2-ancestry mixing, positive discriminant

Percentage Accurate: 43.5% → 95.7%

Time: 18.2s

Alternatives: 10

Speedup: 2.6×

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: 43.5% 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: 95.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 43.3%

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

4. Applied egg-rr 45.6%

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

5. Taylor expanded in g around inf

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

7. Simplified 25.4%

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

8. Taylor expanded in g around inf

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

10. Simplified 95.1%

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

11. Taylor expanded in a around 0

    \[\leadsto \color{blue}{0} + \frac{\sqrt[3]{g + g}}{\sqrt[3]{a \cdot -2}} \]
12. Step-by-step derivation

13. Simplified 95.1%

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

14. Final simplification 95.1%

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

Alternative 2: 88.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+116}:\\ \;\;\;\;{\left(-a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+72}:\\ \;\;\;\;-\sqrt[3]{g \cdot \frac{1}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-g} \cdot {a}^{-0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* a 2.0))))
   (if (<= t_0 -4e+116)
     (* (pow (- a) -0.3333333333333333) (cbrt g))
     (if (<= t_0 5e+72)
       (- (cbrt (* g (/ 1.0 a))))
       (* (cbrt (- g)) (pow a -0.3333333333333333))))))

C

double code(double g, double h, double a) {
	double t_0 = 1.0 / (a * 2.0);
	double tmp;
	if (t_0 <= -4e+116) {
		tmp = pow(-a, -0.3333333333333333) * cbrt(g);
	} else if (t_0 <= 5e+72) {
		tmp = -cbrt((g * (1.0 / a)));
	} else {
		tmp = cbrt(-g) * pow(a, -0.3333333333333333);
	}
	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 <= -4e+116) {
		tmp = Math.pow(-a, -0.3333333333333333) * Math.cbrt(g);
	} else if (t_0 <= 5e+72) {
		tmp = -Math.cbrt((g * (1.0 / a)));
	} else {
		tmp = Math.cbrt(-g) * Math.pow(a, -0.3333333333333333);
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64(1.0 / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -4e+116)
		tmp = Float64((Float64(-a) ^ -0.3333333333333333) * cbrt(g));
	elseif (t_0 <= 5e+72)
		tmp = Float64(-cbrt(Float64(g * Float64(1.0 / a))));
	else
		tmp = Float64(cbrt(Float64(-g)) * (a ^ -0.3333333333333333));
	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, -4e+116], N[(N[Power[(-a), -0.3333333333333333], $MachinePrecision] * N[Power[g, 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+72], (-N[Power[N[(g * N[(1.0 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), N[(N[Power[(-g), 1/3], $MachinePrecision] * N[Power[a, -0.3333333333333333], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) < -4.00000000000000006e116

   1. Initial program 42.8%

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

   3. Taylor expanded in g around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. neg-lowering-neg.f64 22.9

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

   5. Simplified 22.9%

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

   6. Taylor expanded in g around inf

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

      1. *-lowering-*.f64 N/A

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

      2. cbrt-lowering-cbrt.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. cbrt-lowering-cbrt.f64 48.1

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

   8. Simplified 48.1%

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

      1. *-commutative N/A

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

      2. cbrt-unprod N/A

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

      3. neg-mul-1 N/A

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

      4. distribute-frac-neg2 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. cbrt-prod N/A

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

      8. pow1/3 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. inv-pow N/A

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

      11. pow-pow N/A

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

      12. pow-lowering-pow.f64 N/A

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

      13. neg-lowering-neg.f64 N/A

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

      14. metadata-eval N/A

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

      15. cbrt-lowering-cbrt.f64 86.1

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

   10. Applied egg-rr 86.1%

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

   if -4.00000000000000006e116 < (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) < 4.99999999999999992e72

   1. Initial program 46.0%

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

   3. Taylor expanded in g around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. neg-lowering-neg.f64 31.0

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

   5. Simplified 31.0%

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

   6. Taylor expanded in g around inf

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

      1. *-lowering-*.f64 N/A

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

      2. cbrt-lowering-cbrt.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. cbrt-lowering-cbrt.f64 92.8

