2-ancestry mixing, positive discriminant

Percentage Accurate: 43.4% → 95.8%

Time: 12.5s

Alternatives: 5

Speedup: 2.4×

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: 43.4% 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: 95.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{0}{a}} + \sqrt[3]{0.5 \cdot g} \cdot \sqrt[3]{\frac{-2}{a}} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (+ (cbrt (/ 0.0 a)) (* (cbrt (* 0.5 g)) (cbrt (/ -2.0 a)))))

C

double code(double g, double h, double a) {
	return cbrt((0.0 / a)) + (cbrt((0.5 * g)) * cbrt((-2.0 / a)));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt((0.0 / a)) + (Math.cbrt((0.5 * g)) * Math.cbrt((-2.0 / a)));
}

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(0.0 / a)) + Float64(cbrt(Float64(0.5 * g)) * cbrt(Float64(-2.0 / a))))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(0.0 / a), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(0.5 * g), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(-2.0 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.2%

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

3. Taylor expanded in g around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 25.3

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

5. Simplified 25.3%

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

   1. difference-of-squares N/A

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

   2. lift-+.f64 N/A

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

   3. lift--.f64 N/A

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

   4. *-commutative N/A

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

   5. sqrt-prod N/A

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

   6. pow1/2 N/A

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

   7. pow1/2 N/A

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

   8. lower-*.f64 N/A

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

   9. pow1/2 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. pow1/2 N/A

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

   12. lower-sqrt.f64 23.9

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

7. Applied egg-rr 23.9%

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

8. Taylor expanded in g around inf

   \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{4} \cdot \frac{h + -1 \cdot h}{a}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]
9. Step-by-step derivation

   1. associate-*r/ N/A

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

   2. distribute-rgt1-in N/A

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

   3. metadata-eval N/A

      \[\leadsto \sqrt[3]{\frac{\frac{1}{4} \cdot \left(\color{blue}{0} \cdot h\right)}{a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]

   4. mul0-lft N/A

      \[\leadsto \sqrt[3]{\frac{\frac{1}{4} \cdot \color{blue}{0}}{a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]

   5. metadata-eval N/A

      \[\leadsto \sqrt[3]{\frac{\color{blue}{0}}{a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]

   6. lower-/.f64 71.4

      \[\leadsto \sqrt[3]{\color{blue}{\frac{0}{a}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]

10. Simplified 71.4%

    \[\leadsto \sqrt[3]{\color{blue}{\frac{0}{a}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \sqrt[3]{\frac{0}{a}} + \sqrt[3]{\frac{1}{\color{blue}{2 \cdot a}} \cdot \left(g \cdot -2\right)} \]

    2. lift-/.f64 N/A

       \[\leadsto \sqrt[3]{\frac{0}{a}} + \sqrt[3]{\color{blue}{\frac{1}{2 \cdot a}} \cdot \left(g \cdot -2\right)} \]

    3. lift-*.f64 N/A

       \[\leadsto \sqrt[3]{\frac{0}{a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(g \cdot -2\right)}} \]

    4. lift-/.f64 N/A

       \[\leadsto \sqrt[3]{\frac{0}{a}} + \sqrt[3]{\color{blue}{\frac{1}{2 \cdot a}} \cdot \left(g \cdot -2\right)} \]

    5. associate-*l/ N/A

       \[\leadsto \sqrt[3]{\frac{0}{a}} + \sqrt[3]{\color{blue}{\frac{1 \cdot \left(g \cdot -2\right)}{2 \cdot a}}} \]

    6. lift-*.f64 N/A

       \[\leadsto \sqrt[3]{\frac{0}{a}} + \sqrt[3]{\frac{1 \cdot \left(g \cdot -2\right)}{\color{blue}{2 \cdot a}}} \]

    7. times-frac N/A

       \[\leadsto \sqrt[3]{\frac{0}{a}} + \sqrt[3]{\color{blue}{\frac{1}{2} \cdot \frac{g \cdot -2}{a}}} \]

    8. metadata-eval N/A

       \[\leadsto \sqrt[3]{\frac{0}{a}} + \sqrt[3]{\color{blue}{\frac{1}{2}} \cdot \frac{g \cdot -2}{a}} \]

    9. lift-*.f64 N/A

       \[\leadsto \sqrt[3]{\frac{0}{a}} + \sqrt[3]{\frac{1}{2} \cdot \frac{\color{blue}{g \cdot -2}}{a}} \]

