2-ancestry mixing, positive discriminant

Percentage Accurate: 44.8% → 95.9%

Time: 25.9s

Alternatives: 7

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 44.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 95.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.6%

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

3. Simplified 42.6%

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

5. Taylor expanded in g around -inf 24.8%

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

   1. *-commutative 24.8%

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

7. Simplified 24.8%

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

8. Taylor expanded in g around -inf 73.0%

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

   1. neg-mul-1 73.0%

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

10. Simplified 73.0%

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

    1. cbrt-prod 96.0%

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

12. Applied egg-rr 96.0%

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

13. Final simplification 96.0%

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

Alternative 2: 89.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}}\\ \mathbf{if}\;a \leq -2.9 \cdot 10^{-55}:\\ \;\;\;\;t\_0 + \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)}\\ \mathbf{elif}\;a \leq 2.75 \cdot 10^{-24}:\\ \;\;\;\;\sqrt[3]{-1} + \frac{\sqrt[3]{-g}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \sqrt[3]{-\frac{g}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (* (- g g) (/ -0.5 a)))))
   (if (<= a -2.9e-55)
     (+ t_0 (cbrt (* (/ 0.5 a) (* g -2.0))))
     (if (<= a 2.75e-24)
       (+ (cbrt -1.0) (/ (cbrt (- g)) (cbrt a)))
       (+ t_0 (cbrt (- (/ g a))))))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt(((g - g) * (-0.5 / a)));
	double tmp;
	if (a <= -2.9e-55) {
		tmp = t_0 + cbrt(((0.5 / a) * (g * -2.0)));
	} else if (a <= 2.75e-24) {
		tmp = cbrt(-1.0) + (cbrt(-g) / cbrt(a));
	} else {
		tmp = t_0 + cbrt(-(g / a));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.cbrt(((g - g) * (-0.5 / a)));
	double tmp;
	if (a <= -2.9e-55) {
		tmp = t_0 + Math.cbrt(((0.5 / a) * (g * -2.0)));
	} else if (a <= 2.75e-24) {
		tmp = Math.cbrt(-1.0) + (Math.cbrt(-g) / Math.cbrt(a));
	} else {
		tmp = t_0 + Math.cbrt(-(g / a));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(Float64(g - g) * Float64(-0.5 / a)))
	tmp = 0.0
	if (a <= -2.9e-55)
		tmp = Float64(t_0 + cbrt(Float64(Float64(0.5 / a) * Float64(g * -2.0))));
	elseif (a <= 2.75e-24)
		tmp = Float64(cbrt(-1.0) + Float64(cbrt(Float64(-g)) / cbrt(a)));
	else
		tmp = Float64(t_0 + cbrt(Float64(-Float64(g / a))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[(N[(g - g), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[a, -2.9e-55], N[(t$95$0 + N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(g * -2.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.75e-24], N[(N[Power[-1.0, 1/3], $MachinePrecision] + N[(N[Power[(-g), 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[Power[(-N[(g / a), $MachinePrecision]), 1/3], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.9e-55