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

   8. Simplified 92.8%

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

   9. Taylor expanded in a around -inf

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

       1. rem-cube-cbrt N/A

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

       2. *-commutative N/A

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

       3. mul-1-neg N/A

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

       4. neg-lowering-neg.f64 N/A

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

       5. cbrt-lowering-cbrt.f64 N/A

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

       6. /-lowering-/.f64 92.8

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

   11. Simplified 92.8%

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

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

       3. *-lowering-*.f64 N/A

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

       4. /-lowering-/.f64 92.9

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

   13. Applied egg-rr 92.9%

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

   if 4.99999999999999992e72 < (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 31.4%

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

   3. Taylor expanded in g around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. neg-lowering-neg.f64 10.5

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

   5. Simplified 10.5%

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

   6. Taylor expanded in g around inf

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

      1. *-lowering-*.f64 N/A

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

      2. cbrt-lowering-cbrt.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. cbrt-lowering-cbrt.f64 38.9

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

   8. Simplified 38.9%

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

      1. *-commutative N/A

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

      2. cbrt-unprod N/A

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

      3. neg-mul-1 N/A

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

      4. div-inv N/A

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

      5. distribute-lft-neg-in N/A

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

      6. cbrt-prod N/A

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

      7. pow1/3 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. cbrt-lowering-cbrt.f64 N/A

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

      10. neg-lowering-neg.f64 N/A

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

      11. inv-pow N/A

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

      12. pow-pow N/A

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

      13. pow-lowering-pow.f64 N/A

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

      14. metadata-eval 85.6

          \[\leadsto \sqrt[3]{-g} \cdot {a}^{\color{blue}{-0.3333333333333333}} \]

   10. Applied egg-rr 85.6%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{a \cdot 2} \leq -4 \cdot 10^{+116}:\\ \;\;\;\;{\left(-a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g}\\ \mathbf{elif}\;\frac{1}{a \cdot 2} \leq 5 \cdot 10^{+72}:\\ \;\;\;\;-\sqrt[3]{g \cdot \frac{1}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-g} \cdot {a}^{-0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{a \cdot 2} \leq 5 \cdot 10^{+72}:\\ \;\;\;\;-\sqrt[3]{g \cdot \frac{1}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-g} \cdot {a}^{-0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= (/ 1.0 (* a 2.0)) 5e+72)
   (- (cbrt (* g (/ 1.0 a))))
   (* (cbrt (- g)) (pow a -0.3333333333333333))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((1.0 / (a * 2.0)) <= 5e+72) {
		tmp = -cbrt((g * (1.0 / a)));
	} else {
		tmp = cbrt(-g) * pow(a, -0.3333333333333333);
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if ((1.0 / (a * 2.0)) <= 5e+72) {
		tmp = -Math.cbrt((g * (1.0 / a)));
	} else {
		tmp = Math.cbrt(-g) * Math.pow(a, -0.3333333333333333);
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (Float64(1.0 / Float64(a * 2.0)) <= 5e+72)
		tmp = Float64(-cbrt(Float64(g * Float64(1.0 / a))));
	else
		tmp = Float64(cbrt(Float64(-g)) * (a ^ -0.3333333333333333));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[N[(1.0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], 5e+72], (-N[Power[N[(g * N[(1.0 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), N[(N[Power[(-g), 1/3], $MachinePrecision] * N[Power[a, -0.3333333333333333], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) < 4.99999999999999992e72

   1. Initial program 45.5%

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

   3. Taylor expanded in g around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. neg-lowering-neg.f64 29.6

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

   5. Simplified 29.6%

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

   6. Taylor expanded in g around inf

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

      1. *-lowering-*.f64 N/A

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

      2. cbrt-lowering-cbrt.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. cbrt-lowering-cbrt.f64 85.0

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

   8. Simplified 85.0%

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

   9. Taylor expanded in a around -inf

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

       1. rem-cube-cbrt N/A

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

       2. *-commutative N/A

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

       3. mul-1-neg N/A

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

       4. neg-lowering-neg.f64 N/A

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

       5. cbrt-lowering-cbrt.f64 N/A

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

       6. /-lowering-/.f64 85.0

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

   11. Simplified 85.0%

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

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

       3. *-lowering-*.f64 N/A

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

       4. /-lowering-/.f64 85.0

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

   13. Applied egg-rr 85.0%

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

   if 4.99999999999999992e72 < (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 31.4%

