2-ancestry mixing, positive discriminant

Percentage Accurate: 44.0% → 94.1%

Time: 11.7s

Alternatives: 4

Speedup: N/A×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{2 \cdot a}\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ \sqrt[3]{t\_0 \cdot \left(\left(-g\right) + t\_1\right)} + \sqrt[3]{t\_0 \cdot \left(\left(-g\right) - t\_1\right)} \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 44.0% 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: 94.1% accurate, N/A× speedup

Math

\[\begin{array}{l} h_m = \left|h\right| \\ \begin{array}{l} \mathbf{if}\;h\_m \leq 4.5 \cdot 10^{+153}:\\ \;\;\;\;\sqrt[3]{0.5 \cdot {a}^{-1}} \cdot \sqrt[3]{-1 \cdot g + -1 \cdot g} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{h\_m \cdot h\_m}{g} \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{g}{a}} \cdot -1\\ \end{array} \end{array} \]

FPCore

h_m = (fabs.f64 h)
(FPCore (g h_m a)
 :precision binary64
 (if (<= h_m 4.5e+153)
   (+
    (* (cbrt (* 0.5 (pow a -1.0))) (cbrt (+ (* -1.0 g) (* -1.0 g))))
    (cbrt (* (/ 1.0 (* 2.0 a)) (* (/ (* h_m h_m) g) -0.5))))
   (* (cbrt (/ g a)) -1.0)))

C

h_m = fabs(h);
double code(double g, double h_m, double a) {
	double tmp;
	if (h_m <= 4.5e+153) {
		tmp = (cbrt((0.5 * pow(a, -1.0))) * cbrt(((-1.0 * g) + (-1.0 * g)))) + cbrt(((1.0 / (2.0 * a)) * (((h_m * h_m) / g) * -0.5)));
	} else {
		tmp = cbrt((g / a)) * -1.0;
	}
	return tmp;
}

Java

h_m = Math.abs(h);
public static double code(double g, double h_m, double a) {
	double tmp;
	if (h_m <= 4.5e+153) {
		tmp = (Math.cbrt((0.5 * Math.pow(a, -1.0))) * Math.cbrt(((-1.0 * g) + (-1.0 * g)))) + Math.cbrt(((1.0 / (2.0 * a)) * (((h_m * h_m) / g) * -0.5)));
	} else {
		tmp = Math.cbrt((g / a)) * -1.0;
	}
	return tmp;
}

Julia

h_m = abs(h)
function code(g, h_m, a)
	tmp = 0.0
	if (h_m <= 4.5e+153)
		tmp = Float64(Float64(cbrt(Float64(0.5 * (a ^ -1.0))) * cbrt(Float64(Float64(-1.0 * g) + Float64(-1.0 * g)))) + cbrt(Float64(Float64(1.0 / Float64(2.0 * a)) * Float64(Float64(Float64(h_m * h_m) / g) * -0.5))));
	else
		tmp = Float64(cbrt(Float64(g / a)) * -1.0);
	end
	return tmp
end

Wolfram

h_m = N[Abs[h], $MachinePrecision]
code[g_, h$95$m_, a_] := If[LessEqual[h$95$m, 4.5e+153], N[(N[(N[Power[N[(0.5 * N[Power[a, -1.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[(-1.0 * g), $MachinePrecision] + N[(-1.0 * g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(h$95$m * h$95$m), $MachinePrecision] / g), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision] * -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 4.5000000000000001e153

   1. Initial program 45.6%

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

   3. Taylor expanded in g around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 23.3

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

   5. Applied rewrites 23.3%

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

   6. Taylor expanded in g around -inf

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

      1. lift-*.f64 76.4

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

   8. Applied rewrites 76.4%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. cbrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. inv-pow N/A

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

      9. unpow-prod-down N/A

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

      10. metadata-eval N/A

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

      11. inv-pow N/A

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

      12. lower-*.f64 N/A

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

      13. inv-pow N/A

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

      14. lift-pow.f64 N/A

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

      15. lower-cbrt.f64 91.2

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

   10. Applied rewrites 91.2%

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

   if 4.5000000000000001e153 < h

   1. Initial program 0.0%

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

   3. Taylor expanded in g around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cbrt-unprod N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. pow1/3 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 21.6

         \[\leadsto \left({\left(\frac{g}{a}\right)}^{0.3333333333333333} \cdot 1\right) \cdot -1 \]

   5. Applied rewrites 21.6%

      \[\leadsto \color{blue}{\left({\left(\frac{g}{a}\right)}^{0.3333333333333333} \cdot 1\right) \cdot -1} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow1/3 N/A