   1. Initial program 52.2%

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

   3. Simplified 52.2%

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

   5. Taylor expanded in g around -inf 30.4%

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

      1. *-commutative 30.4%

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

   7. Simplified 30.4%

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

   8. Taylor expanded in g around -inf 87.6%

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

      1. neg-mul-1 87.6%

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

   10. Simplified 87.6%

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

   if -2.9e-55 < a < 2.7499999999999999e-24

   1. Initial program 37.9%

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

   3. Simplified 37.9%

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

   5. Taylor expanded in g around -inf 20.8%

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

      1. *-commutative 20.8%

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

   7. Simplified 20.8%

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

   8. Taylor expanded in g around inf 12.1%

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

   9. Taylor expanded in a around 0 12.1%

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

   11. Simplified 48.9%

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

       1. add-sqr-sqrt 21.2%

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

       2. sqrt-unprod 8.4%

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

       3. *-commutative 8.4%

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

       4. *-commutative 8.4%

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

       5. swap-sqr 5.6%

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

       6. frac-times 5.6%

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

       7. metadata-eval 5.6%

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

       8. metadata-eval 5.6%

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

       9. frac-times 5.6%

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

       10. count-2 5.6%

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

       11. count-2 5.6%

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

       12. swap-sqr 5.6%

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

       13. metadata-eval 5.6%

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

       14. metadata-eval 5.6%

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

       15. swap-sqr 5.6%

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

       16. *-commutative 5.6%

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

       17. *-commutative 5.6%

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

       18. swap-sqr 8.4%

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

       19. sqrt-unprod 21.2%

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

       20. add-sqr-sqrt 48.9%

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

   13. Applied egg-rr 91.5%

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

   if 2.7499999999999999e-24 < a

   1. Initial program 40.2%

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

   3. Simplified 40.2%

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

   5. Taylor expanded in g around -inf 25.6%

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

      1. *-commutative 25.6%

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

   7. Simplified 25.6%

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

   8. Taylor expanded in g around -inf 91.2%

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

      1. neg-mul-1 91.2%

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

   10. Simplified 91.2%

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

       1. associate-*l/ 91.3%

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

       2. *-commutative 91.3%

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

       3. associate-*r* 91.3%

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

       4. metadata-eval 91.3%

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

       5. neg-mul-1 91.3%

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

   12. Applied egg-rr 91.3%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.9 \cdot 10^{-55}:\\ \;\;\;\;\sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)}\\ \mathbf{elif}\;a \leq 2.75 \cdot 10^{-24}:\\ \;\;\;\;\sqrt[3]{-1} + \frac{\sqrt[3]{-g}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{-\frac{g}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.05 \cdot 10^{-35} \lor \neg \left(a \leq 4.2 \cdot 10^{-9}\right):\\ \;\;\;\;\sqrt[3]{\frac{-0.25}{a \cdot g}} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + g\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} - \sqrt[3]{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (or (<= a -1.05e-35) (not (<= a 4.2e-9)))
   (+ (cbrt (/ -0.25 (* a g))) (cbrt (* (/ -0.5 a) (+ g g))))
   (- (cbrt (* (/ 0.5 a) (* g -2.0))) (cbrt -2.0))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((a <= -1.05e-35) || !(a <= 4.2e-9)) {
		tmp = cbrt((-0.25 / (a * g))) + cbrt(((-0.5 / a) * (g + g)));
	} else {
		tmp = cbrt(((0.5 / a) * (g * -2.0))) - cbrt(-2.0);
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if ((a <= -1.05e-35) || !(a <= 4.2e-9)) {
		tmp = Math.cbrt((-0.25 / (a * g))) + Math.cbrt(((-0.5 / a) * (g + g)));
	} else {
		tmp = Math.cbrt(((0.5 / a) * (g * -2.0))) - Math.cbrt(-2.0);
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if ((a <= -1.05e-35) || !(a <= 4.2e-9))
		tmp = Float64(cbrt(Float64(-0.25 / Float64(a * g))) + cbrt(Float64(Float64(-0.5 / a) * Float64(g + g))));
	else
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(g * -2.0))) - cbrt(-2.0));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[Or[LessEqual[a, -1.05e-35], N[Not[LessEqual[a, 4.2e-9]], $MachinePrecision]], N[(N[Power[N[(-0.25 / N[(a * g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(-0.5 / a), $MachinePrecision] * N[(g + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(g * -2.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[-2.0, 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.05e-35 or 4.20000000000000039e-9 < a

   1. Initial program 44.2%

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

   3. Simplified 44.2%

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

   5. Taylor expanded in g around -inf 28.2%

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

      1. *-commutative 28.2%

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

   7. Simplified 28.2%

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

   8. Taylor expanded in g around inf 17.8%

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

   10. Applied egg-rr 73.6%

       \[\leadsto \sqrt[3]{\color{blue}{\frac{-0.25}{a \cdot g}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   if -1.05e-35 < a < 4.20000000000000039e-9