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

   3. Taylor expanded in g around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. neg-lowering-neg.f64 10.5

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

   5. Simplified 10.5%

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

   6. Taylor expanded in g around inf

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

      1. *-lowering-*.f64 N/A

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

      2. cbrt-lowering-cbrt.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. cbrt-lowering-cbrt.f64 38.9

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

   8. Simplified 38.9%

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

      1. *-commutative N/A

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

      2. cbrt-unprod N/A

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

      3. neg-mul-1 N/A

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

      4. div-inv N/A

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

      5. distribute-lft-neg-in N/A

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

      6. cbrt-prod N/A

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

      7. pow1/3 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. cbrt-lowering-cbrt.f64 N/A

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

      10. neg-lowering-neg.f64 N/A

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

      11. inv-pow N/A

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

      12. pow-pow N/A

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

      13. pow-lowering-pow.f64 N/A

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

      14. metadata-eval 85.6

          \[\leadsto \sqrt[3]{-g} \cdot {a}^{\color{blue}{-0.3333333333333333}} \]

   10. Applied egg-rr 85.6%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{a \cdot 2} \leq 5 \cdot 10^{+72}:\\ \;\;\;\;-\sqrt[3]{g \cdot \frac{1}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-g} \cdot {a}^{-0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 43.3%

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

3. Taylor expanded in g around inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. neg-lowering-neg.f64 26.6

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

5. Simplified 26.6%

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

6. Taylor expanded in g around inf

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

   1. *-lowering-*.f64 N/A

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

   2. cbrt-lowering-cbrt.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. cbrt-lowering-cbrt.f64 78.0

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

8. Simplified 78.0%

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

   1. *-commutative N/A

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

   2. cbrt-unprod N/A

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

   3. neg-mul-1 N/A

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

   4. distribute-frac-neg N/A

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

   5. cbrt-div N/A

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

   6. /-lowering-/.f64 N/A

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

   7. cbrt-lowering-cbrt.f64 N/A

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

   8. neg-lowering-neg.f64 N/A

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

   9. cbrt-lowering-cbrt.f64 95.0

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

10. Applied egg-rr 95.0%

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

Alternative 5: 73.6% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 43.3%

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

3. Taylor expanded in g around inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. neg-lowering-neg.f64 26.6

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

5. Simplified 26.6%

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

6. Taylor expanded in g around inf

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

   1. *-lowering-*.f64 N/A

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

   2. cbrt-lowering-cbrt.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. cbrt-lowering-cbrt.f64 78.0

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

8. Simplified 78.0%

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

9. Taylor expanded in a around -inf

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

    1. rem-cube-cbrt N/A

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

    2. *-commutative N/A

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

    3. mul-1-neg N/A

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

    4. neg-lowering-neg.f64 N/A

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

    5. cbrt-lowering-cbrt.f64 N/A

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

    6. /-lowering-/.f64 78.0

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

11. Simplified 78.0%

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

    1. clear-num N/A

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

    2. associate-/r/ N/A

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

    3. *-lowering-*.f64 N/A

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

    4. /-lowering-/.f64 78.0

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

13. Applied egg-rr 78.0%

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

14. Final simplification 78.0%

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

Alternative 6: 7.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq -4 \cdot 10^{-306}:\\ \;\;\;\;\sqrt[3]{g + g}\\ \mathbf{else}:\\ \;\;\;\;\left(-g\right) - g\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= (* a 2.0) -4e-306) (cbrt (+ g g)) (- (- g) g)))

C

double code(double g, double h, double a) {
	double tmp;
	if ((a * 2.0) <= -4e-306) {
		tmp = cbrt((g + g));
	} else {
		tmp = -g - g;
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if ((a * 2.0) <= -4e-306) {
		tmp = Math.cbrt((g + g));
	} else {
		tmp = -g - g;
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (Float64(a * 2.0) <= -4e-306)
		tmp = cbrt(Float64(g + g));
	else
		tmp = Float64(Float64(-g) - g);
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[N[(a * 2.0), $MachinePrecision], -4e-306], N[Power[N[(g + g), $MachinePrecision], 1/3], $MachinePrecision], N[((-g) - g), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) a) < -4.00000000000000011e-306

   1. Initial program 45.1%

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

   3. Taylor expanded in g around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. neg-lowering-neg.f64 31.3

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

   5. Simplified 31.3%

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

   6. Taylor expanded in g around -inf

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. neg-lowering-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. cbrt-lowering-cbrt.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. cbrt-lowering-cbrt.f64 N/A

