2-ancestry mixing, positive discriminant

Percentage Accurate: 44.2% → 95.8%

Time: 31.7s

Alternatives: 10

Speedup: 2.1×

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: 44.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 95.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[g_, h_, a_] := N[(N[(N[Power[(-g), 1/3], $MachinePrecision] / N[Power[a, 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 45.9%

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

3. Simplified 45.9%

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

5. Taylor expanded in g around -inf 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 75.3%

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

   1. neg-mul-1 75.3%

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

10. Simplified 75.3%

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

    1. associate-*l/ 75.3%

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

    2. cbrt-div 95.9%

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

12. Applied egg-rr 95.9%

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

    1. *-commutative 95.9%

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

    2. associate-*r* 96.3%

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

    3. metadata-eval 96.3%

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

    4. neg-mul-1 96.3%

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

14. Simplified 96.3%

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

15. Final simplification 96.3%

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

Alternative 2: 88.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.3 \cdot 10^{-129} \lor \neg \left(a \leq 1.1 \cdot 10^{-23}\right):\\ \;\;\;\;\sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{-1}{\frac{a}{g}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (or (<= a -4.3e-129) (not (<= a 1.1e-23)))
   (+ (cbrt (* (- g g) (/ -0.5 a))) (cbrt (/ -1.0 (/ a g))))
   (+ (/ (cbrt (- g)) (cbrt a)) (cbrt -1.0))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((a <= -4.3e-129) || !(a <= 1.1e-23)) {
		tmp = cbrt(((g - g) * (-0.5 / a))) + cbrt((-1.0 / (a / g)));
	} else {
		tmp = (cbrt(-g) / cbrt(a)) + cbrt(-1.0);
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if ((a <= -4.3e-129) || !(a <= 1.1e-23)) {
		tmp = Math.cbrt(((g - g) * (-0.5 / a))) + Math.cbrt((-1.0 / (a / g)));
	} else {
		tmp = (Math.cbrt(-g) / Math.cbrt(a)) + Math.cbrt(-1.0);
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if ((a <= -4.3e-129) || !(a <= 1.1e-23))
		tmp = Float64(cbrt(Float64(Float64(g - g) * Float64(-0.5 / a))) + cbrt(Float64(-1.0 / Float64(a / g))));
	else
		tmp = Float64(Float64(cbrt(Float64(-g)) / cbrt(a)) + cbrt(-1.0));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[Or[LessEqual[a, -4.3e-129], N[Not[LessEqual[a, 1.1e-23]], $MachinePrecision]], N[(N[Power[N[(N[(g - g), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(-1.0 / N[(a / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[(-g), 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[-1.0, 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.29999999999999981e-129 or 1.1e-23 < a

   1. Initial program 51.0%

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

   3. Simplified 51.0%

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

   5. Taylor expanded in g around -inf 30.1%

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

      1. *-commutative 30.1%

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

   7. Simplified 30.1%

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

   8. Taylor expanded in g around -inf 91.9%

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

      1. neg-mul-1 91.9%

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

   10. Simplified 91.9%

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

       1. associate-*l/ 91.9%

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

       2. clear-num 91.9%

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

       3. *-commutative 91.9%

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

       4. associate-*r* 91.9%

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

       5. metadata-eval 91.9%

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

       6. neg-mul-1 91.9%

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

   12. Applied egg-rr 91.9%

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

   if -4.29999999999999981e-129 < a < 1.1e-23

   1. Initial program 38.7%

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

   3. Simplified 38.7%

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

   5. Taylor expanded in g around -inf 25.5%

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

      1. *-commutative 25.5%

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

   7. Simplified 25.5%

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

   8. Taylor expanded in g around inf 11.8%

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

      1. add-sqr-sqrt 7.2%

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

      2. sqrt-unprod 5.0%

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

      3. *-commutative 5.0%

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

      4. *-commutative 5.0%

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

      5. swap-sqr 5.4%

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

      6. *-commutative 5.4%

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

      7. *-commutative 5.4%

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

      8. swap-sqr 5.4%

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

      9. metadata-eval 5.4%

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

      10. metadata-eval 5.4%

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

      11. swap-sqr 5.4%

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

      12. count-2 5.4%

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

      13. count-2 5.4%

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

      14. frac-times 5.5%

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

      15. metadata-eval 5.5%

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

      16. metadata-eval 5.5%

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

      17. frac-times 5.4%

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

      18. swap-sqr 5.0%

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

      19. sqrt-unprod 7.2%

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

      20. add-sqr-sqrt 11.8%

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

      21. associate-*r/ 11.8%

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

   10. Applied egg-rr 0.0%

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

   12. Simplified 45.9%

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

       1. add-sqr-sqrt 28.7%

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

       2. sqrt-unprod 10.0%

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

       3. swap-sqr 5.2%

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

       4. count-2 5.2%

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

       5. count-2 5.2%

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

       6. swap-sqr 5.2%

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

       7. metadata-eval 5.2%

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

       8. metadata-eval 5.2%

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

       9. swap-sqr 5.2%

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

       10. *-commutative 5.2%

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

       11. *-commutative 5.2%

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

       12. frac-times 5.2%

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

       13. metadata-eval 5.2%

           \[\leadsto \sqrt[3]{-1} + \sqrt[3]{\sqrt{\left(\left(g \cdot -2\right) \cdot \left(g \cdot -2\right)\right) \cdot \frac{\color{blue}{0.25}}{a \cdot a}}} \]

