2-ancestry mixing, positive discriminant

Percentage Accurate: 43.3% → 95.0%

Time: 41.1s

Alternatives: 9

Speedup: 4.2×

• could not determine a ground truth

  1. g = -1.6017504128875872e+195
  2. h = -3.775352292797031e-100
  3. a = 1.3523838023818391e+49

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: 43.3% 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: 95.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;g \leq 6.8 \cdot 10^{+77}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot \left(h \cdot \frac{h}{g}\right)\right)} + \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right) \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{g \cdot -2} + \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= g 6.8e+77)
   (+
    (cbrt (* (/ 0.5 a) (* -0.5 (* h (/ h g)))))
    (* (* (cbrt g) (cbrt (/ 1.0 a))) (* (cbrt -0.5) (cbrt 2.0))))
   (+ (* (cbrt (/ 0.5 a)) (cbrt (* g -2.0))) (cbrt (* (- g g) (/ -0.5 a))))))

C

double code(double g, double h, double a) {
	double tmp;
	if (g <= 6.8e+77) {
		tmp = cbrt(((0.5 / a) * (-0.5 * (h * (h / g))))) + ((cbrt(g) * cbrt((1.0 / a))) * (cbrt(-0.5) * cbrt(2.0)));
	} else {
		tmp = (cbrt((0.5 / a)) * cbrt((g * -2.0))) + cbrt(((g - g) * (-0.5 / a)));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if (g <= 6.8e+77) {
		tmp = Math.cbrt(((0.5 / a) * (-0.5 * (h * (h / g))))) + ((Math.cbrt(g) * Math.cbrt((1.0 / a))) * (Math.cbrt(-0.5) * Math.cbrt(2.0)));
	} else {
		tmp = (Math.cbrt((0.5 / a)) * Math.cbrt((g * -2.0))) + Math.cbrt(((g - g) * (-0.5 / a)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (g <= 6.8e+77)
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(-0.5 * Float64(h * Float64(h / g))))) + Float64(Float64(cbrt(g) * cbrt(Float64(1.0 / a))) * Float64(cbrt(-0.5) * cbrt(2.0))));
	else
		tmp = Float64(Float64(cbrt(Float64(0.5 / a)) * cbrt(Float64(g * -2.0))) + cbrt(Float64(Float64(g - g) * Float64(-0.5 / a))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[g, 6.8e+77], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(-0.5 * N[(h * N[(h / g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[(N[Power[g, 1/3], $MachinePrecision] * N[Power[N[(1.0 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(g * -2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(g - g), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if g < 6.79999999999999993e77

   1. Initial program 66.8%

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

   3. Simplified 66.8%

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

   5. Taylor expanded in g around inf 62.0%

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

   6. Taylor expanded in g around inf 73.6%

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

      1. pow2 73.6%

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

      2. associate-/l* 81.3%

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

   8. Applied egg-rr 81.3%

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

      1. pow1/3 42.2%

         \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot \left(h \cdot \frac{h}{g}\right)\right)} + \color{blue}{{\left(\frac{g}{a}\right)}^{0.3333333333333333}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \]

      2. div-inv 42.2%

         \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot \left(h \cdot \frac{h}{g}\right)\right)} + {\color{blue}{\left(g \cdot \frac{1}{a}\right)}}^{0.3333333333333333} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \]

      3. unpow-prod-down 40.8%

         \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot \left(h \cdot \frac{h}{g}\right)\right)} + \color{blue}{\left({g}^{0.3333333333333333} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right)} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \]

      4. pow1/3 48.4%

         \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot \left(h \cdot \frac{h}{g}\right)\right)} + \left(\color{blue}{\sqrt[3]{g}} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right) \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \]

   10. Applied egg-rr 48.4%

       \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot \left(h \cdot \frac{h}{g}\right)\right)} + \color{blue}{\left(\sqrt[3]{g} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right)} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \]
   11. Step-by-step derivation

       1. unpow1/3 92.8%

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

   12. Simplified 92.8%

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

   if 6.79999999999999993e77 < g

   1. Initial program 19.7%

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

   3. Simplified 19.7%

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

   5. Taylor expanded in g around -inf 6.4%

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

      1. *-commutative 6.4%

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

   7. Simplified 6.4%

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

   8. Taylor expanded in g around -inf 60.0%

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

      1. neg-mul-1 60.0%

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

   10. Simplified 60.0%

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

       1. cbrt-prod 96.4%

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

   12. Applied egg-rr 96.4%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq 6.8 \cdot 10^{+77}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot \left(h \cdot \frac{h}{g}\right)\right)} + \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right) \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{g \cdot -2} + \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;g \leq 2.5 \cdot 10^{+77}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot \left(h \cdot \frac{h}{g}\right)\right)} + \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot \frac{\sqrt[3]{g}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{g \cdot -2} + \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= g 2.5e+77)
   (+
    (cbrt (* (/ 0.5 a) (* -0.5 (* h (/ h g)))))
    (* (* (cbrt -0.5) (cbrt 2.0)) (/ (cbrt g) (cbrt a))))
   (+ (* (cbrt (/ 0.5 a)) (cbrt (* g -2.0))) (cbrt (* (- g g) (/ -0.5 a))))))