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

      14. cbrt-lowering-cbrt.f64 N/A

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

      15. cbrt-lowering-cbrt.f64 N/A

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

      16. /-lowering-/.f64 N/A

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

      17. *-lowering-*.f64 N/A

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 9.6

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

   8. Simplified 9.6%

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

      1. cbrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-rgt-identity N/A

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

      5. pow1/3 N/A

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

      6. inv-pow N/A

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

      7. pow-pow N/A

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

      8. *-commutative N/A

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

      9. unpow-prod-down N/A

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

      10. pow-pow N/A

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

      11. inv-pow N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

   10. Applied egg-rr 0.0%

       \[\leadsto \left(-g\right) \cdot \color{blue}{\mathsf{fma}\left({\left(g \cdot g\right)}^{-0.3333333333333333}, {a}^{-0.3333333333333333}, {\left(g \cdot \left(g \cdot a\right)\right)}^{-0.3333333333333333}\right)} \]
   11. Step-by-step derivation

       1. distribute-lft-neg-out N/A

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

       2. neg-sub0 N/A

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

       3. associate-*r* N/A

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

       4. pow-prod-down N/A

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

       5. flip-+ N/A

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

       6. +-inverses N/A

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

       7. +-inverses N/A

          \[\leadsto 0 - g \cdot \frac{0}{\color{blue}{0}} \]

       8. associate-*r/ N/A

          \[\leadsto 0 - \color{blue}{\frac{g \cdot 0}{0}} \]

       9. +-inverses N/A

          \[\leadsto 0 - \frac{g \cdot \color{blue}{\left(g - g\right)}}{0} \]

       10. distribute-lft-out-- N/A

           \[\leadsto 0 - \frac{\color{blue}{g \cdot g - g \cdot g}}{0} \]

       11. +-inverses N/A

           \[\leadsto 0 - \frac{g \cdot g - g \cdot g}{\color{blue}{g - g}} \]

       12. flip-+ N/A

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

   12. Applied egg-rr 7.8%

       \[\leadsto \color{blue}{\sqrt[3]{g + g}} \]

   if -4.00000000000000011e-306 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 41.7%

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

   3. Taylor expanded in g around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. neg-lowering-neg.f64 22.6

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

   5. Simplified 22.6%

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

   6. Taylor expanded in g around -inf

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. neg-lowering-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. cbrt-lowering-cbrt.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. cbrt-lowering-cbrt.f64 N/A

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

      14. cbrt-lowering-cbrt.f64 N/A

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

      15. cbrt-lowering-cbrt.f64 N/A

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

      16. /-lowering-/.f64 N/A

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

      17. *-lowering-*.f64 N/A

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 8.0

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

   8. Simplified 8.0%

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

      1. cbrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-rgt-identity N/A

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

      5. pow1/3 N/A

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

      6. inv-pow N/A

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

      7. pow-pow N/A

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

      8. *-commutative N/A

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

      9. unpow-prod-down N/A

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

      10. pow-pow N/A

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

      11. inv-pow N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

   10. Applied egg-rr 28.4%

       \[\leadsto \left(-g\right) \cdot \color{blue}{\mathsf{fma}\left({\left(g \cdot g\right)}^{-0.3333333333333333}, {a}^{-0.3333333333333333}, {\left(g \cdot \left(g \cdot a\right)\right)}^{-0.3333333333333333}\right)} \]
   11. Step-by-step derivation

       1. distribute-lft-neg-out N/A

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

       2. neg-sub0 N/A

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

       3. associate-*r* N/A

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

       4. pow-prod-down N/A

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

       5. flip-+ N/A

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

       6. +-inverses N/A

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

       7. +-inverses N/A

          \[\leadsto 0 - g \cdot \frac{0}{\color{blue}{0}} \]

       8. associate-*r/ N/A

          \[\leadsto 0 - \color{blue}{\frac{g \cdot 0}{0}} \]

       9. +-inverses N/A

          \[\leadsto 0 - \frac{g \cdot \color{blue}{\left(g - g\right)}}{0} \]

       10. distribute-lft-out-- N/A

           \[\leadsto 0 - \frac{\color{blue}{g \cdot g - g \cdot g}}{0} \]