    10. associate-/l* N/A

        \[\leadsto \sqrt[3]{\frac{0}{a}} + \sqrt[3]{\frac{1}{2} \cdot \color{blue}{\left(g \cdot \frac{-2}{a}\right)}} \]

    11. associate-*r* N/A

        \[\leadsto \sqrt[3]{\frac{0}{a}} + \sqrt[3]{\color{blue}{\left(\frac{1}{2} \cdot g\right) \cdot \frac{-2}{a}}} \]

    12. cbrt-prod N/A

        \[\leadsto \sqrt[3]{\frac{0}{a}} + \color{blue}{\sqrt[3]{\frac{1}{2} \cdot g} \cdot \sqrt[3]{\frac{-2}{a}}} \]

    13. lower-*.f64 N/A

        \[\leadsto \sqrt[3]{\frac{0}{a}} + \color{blue}{\sqrt[3]{\frac{1}{2} \cdot g} \cdot \sqrt[3]{\frac{-2}{a}}} \]

    14. lower-cbrt.f64 N/A

        \[\leadsto \sqrt[3]{\frac{0}{a}} + \color{blue}{\sqrt[3]{\frac{1}{2} \cdot g}} \cdot \sqrt[3]{\frac{-2}{a}} \]

    15. lower-*.f64 N/A

        \[\leadsto \sqrt[3]{\frac{0}{a}} + \sqrt[3]{\color{blue}{\frac{1}{2} \cdot g}} \cdot \sqrt[3]{\frac{-2}{a}} \]

    16. lower-cbrt.f64 N/A

        \[\leadsto \sqrt[3]{\frac{0}{a}} + \sqrt[3]{\frac{1}{2} \cdot g} \cdot \color{blue}{\sqrt[3]{\frac{-2}{a}}} \]

    17. lower-/.f64 94.8

        \[\leadsto \sqrt[3]{\frac{0}{a}} + \sqrt[3]{0.5 \cdot g} \cdot \sqrt[3]{\color{blue}{\frac{-2}{a}}} \]

12. Applied egg-rr 94.8%

    \[\leadsto \sqrt[3]{\frac{0}{a}} + \color{blue}{\sqrt[3]{0.5 \cdot g} \cdot \sqrt[3]{\frac{-2}{a}}} \]
13. Add Preprocessing

Alternative 2: 81.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+112}:\\ \;\;\;\;\sqrt[3]{t\_0 \cdot \left(\left(-0.5 \cdot h\right) \cdot \frac{h}{g}\right)} + \sqrt[3]{t\_0 \cdot \left(g \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0}{a}} + {\left(a \cdot 2\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot -2}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* a 2.0))))
   (if (<= t_0 2e+112)
     (+ (cbrt (* t_0 (* (* -0.5 h) (/ h g)))) (cbrt (* t_0 (* g -2.0))))
     (+
      (cbrt (/ 0.0 a))
      (* (pow (* a 2.0) -0.3333333333333333) (cbrt (* g -2.0)))))))

C

double code(double g, double h, double a) {
	double t_0 = 1.0 / (a * 2.0);
	double tmp;
	if (t_0 <= 2e+112) {
		tmp = cbrt((t_0 * ((-0.5 * h) * (h / g)))) + cbrt((t_0 * (g * -2.0)));
	} else {
		tmp = cbrt((0.0 / a)) + (pow((a * 2.0), -0.3333333333333333) * cbrt((g * -2.0)));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = 1.0 / (a * 2.0);
	double tmp;
	if (t_0 <= 2e+112) {
		tmp = Math.cbrt((t_0 * ((-0.5 * h) * (h / g)))) + Math.cbrt((t_0 * (g * -2.0)));
	} else {
		tmp = Math.cbrt((0.0 / a)) + (Math.pow((a * 2.0), -0.3333333333333333) * Math.cbrt((g * -2.0)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64(1.0 / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= 2e+112)
		tmp = Float64(cbrt(Float64(t_0 * Float64(Float64(-0.5 * h) * Float64(h / g)))) + cbrt(Float64(t_0 * Float64(g * -2.0))));
	else
		tmp = Float64(cbrt(Float64(0.0 / a)) + Float64((Float64(a * 2.0) ^ -0.3333333333333333) * cbrt(Float64(g * -2.0))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(1.0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+112], N[(N[Power[N[(t$95$0 * N[(N[(-0.5 * h), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(t$95$0 * N[(g * -2.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(0.0 / a), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(a * 2.0), $MachinePrecision], -0.3333333333333333], $MachinePrecision] * N[Power[N[(g * -2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) < 1.9999999999999999e112