C

double code(double g, double h, double a) {
	double tmp;
	if (g <= 2.5e+77) {
		tmp = cbrt(((0.5 / a) * (-0.5 * (h * (h / g))))) + ((cbrt(-0.5) * cbrt(2.0)) * (cbrt(g) / cbrt(a)));
	} else {
		tmp = (cbrt((0.5 / a)) * cbrt((g * -2.0))) + cbrt(((g - g) * (-0.5 / a)));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if (g <= 2.5e+77) {
		tmp = Math.cbrt(((0.5 / a) * (-0.5 * (h * (h / g))))) + ((Math.cbrt(-0.5) * Math.cbrt(2.0)) * (Math.cbrt(g) / Math.cbrt(a)));
	} else {
		tmp = (Math.cbrt((0.5 / a)) * Math.cbrt((g * -2.0))) + Math.cbrt(((g - g) * (-0.5 / a)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (g <= 2.5e+77)
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(-0.5 * Float64(h * Float64(h / g))))) + Float64(Float64(cbrt(-0.5) * cbrt(2.0)) * Float64(cbrt(g) / cbrt(a))));
	else
		tmp = Float64(Float64(cbrt(Float64(0.5 / a)) * cbrt(Float64(g * -2.0))) + cbrt(Float64(Float64(g - g) * Float64(-0.5 / a))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[g, 2.5e+77], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(-0.5 * N[(h * N[(h / g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[Power[g, 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(g * -2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(g - g), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if g < 2.50000000000000002e77

   1. Initial program 66.8%

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

   3. Simplified 66.8%

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

   5. Taylor expanded in g around inf 62.0%

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

   6. Taylor expanded in g around inf 73.6%

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

      1. pow2 73.6%

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

      2. associate-/l* 81.3%

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

   8. Applied egg-rr 81.3%

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

      1. cbrt-div 92.5%

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

      2. div-inv 92.5%

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

   10. Applied egg-rr 92.5%

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

       1. associate-*r/ 92.5%

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

       2. *-rgt-identity 92.5%

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

   12. Simplified 92.5%

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

   if 2.50000000000000002e77 < g

   1. Initial program 19.7%

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

   3. Simplified 19.7%

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

   5. Taylor expanded in g around -inf 6.4%

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

      1. *-commutative 6.4%

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

   7. Simplified 6.4%

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

   8. Taylor expanded in g around -inf 60.0%

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

      1. neg-mul-1 60.0%

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

   10. Simplified 60.0%

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

       1. cbrt-prod 96.4%

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

   12. Applied egg-rr 96.4%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq 2.5 \cdot 10^{+77}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot \left(h \cdot \frac{h}{g}\right)\right)} + \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot \frac{\sqrt[3]{g}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{g \cdot -2} + \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{-122}:\\ \;\;\;\;-\sqrt[3]{\frac{g}{a}}\\ \mathbf{elif}\;a \leq 5.1 \cdot 10^{-76}:\\ \;\;\;\;\frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{-1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= a -3.8e-122)
   (- (cbrt (/ g a)))
   (if (<= a 5.1e-76)
     (+ (/ (cbrt (- g)) (cbrt a)) (cbrt -1.0))
     (+ (cbrt (* (- g g) (/ -0.5 a))) (cbrt (* (/ 0.5 a) (* g -2.0)))))))