       11. +-inverses N/A

           \[\leadsto 0 - \frac{g \cdot g - g \cdot g}{\color{blue}{g - g}} \]

       12. flip-+ N/A

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

   12. Applied egg-rr 5.8%

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

4. Final simplification 6.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq -4 \cdot 10^{-306}:\\ \;\;\;\;\sqrt[3]{g + g}\\ \mathbf{else}:\\ \;\;\;\;\left(-g\right) - g\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.6% accurate, 2.6× 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 43.3%

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

3. Taylor expanded in g around inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. neg-lowering-neg.f64 26.6

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

5. Simplified 26.6%

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

6. Taylor expanded in g around inf

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

   1. *-lowering-*.f64 N/A

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

   2. cbrt-lowering-cbrt.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. cbrt-lowering-cbrt.f64 78.0

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

8. Simplified 78.0%

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

9. Taylor expanded in a around -inf

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

    1. rem-cube-cbrt N/A

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

    2. *-commutative N/A

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

    3. mul-1-neg N/A

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

    4. neg-lowering-neg.f64 N/A

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

    5. cbrt-lowering-cbrt.f64 N/A

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

    6. /-lowering-/.f64 78.0

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

11. Simplified 78.0%

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

Alternative 8: 4.4% accurate, 13.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 43.3%

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

3. Taylor expanded in g around inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. neg-lowering-neg.f64 26.6

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

5. Simplified 26.6%

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

6. Taylor expanded in g around -inf

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

   1. associate-*r* N/A

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

   2. *-lowering-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. neg-lowering-neg.f64 N/A

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

   5. +-commutative N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. cbrt-lowering-cbrt.f64 N/A

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

   8. /-lowering-/.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. unpow2 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. cbrt-lowering-cbrt.f64 N/A

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

   14. cbrt-lowering-cbrt.f64 N/A

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

   15. cbrt-lowering-cbrt.f64 N/A

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

   16. /-lowering-/.f64 N/A

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

   17. *-lowering-*.f64 N/A

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

   18. unpow2 N/A

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

   19. *-lowering-*.f64 8.7

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

8. Simplified 8.7%

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

   1. cbrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. *-rgt-identity N/A

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

   5. pow1/3 N/A

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

   6. inv-pow N/A

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

   7. pow-pow N/A

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

   8. *-commutative N/A

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

   9. unpow-prod-down N/A

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

   10. pow-pow N/A

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

   11. inv-pow N/A

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

   12. accelerator-lowering-fma.f64 N/A

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

10. Applied egg-rr 15.1%

    \[\leadsto \left(-g\right) \cdot \color{blue}{\mathsf{fma}\left({\left(g \cdot g\right)}^{-0.3333333333333333}, {a}^{-0.3333333333333333}, {\left(g \cdot \left(g \cdot a\right)\right)}^{-0.3333333333333333}\right)} \]

12. Applied egg-rr 4.8%

    \[\leadsto \left(-g\right) \cdot \color{blue}{\frac{-1}{g + g}} \]
13. Add Preprocessing

Alternative 9: 6.2% accurate, 17.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;g + g\\ \mathbf{else}:\\ \;\;\;\;\left(-g\right) - g\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= (* a 2.0) -5e-310) (+ g g) (- (- g) g)))

C

double code(double g, double h, double a) {
	double tmp;
	if ((a * 2.0) <= -5e-310) {
		tmp = g + g;
	} else {
		tmp = -g - g;
	}
	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 * 2.0d0) <= (-5d-310)) then
        tmp = g + g
    else
        tmp = -g - g
    end if
    code = tmp
end function

Java

public static double code(double g, double h, double a) {
	double tmp;
	if ((a * 2.0) <= -5e-310) {
		tmp = g + g;
	} else {
		tmp = -g - g;
	}
	return tmp;
}

Python

def code(g, h, a):
	tmp = 0
	if (a * 2.0) <= -5e-310:
		tmp = g + g
	else:
		tmp = -g - g
	return tmp

Julia

function code(g, h, a)
	tmp = 0.0
	if (Float64(a * 2.0) <= -5e-310)
		tmp = Float64(g + g);
	else
		tmp = Float64(Float64(-g) - g);
	end
	return tmp
end