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

      4. lower-cbrt.f64 N/A

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

      5. lift-/.f64 42.5

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

   7. Applied rewrites 42.5%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 4.5 \cdot 10^{+153}:\\ \;\;\;\;\sqrt[3]{0.5 \cdot {a}^{-1}} \cdot \sqrt[3]{-1 \cdot g + -1 \cdot g} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{h \cdot h}{g} \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{g}{a}} \cdot -1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.7% accurate, N/A× speedup

Math

\[\begin{array}{l} h_m = \left|h\right| \\ \begin{array}{l} \mathbf{if}\;g \leq 9 \cdot 10^{-206}:\\ \;\;\;\;\sqrt[3]{\frac{g}{a}} \cdot -1\\ \mathbf{else}:\\ \;\;\;\;\frac{{g}^{0.3333333333333333}}{\sqrt[3]{a}} \cdot -1\\ \end{array} \end{array} \]

FPCore

h_m = (fabs.f64 h)
(FPCore (g h_m a)
 :precision binary64
 (if (<= g 9e-206)
   (* (cbrt (/ g a)) -1.0)
   (* (/ (pow g 0.3333333333333333) (cbrt a)) -1.0)))

C

h_m = fabs(h);
double code(double g, double h_m, double a) {
	double tmp;
	if (g <= 9e-206) {
		tmp = cbrt((g / a)) * -1.0;
	} else {
		tmp = (pow(g, 0.3333333333333333) / cbrt(a)) * -1.0;
	}
	return tmp;
}

Java

h_m = Math.abs(h);
public static double code(double g, double h_m, double a) {
	double tmp;
	if (g <= 9e-206) {
		tmp = Math.cbrt((g / a)) * -1.0;
	} else {
		tmp = (Math.pow(g, 0.3333333333333333) / Math.cbrt(a)) * -1.0;
	}
	return tmp;
}

Julia

h_m = abs(h)
function code(g, h_m, a)
	tmp = 0.0
	if (g <= 9e-206)
		tmp = Float64(cbrt(Float64(g / a)) * -1.0);
	else
		tmp = Float64(Float64((g ^ 0.3333333333333333) / cbrt(a)) * -1.0);
	end
	return tmp
end

Wolfram

h_m = N[Abs[h], $MachinePrecision]
code[g_, h$95$m_, a_] := If[LessEqual[g, 9e-206], N[(N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision] * -1.0), $MachinePrecision], N[(N[(N[Power[g, 0.3333333333333333], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] * -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if g < 8.9999999999999996e-206

   1. Initial program 37.7%

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

   3. Taylor expanded in g around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cbrt-unprod N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. pow1/3 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 35.7

         \[\leadsto \left({\left(\frac{g}{a}\right)}^{0.3333333333333333} \cdot 1\right) \cdot -1 \]

   5. Applied rewrites 35.7%

      \[\leadsto \color{blue}{\left({\left(\frac{g}{a}\right)}^{0.3333333333333333} \cdot 1\right) \cdot -1} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow1/3 N/A

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

      4. lower-cbrt.f64 N/A

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

      5. lift-/.f64 71.7

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

   7. Applied rewrites 71.7%

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

   if 8.9999999999999996e-206 < g

   1. Initial program 47.3%

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

   3. Taylor expanded in g around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cbrt-unprod N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. pow1/3 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 40.4

         \[\leadsto \left({\left(\frac{g}{a}\right)}^{0.3333333333333333} \cdot 1\right) \cdot -1 \]

   5. Applied rewrites 40.4%

      \[\leadsto \color{blue}{\left({\left(\frac{g}{a}\right)}^{0.3333333333333333} \cdot 1\right) \cdot -1} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow1/3 N/A

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

      4. cbrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. pow1/3 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cbrt.f64 90.2

         \[\leadsto \left(\frac{{g}^{0.3333333333333333}}{\sqrt[3]{a}} \cdot 1\right) \cdot -1 \]

   7. Applied rewrites 90.2%

      \[\leadsto \left(\frac{{g}^{0.3333333333333333}}{\sqrt[3]{a}} \cdot 1\right) \cdot -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq 9 \cdot 10^{-206}:\\ \;\;\;\;\sqrt[3]{\frac{g}{a}} \cdot -1\\ \mathbf{else}:\\ \;\;\;\;\frac{{g}^{0.3333333333333333}}{\sqrt[3]{a}} \cdot -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.1% accurate, N/A× speedup

Math

\[\begin{array}{l} h_m = \left|h\right| \\ \sqrt[3]{\frac{g}{a}} \cdot -1 \end{array} \]