       14. metadata-eval 5.2%

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

       15. frac-times 5.2%

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

       16. swap-sqr 10.0%

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

       17. *-commutative 10.0%

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

       18. *-commutative 10.0%

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

       19. sqrt-unprod 28.7%

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

       20. add-sqr-sqrt 45.9%

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

   14. Applied egg-rr 91.0%

       \[\leadsto \sqrt[3]{-1} + \color{blue}{\frac{\sqrt[3]{-g}}{\sqrt[3]{a}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.3 \cdot 10^{-129} \lor \neg \left(a \leq 1.1 \cdot 10^{-23}\right):\\ \;\;\;\;\sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{-1}{\frac{a}{g}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

function code(g, h, a)
	return Float64(Float64(cbrt(Float64(0.5 / a)) * cbrt(Float64(g * -2.0))) + cbrt(0.0))
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[0.0, 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 45.9%

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

3. Simplified 45.9%

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

5. Taylor expanded in g around -inf 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 15.5%

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

   1. cbrt-prod 17.9%

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

10. Applied egg-rr 17.9%

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

12. Applied egg-rr 96.1%

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

13. Final simplification 96.1%

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

Alternative 4: 61.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)}\\ \mathbf{if}\;g \leq -1 \cdot 10^{+30} \lor \neg \left(g \leq 19000000000\right):\\ \;\;\;\;t\_0 + \frac{-1}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \sqrt[3]{g}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (* (/ 0.5 a) (* g -2.0)))))
   (if (or (<= g -1e+30) (not (<= g 19000000000.0)))
     (+ t_0 (/ -1.0 (cbrt a)))
     (+ t_0 (cbrt g)))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt(((0.5 / a) * (g * -2.0)));
	double tmp;
	if ((g <= -1e+30) || !(g <= 19000000000.0)) {
		tmp = t_0 + (-1.0 / cbrt(a));
	} else {
		tmp = t_0 + cbrt(g);
	}
	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 ((g <= -1e+30) || !(g <= 19000000000.0)) {
		tmp = t_0 + (-1.0 / Math.cbrt(a));
	} else {
		tmp = t_0 + Math.cbrt(g);
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(Float64(0.5 / a) * Float64(g * -2.0)))
	tmp = 0.0
	if ((g <= -1e+30) || !(g <= 19000000000.0))
		tmp = Float64(t_0 + Float64(-1.0 / cbrt(a)));
	else
		tmp = Float64(t_0 + cbrt(g));
	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[g, -1e+30], N[Not[LessEqual[g, 19000000000.0]], $MachinePrecision]], N[(t$95$0 + N[(-1.0 / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[Power[g, 1/3], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if g < -1e30 or 1.9e10 < g

   1. Initial program 34.7%

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

   3. Simplified 34.7%

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

   5. Taylor expanded in g around -inf 19.2%

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

      1. *-commutative 19.2%

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

   7. Simplified 19.2%

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

   8. Taylor expanded in g around inf 14.7%

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

      1. associate-*r/ 44.7%

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

      2. cbrt-div 70.2%

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

   10. Applied egg-rr 0.0%

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

   12. Simplified 69.6%

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

   if -1e30 < g < 1.9e10

   1. Initial program 75.1%

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

   3. Simplified 75.1%

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

   5. Taylor expanded in g around -inf 51.7%

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

      1. *-commutative 51.7%

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

   7. Simplified 51.7%

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

   8. Taylor expanded in g around inf 17.3%

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

   9. Taylor expanded in g around 0 17.3%

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

   11. Simplified 48.7%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -1 \cdot 10^{+30} \lor \neg \left(g \leq 19000000000\right):\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \frac{-1}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \sqrt[3]{g}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 46.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;g \leq -2.2 \lor \neg \left(g \leq 1.1 \cdot 10^{-11}\right):\\ \;\;\;\;\sqrt[3]{-1} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \sqrt[3]{g}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (or (<= g -2.2) (not (<= g 1.1e-11)))
   (+ (cbrt -1.0) (cbrt (/ (- g) a)))
   (+ (cbrt (* (/ 0.5 a) (* g -2.0))) (cbrt g))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((g <= -2.2) || !(g <= 1.1e-11)) {
		tmp = cbrt(-1.0) + cbrt((-g / a));
	} else {
		tmp = cbrt(((0.5 / a) * (g * -2.0))) + cbrt(g);
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if ((g <= -2.2) || !(g <= 1.1e-11)) {
		tmp = Math.cbrt(-1.0) + Math.cbrt((-g / a));
	} else {
		tmp = Math.cbrt(((0.5 / a) * (g * -2.0))) + Math.cbrt(g);
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if ((g <= -2.2) || !(g <= 1.1e-11))
		tmp = Float64(cbrt(-1.0) + cbrt(Float64(Float64(-g) / a)));
	else
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(g * -2.0))) + cbrt(g));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[Or[LessEqual[g, -2.2], N[Not[LessEqual[g, 1.1e-11]], $MachinePrecision]], N[(N[Power[-1.0, 1/3], $MachinePrecision] + N[Power[N[((-g) / a), $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[g, 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if g < -2.2000000000000002 or 1.1000000000000001e-11 < g