   1. Initial program 42.9%

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

   3. Taylor expanded in g around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 26.7

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

   5. Simplified 26.7%

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

   6. Taylor expanded in g around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 73.6

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

   8. Simplified 73.6%

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 78.5

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

   10. Applied egg-rr 78.5%

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

   if 1.9999999999999999e112 < (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 30.3%

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

   3. Taylor expanded in g around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 15.9

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

   5. Simplified 15.9%

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

      1. difference-of-squares N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. *-commutative N/A

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

      5. sqrt-prod N/A

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

      6. pow1/2 N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 6.2

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

   7. Applied egg-rr 6.2%

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

   8. Taylor expanded in g around inf

      \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{4} \cdot \frac{h + -1 \cdot h}{a}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \sqrt[3]{\frac{\frac{1}{4} \cdot \left(\color{blue}{0} \cdot h\right)}{a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]

      4. mul0-lft N/A

         \[\leadsto \sqrt[3]{\frac{\frac{1}{4} \cdot \color{blue}{0}}{a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]

      5. metadata-eval N/A

         \[\leadsto \sqrt[3]{\frac{\color{blue}{0}}{a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]

      6. lower-/.f64 38.9

         \[\leadsto \sqrt[3]{\color{blue}{\frac{0}{a}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]

   10. Simplified 38.9%

       \[\leadsto \sqrt[3]{\color{blue}{\frac{0}{a}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt[3]{\frac{0}{a}} + \sqrt[3]{\frac{1}{\color{blue}{2 \cdot a}} \cdot \left(g \cdot -2\right)} \]

       2. lift-/.f64 N/A

          \[\leadsto \sqrt[3]{\frac{0}{a}} + \sqrt[3]{\color{blue}{\frac{1}{2 \cdot a}} \cdot \left(g \cdot -2\right)} \]

       3. lift-*.f64 N/A

          \[\leadsto \sqrt[3]{\frac{0}{a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(g \cdot -2\right)}} \]

       4. cbrt-prod N/A

          \[\leadsto \sqrt[3]{\frac{0}{a}} + \color{blue}{\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{g \cdot -2}} \]

       5. pow1/3 N/A

          \[\leadsto \sqrt[3]{\frac{0}{a}} + \color{blue}{{\left(\frac{1}{2 \cdot a}\right)}^{\frac{1}{3}}} \cdot \sqrt[3]{g \cdot -2} \]

       6. lift-cbrt.f64 N/A

          \[\leadsto \sqrt[3]{\frac{0}{a}} + {\left(\frac{1}{2 \cdot a}\right)}^{\frac{1}{3}} \cdot \color{blue}{\sqrt[3]{g \cdot -2}} \]

       7. lower-*.f64 N/A

          \[\leadsto \sqrt[3]{\frac{0}{a}} + \color{blue}{{\left(\frac{1}{2 \cdot a}\right)}^{\frac{1}{3}} \cdot \sqrt[3]{g \cdot -2}} \]

       8. lift-/.f64 N/A

          \[\leadsto \sqrt[3]{\frac{0}{a}} + {\color{blue}{\left(\frac{1}{2 \cdot a}\right)}}^{\frac{1}{3}} \cdot \sqrt[3]{g \cdot -2} \]

       9. inv-pow N/A

          \[\leadsto \sqrt[3]{\frac{0}{a}} + {\color{blue}{\left({\left(2 \cdot a\right)}^{-1}\right)}}^{\frac{1}{3}} \cdot \sqrt[3]{g \cdot -2} \]

       10. pow-pow N/A

           \[\leadsto \sqrt[3]{\frac{0}{a}} + \color{blue}{{\left(2 \cdot a\right)}^{\left(-1 \cdot \frac{1}{3}\right)}} \cdot \sqrt[3]{g \cdot -2} \]

       11. metadata-eval N/A

           \[\leadsto \sqrt[3]{\frac{0}{a}} + {\left(2 \cdot a\right)}^{\color{blue}{\frac{-1}{3}}} \cdot \sqrt[3]{g \cdot -2} \]