C

double code(double g, double h, double a) {
	double tmp;
	if (a <= -3.8e-122) {
		tmp = -cbrt((g / a));
	} else if (a <= 5.1e-76) {
		tmp = (cbrt(-g) / cbrt(a)) + cbrt(-1.0);
	} else {
		tmp = cbrt(((g - g) * (-0.5 / a))) + cbrt(((0.5 / a) * (g * -2.0)));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if (a <= -3.8e-122) {
		tmp = -Math.cbrt((g / a));
	} else if (a <= 5.1e-76) {
		tmp = (Math.cbrt(-g) / Math.cbrt(a)) + Math.cbrt(-1.0);
	} else {
		tmp = Math.cbrt(((g - g) * (-0.5 / a))) + Math.cbrt(((0.5 / a) * (g * -2.0)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (a <= -3.8e-122)
		tmp = Float64(-cbrt(Float64(g / a)));
	elseif (a <= 5.1e-76)
		tmp = Float64(Float64(cbrt(Float64(-g)) / cbrt(a)) + cbrt(-1.0));
	else
		tmp = Float64(cbrt(Float64(Float64(g - g) * Float64(-0.5 / a))) + cbrt(Float64(Float64(0.5 / a) * Float64(g * -2.0))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[a, -3.8e-122], (-N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision]), If[LessEqual[a, 5.1e-76], N[(N[(N[Power[(-g), 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[-1.0, 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(g - g), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(g * -2.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.8000000000000001e-122

   1. Initial program 47.8%

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

   3. Simplified 47.8%

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

   5. Taylor expanded in g around -inf 12.8%

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

      1. *-commutative 12.8%

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

   7. Simplified 12.8%

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

   8. Taylor expanded in g around inf 17.9%

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

   10. Applied egg-rr 0.0%

       \[\leadsto \sqrt[3]{\color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{0}{0}}{a}\right)} - 1}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   12. Simplified 41.9%

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

   13. Taylor expanded in g around -inf 89.2%

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

       1. mul-1-neg 89.2%

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

   15. Simplified 89.2%

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

   if -3.8000000000000001e-122 < a < 5.09999999999999986e-76

   1. Initial program 33.6%

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

   3. Simplified 33.6%

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

   5. Taylor expanded in g around -inf 11.6%

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

      1. *-commutative 11.6%

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

   7. Simplified 11.6%

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

   8. Taylor expanded in g around inf 10.7%

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

   10. Applied egg-rr 0.0%

       \[\leadsto \sqrt[3]{\color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{0}{0}}{a}\right)} - 1}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   12. Simplified 36.0%

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

       1. add-sqr-sqrt 12.9%

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

       2. sqrt-unprod 4.9%

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

       3. *-commutative 4.9%

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

       4. *-commutative 4.9%

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

       5. swap-sqr 3.8%

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

       6. frac-times 3.8%

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

       7. metadata-eval 3.8%

          \[\leadsto \sqrt[3]{-1} + \sqrt[3]{\sqrt{\frac{\color{blue}{0.25}}{a \cdot a} \cdot \left(\left(g + g\right) \cdot \left(g + g\right)\right)}} \]

       8. metadata-eval 3.8%

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

       9. frac-times 3.8%

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

       10. count-2 3.8%

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

       11. count-2 3.8%

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

       12. swap-sqr 3.8%

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

       13. metadata-eval 3.8%

           \[\leadsto \sqrt[3]{-1} + \sqrt[3]{\sqrt{\left(\frac{0.5}{a} \cdot \frac{0.5}{a}\right) \cdot \left(\color{blue}{4} \cdot \left(g \cdot g\right)\right)}} \]

       14. metadata-eval 3.8%

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

       15. swap-sqr 3.8%

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

       16. *-commutative 3.8%

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

       17. *-commutative 3.8%

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

       18. swap-sqr 4.9%

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

       19. sqrt-unprod 12.9%

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

       20. add-sqr-sqrt 36.0%

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

   14. Applied egg-rr 88.8%

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

   if 5.09999999999999986e-76 < a

   1. Initial program 37.0%

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

   3. Simplified 37.0%

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

   5. Taylor expanded in g around -inf 9.5%

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

      1. *-commutative 9.5%

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

   7. Simplified 9.5%

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

   8. Taylor expanded in g around -inf 76.1%

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

      1. neg-mul-1 76.1%

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

   10. Simplified 76.1%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{-122}:\\ \;\;\;\;-\sqrt[3]{\frac{g}{a}}\\ \mathbf{elif}\;a \leq 5.1 \cdot 10^{-76}:\\ \;\;\;\;\frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{-1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 39.0%