   1. Initial program 40.8%

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

   3. Simplified 40.8%

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

   5. Taylor expanded in g around -inf 21.1%

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

      1. *-commutative 21.1%

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

   7. Simplified 21.1%

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

   8. Taylor expanded in g around inf 12.4%

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

      1. add-sqr-sqrt 24.3%

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

      2. sqrt-unprod 10.1%

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

      3. *-commutative 10.1%

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

      4. *-commutative 10.1%

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

      5. swap-sqr 7.6%

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

      6. frac-times 7.6%

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

      7. metadata-eval 7.6%

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

      8. metadata-eval 7.6%

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

      9. frac-times 7.6%

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

      10. count-2 7.6%

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

      11. count-2 7.6%

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

      12. swap-sqr 7.6%

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

      13. metadata-eval 7.6%

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

      14. metadata-eval 7.6%

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

      15. swap-sqr 7.6%

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

      16. *-commutative 7.6%

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

      17. *-commutative 7.6%

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

      18. swap-sqr 10.1%

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

      19. sqrt-unprod 24.3%

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

      20. add-sqr-sqrt 50.0%

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

   10. Applied egg-rr 12.4%

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

   12. Simplified 50.2%

       \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \color{blue}{-1 \cdot \sqrt[3]{-2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.05 \cdot 10^{-35} \lor \neg \left(a \leq 4.2 \cdot 10^{-9}\right):\\ \;\;\;\;\sqrt[3]{\frac{-0.25}{a \cdot g}} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + g\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} - \sqrt[3]{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)}\\ \mathbf{if}\;a \leq -0.25 \lor \neg \left(a \leq 0.135\right):\\ \;\;\;\;t\_0 - \sqrt[3]{\frac{0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \sqrt[3]{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (* (/ 0.5 a) (* g -2.0)))))
   (if (or (<= a -0.25) (not (<= a 0.135)))
     (- t_0 (cbrt (/ 0.5 a)))
     (- t_0 (cbrt -2.0)))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt(((0.5 / a) * (g * -2.0)));
	double tmp;
	if ((a <= -0.25) || !(a <= 0.135)) {
		tmp = t_0 - cbrt((0.5 / a));
	} else {
		tmp = t_0 - cbrt(-2.0);
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.cbrt(((0.5 / a) * (g * -2.0)));
	double tmp;
	if ((a <= -0.25) || !(a <= 0.135)) {
		tmp = t_0 - Math.cbrt((0.5 / a));
	} else {
		tmp = t_0 - Math.cbrt(-2.0);
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(Float64(0.5 / a) * Float64(g * -2.0)))
	tmp = 0.0
	if ((a <= -0.25) || !(a <= 0.135))
		tmp = Float64(t_0 - cbrt(Float64(0.5 / a)));
	else
		tmp = Float64(t_0 - cbrt(-2.0));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(g * -2.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[Or[LessEqual[a, -0.25], N[Not[LessEqual[a, 0.135]], $MachinePrecision]], N[(t$95$0 - N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[Power[-2.0, 1/3], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.25 or 0.13500000000000001 < a