       12. lift-pow.f64 87.8

           \[\leadsto \sqrt[3]{\frac{0}{a}} + \color{blue}{{\left(2 \cdot a\right)}^{-0.3333333333333333}} \cdot \sqrt[3]{g \cdot -2} \]

   12. Applied egg-rr 87.8%

       \[\leadsto \sqrt[3]{\frac{0}{a}} + \color{blue}{{\left(2 \cdot a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot -2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{a \cdot 2} \leq 2 \cdot 10^{+112}:\\ \;\;\;\;\sqrt[3]{\frac{1}{a \cdot 2} \cdot \left(\left(-0.5 \cdot h\right) \cdot \frac{h}{g}\right)} + \sqrt[3]{\frac{1}{a \cdot 2} \cdot \left(g \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0}{a}} + {\left(a \cdot 2\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot -2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{a \cdot 2}\\ \sqrt[3]{t\_0 \cdot \left(\left(-0.5 \cdot h\right) \cdot \frac{h}{g}\right)} + \sqrt[3]{t\_0 \cdot \left(g \cdot -2\right)} \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* a 2.0))))
   (+ (cbrt (* t_0 (* (* -0.5 h) (/ h g)))) (cbrt (* t_0 (* g -2.0))))))

C

double code(double g, double h, double a) {
	double t_0 = 1.0 / (a * 2.0);
	return cbrt((t_0 * ((-0.5 * h) * (h / g)))) + cbrt((t_0 * (g * -2.0)));
}

Java

public static double code(double g, double h, double a) {
	double t_0 = 1.0 / (a * 2.0);
	return Math.cbrt((t_0 * ((-0.5 * h) * (h / g)))) + Math.cbrt((t_0 * (g * -2.0)));
}

Julia

function code(g, h, a)
	t_0 = Float64(1.0 / Float64(a * 2.0))
	return Float64(cbrt(Float64(t_0 * Float64(Float64(-0.5 * h) * Float64(h / g)))) + cbrt(Float64(t_0 * Float64(g * -2.0))))
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(1.0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[Power[N[(t$95$0 * N[(N[(-0.5 * h), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(t$95$0 * N[(g * -2.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 41.2%

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

3. Taylor expanded in g around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 25.3

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

5. Simplified 25.3%

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

6. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 69.2

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

8. Simplified 69.2%

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

   1. associate-*r* N/A

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

   2. associate-/l* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 73.5

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

10. Applied egg-rr 73.5%

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

11. Final simplification 73.5%

    \[\leadsto \sqrt[3]{\frac{1}{a \cdot 2} \cdot \left(\left(-0.5 \cdot h\right) \cdot \frac{h}{g}\right)} + \sqrt[3]{\frac{1}{a \cdot 2} \cdot \left(g \cdot -2\right)} \]
12. Add Preprocessing

Alternative 4: 72.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{0}{a}} - \sqrt[3]{\frac{g}{a}} \end{array} \]

FPCore

(FPCore (g h a) :precision binary64 (- (cbrt (/ 0.0 a)) (cbrt (/ g a))))

C

double code(double g, double h, double a) {
	return cbrt((0.0 / a)) - cbrt((g / a));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt((0.0 / a)) - Math.cbrt((g / a));
}

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(0.0 / a)) - cbrt(Float64(g / a)))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(0.0 / a), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.2%

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

3. Taylor expanded in g around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 25.3

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

5. Simplified 25.3%

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

   1. difference-of-squares N/A

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

   2. lift-+.f64 N/A

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

   3. lift--.f64 N/A

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

   4. *-commutative N/A

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

   5. sqrt-prod N/A

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

   6. pow1/2 N/A

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

   7. pow1/2 N/A

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

   8. lower-*.f64 N/A

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

   9. pow1/2 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. pow1/2 N/A

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

   12. lower-sqrt.f64 23.9

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

7. Applied egg-rr 23.9%

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

8. Taylor expanded in g around inf

   \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{4} \cdot \frac{h + -1 \cdot h}{a}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]
9. Step-by-step derivation

   1. associate-*r/ N/A

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

   2. distribute-rgt1-in N/A

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

   3. metadata-eval N/A

      \[\leadsto \sqrt[3]{\frac{\frac{1}{4} \cdot \left(\color{blue}{0} \cdot h\right)}{a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]