FPCore

h_m = (fabs.f64 h)
(FPCore (g h_m a) :precision binary64 (* (cbrt (/ g a)) -1.0))

C

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

Java

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

Julia

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

Wolfram

h_m = N[Abs[h], $MachinePrecision]
code[g_, h$95$m_, a_] := N[(N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision] * -1.0), $MachinePrecision]

Derivation

1. Initial program 42.6%

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

3. Taylor expanded in g around -inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. cbrt-unprod N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. lower-*.f64 N/A

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

   7. pow1/3 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-/.f64 38.1

      \[\leadsto \left({\left(\frac{g}{a}\right)}^{0.3333333333333333} \cdot 1\right) \cdot -1 \]

5. Applied rewrites 38.1%

   \[\leadsto \color{blue}{\left({\left(\frac{g}{a}\right)}^{0.3333333333333333} \cdot 1\right) \cdot -1} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow1/3 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lift-/.f64 75.4

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

7. Applied rewrites 75.4%

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

8. Final simplification 75.4%

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

Alternative 4: 22.3% accurate, N/A× speedup

Math

\[\begin{array}{l} h_m = \left|h\right| \\ {\left(e^{0.3333333333333333}\right)}^{\left(\log g - \log a\right)} \cdot -1 \end{array} \]

FPCore

h_m = (fabs.f64 h)
(FPCore (g h_m a)
 :precision binary64
 (* (pow (exp 0.3333333333333333) (- (log g) (log a))) -1.0))

C

h_m = fabs(h);
double code(double g, double h_m, double a) {
	return pow(exp(0.3333333333333333), (log(g) - log(a))) * -1.0;
}

Fortran

h_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(g, h_m, a)
use fmin_fmax_functions
    real(8), intent (in) :: g
    real(8), intent (in) :: h_m
    real(8), intent (in) :: a
    code = (exp(0.3333333333333333d0) ** (log(g) - log(a))) * (-1.0d0)
end function

Java

h_m = Math.abs(h);
public static double code(double g, double h_m, double a) {
	return Math.pow(Math.exp(0.3333333333333333), (Math.log(g) - Math.log(a))) * -1.0;
}

Python

h_m = math.fabs(h)
def code(g, h_m, a):
	return math.pow(math.exp(0.3333333333333333), (math.log(g) - math.log(a))) * -1.0

Julia

h_m = abs(h)
function code(g, h_m, a)
	return Float64((exp(0.3333333333333333) ^ Float64(log(g) - log(a))) * -1.0)
end

MATLAB

h_m = abs(h);
function tmp = code(g, h_m, a)
	tmp = (exp(0.3333333333333333) ^ (log(g) - log(a))) * -1.0;
end

Wolfram

h_m = N[Abs[h], $MachinePrecision]
code[g_, h$95$m_, a_] := N[(N[Power[N[Exp[0.3333333333333333], $MachinePrecision], N[(N[Log[g], $MachinePrecision] - N[Log[a], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -1.0), $MachinePrecision]

Derivation

1. Initial program 42.6%

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

3. Taylor expanded in g around -inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. cbrt-unprod N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. lower-*.f64 N/A

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

   7. pow1/3 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-/.f64 38.1

      \[\leadsto \left({\left(\frac{g}{a}\right)}^{0.3333333333333333} \cdot 1\right) \cdot -1 \]

5. Applied rewrites 38.1%

   \[\leadsto \color{blue}{\left({\left(\frac{g}{a}\right)}^{0.3333333333333333} \cdot 1\right) \cdot -1} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow1/3 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lift-/.f64 75.4

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

7. Applied rewrites 75.4%

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

   1. lift-/.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. pow1/3 N/A

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

   4. exp-to-pow N/A

      \[\leadsto \left(e^{\log \left(\frac{g}{a}\right) \cdot \frac{1}{3}} \cdot 1\right) \cdot -1 \]

   5. lift-log.f64 N/A

      \[\leadsto \left(e^{\log \left(\frac{g}{a}\right) \cdot \frac{1}{3}} \cdot 1\right) \cdot -1 \]

   6. lift-/.f64 N/A

      \[\leadsto \left(e^{\log \left(\frac{g}{a}\right) \cdot \frac{1}{3}} \cdot 1\right) \cdot -1 \]

   7. lift-*.f64 N/A

      \[\leadsto \left(e^{\log \left(\frac{g}{a}\right) \cdot \frac{1}{3}} \cdot 1\right) \cdot -1 \]

   8. lift-exp.f64 38.1

      \[\leadsto \left(e^{\log \left(\frac{g}{a}\right) \cdot 0.3333333333333333} \cdot 1\right) \cdot -1 \]

