, \sqrt{h + g}, -g\right) \cdot \frac{0.5}{a}}\right|} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   5. Step-by-step derivation

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow1/3 N/A

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

      12. lower-pow.f64 N/A

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

   6. Applied rewrites 8.9%

      \[\leadsto \left|\sqrt[3]{\mathsf{fma}\left(\sqrt{g - h}, \sqrt{h + g}, -g\right) \cdot \frac{0.5}{a}}\right| + \color{blue}{{\left(\left(\left(-g\right) - \sqrt{\left(h + g\right) \cdot \left(g - h\right)}\right) \cdot \frac{0.5}{a}\right)}^{0.3333333333333333}} \]

   7. Taylor expanded in g around inf

      \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}} \]

      2. lower-cbrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a}}} \cdot \sqrt[3]{-1} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{a}}} \cdot \sqrt[3]{-1} \]

      4. lower-cbrt.f64 86.9

         \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \color{blue}{\sqrt[3]{-1}} \]

   9. Applied rewrites 86.9%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}} \]

   if 2.0000000000000001e76 < (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 24.0%

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

   4. Applied rewrites 8.2%

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

   5. Taylor expanded in g around inf

      \[\leadsto \sqrt[3]{\frac{\frac{1}{2}}{a}} \cdot \color{blue}{\left(\sqrt[3]{g} \cdot \left(\sqrt[3]{-1} \cdot \sqrt[3]{2}\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-cbrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cbrt.f64 N/A

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

      5. lower-cbrt.f64 97.0

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

   7. Applied rewrites 97.0%

      \[\leadsto \sqrt[3]{\frac{0.5}{a}} \cdot \color{blue}{\left(\sqrt[3]{g} \cdot \left(\sqrt[3]{-1} \cdot \sqrt[3]{2}\right)\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 96.0%

      \[\leadsto \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{g \cdot -2} \]
   10. Step-by-step derivation

       1. lift-cbrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt[3]{\frac{\frac{1}{2}}{a}}} \cdot \sqrt[3]{g \cdot -2} \]

       2. pow1/3 N/A

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

       3. lift-/.f64 N/A

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

       4. metadata-eval N/A

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

       5. associate-/r* N/A

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

       6. lift-*.f64 N/A

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

       7. inv-pow N/A

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

       8. pow-pow N/A

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

       9. lower-pow.f64 N/A

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

       10. metadata-eval 88.3

           \[\leadsto {\left(2 \cdot a\right)}^{\color{blue}{-0.3333333333333333}} \cdot \sqrt[3]{g \cdot -2} \]

   11. Applied rewrites 88.3%

       \[\leadsto \color{blue}{{\left(2 \cdot a\right)}^{-0.3333333333333333}} \cdot \sqrt[3]{g \cdot -2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(2 \cdot a\right)}^{-1} \leq 2 \cdot 10^{+76}:\\ \;\;\;\;\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot -2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{g \cdot -2} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 46.7%

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

4. Applied rewrites 13.5%

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

5. Taylor expanded in g around inf

   \[\leadsto \sqrt[3]{\frac{\frac{1}{2}}{a}} \cdot \color{blue}{\left(\sqrt[3]{g} \cdot \left(\sqrt[3]{-1} \cdot \sqrt[3]{2}\right)\right)} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-cbrt.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-cbrt.f64 96.1

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

7. Applied rewrites 96.1%

   \[\leadsto \sqrt[3]{\frac{0.5}{a}} \cdot \color{blue}{\left(\sqrt[3]{g} \cdot \left(\sqrt[3]{-1} \cdot \sqrt[3]{2}\right)\right)} \]
8. Step-by-step derivation

9. Applied rewrites 96.3%

   \[\leadsto \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{g \cdot -2} \]
10. Add Preprocessing

Alternative 4: 73.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 46.7%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-cbrt.f64 N/A

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

   2. pow1/3 N/A

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

   3. sqr-pow N/A

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

   4. fabs-sqr N/A

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

   5. sqr-pow N/A

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

   6. pow1/3 N/A

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

   7. lift-cbrt.f64 N/A

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

   8. lower-fabs.f64 35.1

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

4. Applied rewrites 12.6%

   \[\leadsto \color{blue}{\left|\sqrt[3]{\mathsf{fma}\left(\sqrt{g - h}, \sqrt{h + g}, -g\right) \cdot \frac{0.5}{a}}\right|} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
5. Step-by-step derivation

   1. lift-cbrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-neg.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. pow1/3 N/A

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

   12. lower-pow.f64 N/A

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

6. Applied rewrites 7.4%

   \[\leadsto \left|\sqrt[3]{\mathsf{fma}\left(\sqrt{g - h}, \sqrt{h + g}, -g\right) \cdot \frac{0.5}{a}}\right| + \color{blue}{{\left(\left(\left(-g\right) - \sqrt{\left(h + g\right) \cdot \left(g - h\right)}\right) \cdot \frac{0.5}{a}\right)}^{0.3333333333333333}} \]

7. Taylor expanded in g around inf

   \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}} \]

   2. lower-cbrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a}}} \cdot \sqrt[3]{-1} \]

   3. lower-/.f64 N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{a}}} \cdot \sqrt[3]{-1} \]

   4. lower-cbrt.f64 77.2

      \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \color{blue}{\sqrt[3]{-1}} \]

9. Applied rewrites 77.2%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024339 
(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))))))))