   4. mul0-lft N/A

      \[\leadsto \sqrt[3]{\frac{\frac{1}{4} \cdot \color{blue}{0}}{a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]

   5. metadata-eval N/A

      \[\leadsto \sqrt[3]{\frac{\color{blue}{0}}{a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]

   6. lower-/.f64 71.4

      \[\leadsto \sqrt[3]{\color{blue}{\frac{0}{a}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]

10. Simplified 71.4%

    \[\leadsto \sqrt[3]{\color{blue}{\frac{0}{a}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]

11. Taylor expanded in g around -inf

    \[\leadsto \sqrt[3]{\frac{0}{a}} + \color{blue}{-1 \cdot \sqrt[3]{\frac{g}{a}}} \]
12. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \sqrt[3]{\frac{0}{a}} + \color{blue}{\left(\mathsf{neg}\left(\sqrt[3]{\frac{g}{a}}\right)\right)} \]

    2. lower-neg.f64 N/A

       \[\leadsto \sqrt[3]{\frac{0}{a}} + \color{blue}{\left(\mathsf{neg}\left(\sqrt[3]{\frac{g}{a}}\right)\right)} \]

    3. lower-cbrt.f64 N/A

       \[\leadsto \sqrt[3]{\frac{0}{a}} + \left(\mathsf{neg}\left(\color{blue}{\sqrt[3]{\frac{g}{a}}}\right)\right) \]

    4. lower-/.f64 71.4

       \[\leadsto \sqrt[3]{\frac{0}{a}} + \left(-\sqrt[3]{\color{blue}{\frac{g}{a}}}\right) \]

13. Simplified 71.4%

    \[\leadsto \sqrt[3]{\frac{0}{a}} + \color{blue}{\left(-\sqrt[3]{\frac{g}{a}}\right)} \]

14. Final simplification 71.4%

    \[\leadsto \sqrt[3]{\frac{0}{a}} - \sqrt[3]{\frac{g}{a}} \]
15. Add Preprocessing

Alternative 5: 72.6% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 41.2%

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

3. Taylor expanded in g around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 25.3

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

5. Simplified 25.3%

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

   1. difference-of-squares N/A

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

   2. lift-+.f64 N/A

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

   3. lift--.f64 N/A

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

   4. *-commutative N/A

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

   5. sqrt-prod N/A

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

   6. pow1/2 N/A

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

   7. pow1/2 N/A

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

   8. lower-*.f64 N/A

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

   9. pow1/2 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. pow1/2 N/A

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

   12. lower-sqrt.f64 23.9

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

7. Applied egg-rr 23.9%

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

8. Taylor expanded in g around inf

   \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{4} \cdot \frac{h + -1 \cdot h}{a}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]
9. Step-by-step derivation

   1. associate-*r/ N/A

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

   2. distribute-rgt1-in N/A

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

   3. metadata-eval N/A

      \[\leadsto \sqrt[3]{\frac{\frac{1}{4} \cdot \left(\color{blue}{0} \cdot h\right)}{a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]

   4. mul0-lft N/A

      \[\leadsto \sqrt[3]{\frac{\frac{1}{4} \cdot \color{blue}{0}}{a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]

   5. metadata-eval N/A

      \[\leadsto \sqrt[3]{\frac{\color{blue}{0}}{a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]

   6. lower-/.f64 71.4

      \[\leadsto \sqrt[3]{\color{blue}{\frac{0}{a}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]

10. Simplified 71.4%

    \[\leadsto \sqrt[3]{\color{blue}{\frac{0}{a}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]
11. Step-by-step derivation

    1. lift-/.f64 N/A

       \[\leadsto \sqrt[3]{\color{blue}{\frac{0}{a}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]

    2. pow1/3 N/A

       \[\leadsto \color{blue}{{\left(\frac{0}{a}\right)}^{\frac{1}{3}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]

    3. lift-/.f64 N/A

       \[\leadsto {\color{blue}{\left(\frac{0}{a}\right)}}^{\frac{1}{3}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]

    4. div0 N/A

       \[\leadsto {\color{blue}{0}}^{\frac{1}{3}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]

    5. metadata-eval 71.4

       \[\leadsto \color{blue}{0} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]

12. Applied egg-rr 71.4%

    \[\leadsto \color{blue}{0} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]

13. Final simplification 71.4%

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

Reproduce

herbie shell --seed 2024219 
(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))))))))