   1. Initial program 44.1%

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

   3. Simplified 44.1%

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

   5. Taylor expanded in g around -inf 28.2%

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

      1. *-commutative 28.2%

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

   7. Simplified 28.2%

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

   8. Taylor expanded in g around inf 17.7%

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

   10. Applied egg-rr 0.0%

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

   12. Simplified 71.3%

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

   if -0.25 < a < 0.13500000000000001

   1. Initial program 41.1%

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

   3. Simplified 41.1%

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

   5. Taylor expanded in g around -inf 21.3%

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

      1. *-commutative 21.3%

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

   7. Simplified 21.3%

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

   8. Taylor expanded in g around inf 12.6%

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

      1. add-sqr-sqrt 24.6%

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

      2. sqrt-unprod 10.7%

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

      3. *-commutative 10.7%

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

      4. *-commutative 10.7%

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

      5. swap-sqr 8.2%

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

      6. frac-times 8.2%

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

      7. metadata-eval 8.2%

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

      8. metadata-eval 8.2%

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

      9. frac-times 8.2%

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

      10. count-2 8.2%

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

      11. count-2 8.2%

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

      12. swap-sqr 8.2%

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

      13. metadata-eval 8.2%

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

      14. metadata-eval 8.2%

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

      15. swap-sqr 8.2%

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

      16. *-commutative 8.2%

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

      17. *-commutative 8.2%

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

      18. swap-sqr 10.7%

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

      19. sqrt-unprod 24.6%

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

      20. add-sqr-sqrt 50.0%

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

   10. Applied egg-rr 12.6%

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

   12. Simplified 50.1%

       \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \color{blue}{-1 \cdot \sqrt[3]{-2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.25 \lor \neg \left(a \leq 0.135\right):\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} - \sqrt[3]{\frac{0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} - \sqrt[3]{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.6%

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

3. Simplified 42.6%

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

5. Taylor expanded in g around -inf 24.8%

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

   1. *-commutative 24.8%

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

7. Simplified 24.8%

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

8. Taylor expanded in g around -inf 73.0%

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

   1. neg-mul-1 73.0%

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

10. Simplified 73.0%

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

    1. associate-*l/ 73.0%

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

    2. *-commutative 73.0%

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

    3. associate-*r* 73.0%

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

    4. metadata-eval 73.0%

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

    5. neg-mul-1 73.0%

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

12. Applied egg-rr 73.0%

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

13. Final simplification 73.0%

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

Alternative 6: 44.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.6%

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

3. Simplified 42.6%

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

5. Taylor expanded in g around -inf 24.8%

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

   1. *-commutative 24.8%

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

7. Simplified 24.8%

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

8. Taylor expanded in g around inf 15.2%

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

9. Taylor expanded in a around 0 15.2%

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

11. Simplified 46.2%

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

12. Taylor expanded in g around 0 46.2%

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

    1. neg-mul-1 46.2%

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

    2. distribute-neg-frac 46.2%

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

14. Simplified 46.2%

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

15. Final simplification 46.2%

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

Alternative 7: 4.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{-1} - \sqrt[3]{-2} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.6%

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

3. Simplified 42.6%

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

5. Taylor expanded in g around -inf 24.8%

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

   1. *-commutative 24.8%

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

7. Simplified 24.8%

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

8. Taylor expanded in g around inf 15.2%

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

9. Taylor expanded in a around 0 15.2%

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

11. Simplified 46.2%

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

    1. add-sqr-sqrt 22.6%

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

    2. sqrt-unprod 14.1%

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

    3. *-commutative 14.1%

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

    4. *-commutative 14.1%

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

    5. swap-sqr 6.4%

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

    6. frac-times 6.4%

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

    7. metadata-eval 6.4%

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

    8. metadata-eval 6.4%

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

    9. frac-times 6.4%

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

    10. count-2 6.4%

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

    11. count-2 6.4%

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

    12. swap-sqr 6.4%

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

    13. metadata-eval 6.4%

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

    14. metadata-eval 6.4%

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

    15. swap-sqr 6.4%

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

    16. *-commutative 6.4%

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

    17. *-commutative 6.4%

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

    18. swap-sqr 14.1%

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

    19. sqrt-unprod 22.6%

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

    20. add-sqr-sqrt 46.2%

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

13. Applied egg-rr 45.8%

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

15. Simplified 4.5%

    \[\leadsto \sqrt[3]{-1} + \color{blue}{-1 \cdot \sqrt[3]{-2}} \]

16. Final simplification 4.5%

    \[\leadsto \sqrt[3]{-1} - \sqrt[3]{-2} \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2024034 
(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))))))))