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

3. Simplified 39.0%

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

5. Taylor expanded in g around -inf 11.4%

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

   1. *-commutative 11.4%

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

7. Simplified 11.4%

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

8. Taylor expanded in g around -inf 65.7%

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

   1. neg-mul-1 65.7%

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

10. Simplified 65.7%

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

    1. cbrt-prod 92.2%

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

12. Applied egg-rr 92.2%

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

13. Final simplification 92.2%

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

Alternative 5: 93.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 39.0%

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

3. Simplified 39.0%

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

5. Taylor expanded in g around -inf 11.4%

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

   1. *-commutative 11.4%

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

7. Simplified 11.4%

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

8. Taylor expanded in g around -inf 65.7%

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

   1. neg-mul-1 65.7%

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

10. Simplified 65.7%

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

    1. associate-*l/ 65.6%

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

    2. cbrt-div 92.2%

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

    3. *-commutative 92.2%

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

    4. associate-*r* 92.2%

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

    5. metadata-eval 92.2%

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

    6. neg-mul-1 92.2%

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

12. Applied egg-rr 92.2%

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

13. Final simplification 92.2%

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

Alternative 6: 71.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 39.0%

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

3. Simplified 39.0%

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

5. Taylor expanded in g around -inf 11.4%

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

   1. *-commutative 11.4%

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

7. Simplified 11.4%

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

8. Taylor expanded in g around -inf 65.7%

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

   1. neg-mul-1 65.7%

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

10. Simplified 65.7%

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

11. Final simplification 65.7%

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

Alternative 7: 71.6% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 39.0%

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

3. Simplified 39.0%

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

5. Taylor expanded in g around -inf 11.4%

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

   1. *-commutative 11.4%

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

7. Simplified 11.4%

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

8. Taylor expanded in g around inf 14.5%

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

10. Applied egg-rr 0.0%

    \[\leadsto \sqrt[3]{\color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{0}{0}}{a}\right)} - 1}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

12. Simplified 38.0%

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

13. Taylor expanded in g around -inf 65.6%

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

    1. mul-1-neg 65.6%

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

15. Simplified 65.6%

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

16. Final simplification 65.6%

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

Alternative 8: 7.0% accurate, 72.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-310}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a) :precision binary64 (if (<= a -5e-310) 1.0 -1.0))

C

double code(double g, double h, double a) {
	double tmp;
	if (a <= -5e-310) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(g, h, a)
    real(8), intent (in) :: g
    real(8), intent (in) :: h
    real(8), intent (in) :: a
    real(8) :: tmp
    if (a <= (-5d-310)) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double g, double h, double a) {
	double tmp;
	if (a <= -5e-310) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(g, h, a):
	tmp = 0
	if a <= -5e-310:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(g, h, a)
	tmp = 0.0
	if (a <= -5e-310)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(g, h, a)
	tmp = 0.0;
	if (a <= -5e-310)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[g_, h_, a_] := If[LessEqual[a, -5e-310], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if a < -4.999999999999985e-310

   1. Initial program 41.8%

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

   3. Simplified 41.8%

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

   5. Taylor expanded in g around -inf 11.9%

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

      1. *-commutative 11.9%

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

   7. Simplified 11.9%

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

   8. Taylor expanded in g around inf 15.0%

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

   10. Applied egg-rr 0.0%

       \[\leadsto \sqrt[3]{\color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{0}{0}}{a}\right)} - 1}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   12. Simplified 38.1%

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

   14. Applied egg-rr 7.0%

       \[\leadsto \color{blue}{1} \]

   if -4.999999999999985e-310 < a

   1. Initial program 36.4%

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

   3. Simplified 36.4%

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

   5. Taylor expanded in g around -inf 10.9%

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

      1. *-commutative 10.9%

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

   7. Simplified 10.9%

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

   8. Taylor expanded in g around inf 14.1%

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

   10. Applied egg-rr 0.0%

       \[\leadsto \sqrt[3]{\color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{0}{0}}{a}\right)} - 1}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   12. Simplified 37.9%

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

   14. Applied egg-rr 6.8%

       \[\leadsto \color{blue}{-1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 6.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-310}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 4.5% accurate, 433.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (g h a) :precision binary64 -1.0)

C

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

Fortran

real(8) function code(g, h, a)
    real(8), intent (in) :: g
    real(8), intent (in) :: h
    real(8), intent (in) :: a
    code = -1.0d0
end function

Java

public static double code(double g, double h, double a) {
	return -1.0;
}

Python

def code(g, h, a):
	return -1.0

Julia

function code(g, h, a)
	return -1.0
end

MATLAB

function tmp = code(g, h, a)
	tmp = -1.0;
end

Wolfram

code[g_, h_, a_] := -1.0

Derivation

1. Initial program 39.0%

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

3. Simplified 39.0%

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

5. Taylor expanded in g around -inf 11.4%

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

   1. *-commutative 11.4%

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

7. Simplified 11.4%

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

8. Taylor expanded in g around inf 14.5%

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

10. Applied egg-rr 0.0%

    \[\leadsto \sqrt[3]{\color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{0}{0}}{a}\right)} - 1}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

12. Simplified 38.0%

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

14. Applied egg-rr 4.5%

    \[\leadsto \color{blue}{-1} \]

15. Final simplification 4.5%

    \[\leadsto -1 \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024069 
(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))))))))