MATLAB

function tmp_2 = code(g, h, a)
	tmp = 0.0;
	if ((a * 2.0) <= -5e-310)
		tmp = g + g;
	else
		tmp = -g - g;
	end
	tmp_2 = tmp;
end

Wolfram

code[g_, h_, a_] := If[LessEqual[N[(a * 2.0), $MachinePrecision], -5e-310], N[(g + g), $MachinePrecision], N[((-g) - g), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) a) < -4.999999999999985e-310

   1. Initial program 45.1%

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

   3. Taylor expanded in g around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. neg-lowering-neg.f64 31.3

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

   5. Simplified 31.3%

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

   6. Taylor expanded in g around -inf

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. neg-lowering-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. cbrt-lowering-cbrt.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. cbrt-lowering-cbrt.f64 N/A

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

      14. cbrt-lowering-cbrt.f64 N/A

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

      15. cbrt-lowering-cbrt.f64 N/A

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

      16. /-lowering-/.f64 N/A

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

      17. *-lowering-*.f64 N/A

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 9.6

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

   8. Simplified 9.6%

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

      1. cbrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-rgt-identity N/A

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

      5. pow1/3 N/A

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

      6. inv-pow N/A

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

      7. pow-pow N/A

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

      8. *-commutative N/A

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

      9. unpow-prod-down N/A

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

      10. pow-pow N/A

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

      11. inv-pow N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

   10. Applied egg-rr 0.0%

       \[\leadsto \left(-g\right) \cdot \color{blue}{\mathsf{fma}\left({\left(g \cdot g\right)}^{-0.3333333333333333}, {a}^{-0.3333333333333333}, {\left(g \cdot \left(g \cdot a\right)\right)}^{-0.3333333333333333}\right)} \]
   11. Step-by-step derivation

       1. distribute-lft-neg-out N/A

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

       2. neg-sub0 N/A

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

       3. associate-*r* N/A

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

       4. pow-prod-down N/A

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

       5. flip-+ N/A

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

       6. +-inverses N/A

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

       7. +-inverses N/A

          \[\leadsto 0 - g \cdot \frac{0}{\color{blue}{0}} \]

       8. associate-*r/ N/A

          \[\leadsto 0 - \color{blue}{\frac{g \cdot 0}{0}} \]

       9. +-inverses N/A

          \[\leadsto 0 - \frac{g \cdot \color{blue}{\left(g - g\right)}}{0} \]

       10. distribute-lft-out-- N/A

           \[\leadsto 0 - \frac{\color{blue}{g \cdot g - g \cdot g}}{0} \]

       11. +-inverses N/A

           \[\leadsto 0 - \frac{g \cdot g - g \cdot g}{\color{blue}{g - g}} \]

       12. flip-+ N/A

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

   12. Applied egg-rr 6.2%

       \[\leadsto \color{blue}{g + g} \]

   if -4.999999999999985e-310 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 41.7%

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

   3. Taylor expanded in g around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. neg-lowering-neg.f64 22.6

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

   5. Simplified 22.6%

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

   6. Taylor expanded in g around -inf

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. neg-lowering-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. cbrt-lowering-cbrt.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. cbrt-lowering-cbrt.f64 N/A

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

      14. cbrt-lowering-cbrt.f64 N/A

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

      15. cbrt-lowering-cbrt.f64 N/A

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

      16. /-lowering-/.f64 N/A

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

      17. *-lowering-*.f64 N/A

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 8.0

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

   8. Simplified 8.0%

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

      1. cbrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-rgt-identity N/A

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

      5. pow1/3 N/A

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

      6. inv-pow N/A

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

      7. pow-pow N/A

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

      8. *-commutative N/A

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

      9. unpow-prod-down N/A

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

      10. pow-pow N/A

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

      11. inv-pow N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

   10. Applied egg-rr 28.4%

       \[\leadsto \left(-g\right) \cdot \color{blue}{\mathsf{fma}\left({\left(g \cdot g\right)}^{-0.3333333333333333}, {a}^{-0.3333333333333333}, {\left(g \cdot \left(g \cdot a\right)\right)}^{-0.3333333333333333}\right)} \]
   11. Step-by-step derivation