   9. lift-/.f64 N/A

      \[\leadsto \left(e^{\log \left(\frac{g}{a}\right) \cdot \frac{1}{3}} \cdot 1\right) \cdot -1 \]

   10. lift-log.f64 N/A

       \[\leadsto \left(e^{\log \left(\frac{g}{a}\right) \cdot \frac{1}{3}} \cdot 1\right) \cdot -1 \]

   11. log-div N/A

       \[\leadsto \left(e^{\left(\log g - \log a\right) \cdot \frac{1}{3}} \cdot 1\right) \cdot -1 \]

   12. lower--.f64 N/A

       \[\leadsto \left(e^{\left(\log g - \log a\right) \cdot \frac{1}{3}} \cdot 1\right) \cdot -1 \]

   13. lower-log.f64 N/A

       \[\leadsto \left(e^{\left(\log g - \log a\right) \cdot \frac{1}{3}} \cdot 1\right) \cdot -1 \]

   14. lower-log.f64 23.0

       \[\leadsto \left(e^{\left(\log g - \log a\right) \cdot 0.3333333333333333} \cdot 1\right) \cdot -1 \]

9. Applied rewrites 23.0%

   \[\leadsto \left(e^{\left(\log g - \log a\right) \cdot 0.3333333333333333} \cdot 1\right) \cdot -1 \]

10. Taylor expanded in g around inf

    \[\leadsto \left(e^{\frac{1}{3} \cdot \left(-1 \cdot \log \left(\frac{1}{g}\right) - \log a\right)} \cdot 1\right) \cdot -1 \]
11. Step-by-step derivation

    1. exp-prod N/A

       \[\leadsto \left({\left(e^{\frac{1}{3}}\right)}^{\left(-1 \cdot \log \left(\frac{1}{g}\right) - \log a\right)} \cdot 1\right) \cdot -1 \]

    2. lower-pow.f64 N/A

       \[\leadsto \left({\left(e^{\frac{1}{3}}\right)}^{\left(-1 \cdot \log \left(\frac{1}{g}\right) - \log a\right)} \cdot 1\right) \cdot -1 \]

    3. lower-exp.f64 N/A

       \[\leadsto \left({\left(e^{\frac{1}{3}}\right)}^{\left(-1 \cdot \log \left(\frac{1}{g}\right) - \log a\right)} \cdot 1\right) \cdot -1 \]

    4. lower--.f64 N/A

       \[\leadsto \left({\left(e^{\frac{1}{3}}\right)}^{\left(-1 \cdot \log \left(\frac{1}{g}\right) - \log a\right)} \cdot 1\right) \cdot -1 \]

    5. lower-*.f64 N/A

       \[\leadsto \left({\left(e^{\frac{1}{3}}\right)}^{\left(-1 \cdot \log \left(\frac{1}{g}\right) - \log a\right)} \cdot 1\right) \cdot -1 \]

    6. log-div N/A

       \[\leadsto \left({\left(e^{\frac{1}{3}}\right)}^{\left(-1 \cdot \left(\log 1 - \log g\right) - \log a\right)} \cdot 1\right) \cdot -1 \]

    7. metadata-eval N/A

       \[\leadsto \left({\left(e^{\frac{1}{3}}\right)}^{\left(-1 \cdot \left(0 - \log g\right) - \log a\right)} \cdot 1\right) \cdot -1 \]

    8. lower--.f64 N/A

       \[\leadsto \left({\left(e^{\frac{1}{3}}\right)}^{\left(-1 \cdot \left(0 - \log g\right) - \log a\right)} \cdot 1\right) \cdot -1 \]

    9. lift-log.f64 N/A

       \[\leadsto \left({\left(e^{\frac{1}{3}}\right)}^{\left(-1 \cdot \left(0 - \log g\right) - \log a\right)} \cdot 1\right) \cdot -1 \]

    10. lift-log.f64 23.1

        \[\leadsto \left({\left(e^{0.3333333333333333}\right)}^{\left(-1 \cdot \left(0 - \log g\right) - \log a\right)} \cdot 1\right) \cdot -1 \]

12. Applied rewrites 23.1%

    \[\leadsto \left({\left(e^{0.3333333333333333}\right)}^{\left(-1 \cdot \left(0 - \log g\right) - \log a\right)} \cdot 1\right) \cdot -1 \]

13. Final simplification 23.1%

    \[\leadsto {\left(e^{0.3333333333333333}\right)}^{\left(\log g - \log a\right)} \cdot -1 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(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))))))))