       1. distribute-lft-neg-out N/A

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

       2. neg-sub0 N/A

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

       3. associate-*r* N/A

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

       4. pow-prod-down N/A

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

       5. flip-+ N/A

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

       6. +-inverses N/A

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

       7. +-inverses N/A

          \[\leadsto 0 - g \cdot \frac{0}{\color{blue}{0}} \]

       8. associate-*r/ N/A

          \[\leadsto 0 - \color{blue}{\frac{g \cdot 0}{0}} \]

       9. +-inverses N/A

          \[\leadsto 0 - \frac{g \cdot \color{blue}{\left(g - g\right)}}{0} \]

       10. distribute-lft-out-- N/A

           \[\leadsto 0 - \frac{\color{blue}{g \cdot g - g \cdot g}}{0} \]

       11. +-inverses N/A

           \[\leadsto 0 - \frac{g \cdot g - g \cdot g}{\color{blue}{g - g}} \]

       12. flip-+ N/A

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

   12. Applied egg-rr 5.8%

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

4. Final simplification 6.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;g + g\\ \mathbf{else}:\\ \;\;\;\;\left(-g\right) - g\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 3.7% accurate, 75.5× speedup

Math

\[\begin{array}{l} \\ g + g \end{array} \]

FPCore

(FPCore (g h a) :precision binary64 (+ g g))

C

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

Fortran

real(8) function code(g, h, a)
    real(8), intent (in) :: g
    real(8), intent (in) :: h
    real(8), intent (in) :: a
    code = g + g
end function

Java

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

Python

def code(g, h, a):
	return g + g

Julia

function code(g, h, a)
	return Float64(g + g)
end

MATLAB

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

Wolfram

code[g_, h_, a_] := N[(g + g), $MachinePrecision]

Derivation

1. Initial program 43.3%

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

3. Taylor expanded in g around inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. neg-lowering-neg.f64 26.6

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

5. Simplified 26.6%

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

6. Taylor expanded in g around -inf

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

   1. associate-*r* N/A

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

   2. *-lowering-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. neg-lowering-neg.f64 N/A

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

   5. +-commutative N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. cbrt-lowering-cbrt.f64 N/A

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

   8. /-lowering-/.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. unpow2 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. cbrt-lowering-cbrt.f64 N/A

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

   14. cbrt-lowering-cbrt.f64 N/A

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

   15. cbrt-lowering-cbrt.f64 N/A

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

   16. /-lowering-/.f64 N/A

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

   17. *-lowering-*.f64 N/A

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

   18. unpow2 N/A

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

   19. *-lowering-*.f64 8.7

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

8. Simplified 8.7%

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

   1. cbrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. *-rgt-identity N/A

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

   5. pow1/3 N/A

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

   6. inv-pow N/A

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

   7. pow-pow N/A

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

   8. *-commutative N/A

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

   9. unpow-prod-down N/A

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

   10. pow-pow N/A

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

   11. inv-pow N/A

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

   12. accelerator-lowering-fma.f64 N/A

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

10. Applied egg-rr 15.1%

    \[\leadsto \left(-g\right) \cdot \color{blue}{\mathsf{fma}\left({\left(g \cdot g\right)}^{-0.3333333333333333}, {a}^{-0.3333333333333333}, {\left(g \cdot \left(g \cdot a\right)\right)}^{-0.3333333333333333}\right)} \]
11. Step-by-step derivation

    1. distribute-lft-neg-out N/A

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

    2. neg-sub0 N/A

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

    3. associate-*r* N/A

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

    4. pow-prod-down N/A

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

    5. flip-+ N/A

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

    6. +-inverses N/A

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

    7. +-inverses N/A

       \[\leadsto 0 - g \cdot \frac{0}{\color{blue}{0}} \]

    8. associate-*r/ N/A

       \[\leadsto 0 - \color{blue}{\frac{g \cdot 0}{0}} \]

    9. +-inverses N/A

       \[\leadsto 0 - \frac{g \cdot \color{blue}{\left(g - g\right)}}{0} \]

    10. distribute-lft-out-- N/A

        \[\leadsto 0 - \frac{\color{blue}{g \cdot g - g \cdot g}}{0} \]

    11. +-inverses N/A

        \[\leadsto 0 - \frac{g \cdot g - g \cdot g}{\color{blue}{g - g}} \]

    12. flip-+ N/A

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

12. Applied egg-rr 3.5%

    \[\leadsto \color{blue}{g + g} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(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))))))))