   1. Initial program 37.4%

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

   3. Simplified 37.4%

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

   5. Taylor expanded in g around -inf 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 14.9%

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

      1. add-sqr-sqrt 7.2%

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

      2. sqrt-unprod 9.4%

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

      3. *-commutative 9.4%

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

      4. *-commutative 9.4%

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

      5. swap-sqr 12.2%

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

      6. *-commutative 12.2%

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

      7. *-commutative 12.2%

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

      8. swap-sqr 12.2%

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

      9. metadata-eval 12.2%

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

      10. metadata-eval 12.2%

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

      11. swap-sqr 12.2%

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

      12. count-2 12.2%

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

      13. count-2 12.2%

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

      14. frac-times 12.2%

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

      15. metadata-eval 12.2%

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

      16. metadata-eval 12.2%

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

      17. frac-times 12.2%

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

      18. swap-sqr 9.4%

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

      19. sqrt-unprod 7.2%

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

      20. add-sqr-sqrt 14.9%

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

      21. associate-*r/ 14.9%

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

   10. Applied egg-rr 0.0%

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

   12. Simplified 46.5%

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

   13. Taylor expanded in g around 0 46.5%

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

       1. mul-1-neg 46.5%

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

       2. distribute-neg-frac2 46.5%

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

   15. Simplified 46.5%

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

   if -2.2000000000000002 < g < 1.1000000000000001e-11

   1. Initial program 72.1%

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

   3. Simplified 72.1%

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

   5. Taylor expanded in g around -inf 50.8%

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

      1. *-commutative 50.8%

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

   7. Simplified 50.8%

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

   8. Taylor expanded in g around inf 17.2%

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

   9. Taylor expanded in g around 0 17.2%

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

   11. Simplified 43.8%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -2.2 \lor \neg \left(g \leq 1.1 \cdot 10^{-11}\right):\\ \;\;\;\;\sqrt[3]{-1} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \sqrt[3]{g}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.1% accurate, 2.1× 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(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 45.9%

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

3. Simplified 45.9%

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

5. Taylor expanded in g around -inf 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 75.3%

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

   1. neg-mul-1 75.3%

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

10. Simplified 75.3%

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

11. Taylor expanded in g around -inf 75.3%

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

    1. mul-1-neg 75.3%

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

13. Simplified 75.3%

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

14. Final simplification 75.3%

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

Alternative 7: 43.3% 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 45.9%

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

3. Simplified 45.9%

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

5. Taylor expanded in g around -inf 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 15.5%

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

   1. add-sqr-sqrt 8.0%

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

   2. sqrt-unprod 15.4%

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

   3. *-commutative 15.4%

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

   4. *-commutative 15.4%

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

   5. swap-sqr 18.0%

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

   6. *-commutative 18.0%

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

   7. *-commutative 18.0%

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

   8. swap-sqr 18.0%

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

   9. metadata-eval 18.0%

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

   10. metadata-eval 18.0%

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

   11. swap-sqr 18.0%

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

   12. count-2 18.0%

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

   13. count-2 18.0%

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

   14. frac-times 18.0%

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

   15. metadata-eval 18.0%

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

   16. metadata-eval 18.0%

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

   17. frac-times 18.0%

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

   18. swap-sqr 15.4%

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

   19. sqrt-unprod 8.0%

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

   20. add-sqr-sqrt 15.5%

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

   21. associate-*r/ 15.5%

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

10. Applied egg-rr 0.0%

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

12. Simplified 42.1%

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

13. Taylor expanded in g around 0 42.1%

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

    1. mul-1-neg 42.1%

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

    2. distribute-neg-frac2 42.1%

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

15. Simplified 42.1%

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

16. Final simplification 42.1%

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

Alternative 8: 4.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 45.9%

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

3. Simplified 45.9%

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

5. Taylor expanded in g around -inf 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 15.5%

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

   1. add-sqr-sqrt 8.0%

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

   2. sqrt-unprod 15.4%

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

   3. *-commutative 15.4%

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

   4. *-commutative 15.4%

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

   5. swap-sqr 18.0%

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

   6. *-commutative 18.0%

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

   7. *-commutative 18.0%

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

   8. swap-sqr 18.0%

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

   9. metadata-eval 18.0%

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

   10. metadata-eval 18.0%

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

   11. swap-sqr 18.0%

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

   12. count-2 18.0%

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

   13. count-2 18.0%

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

   14. frac-times 18.0%

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

   15. metadata-eval 18.0%

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

   16. metadata-eval 18.0%

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

   17. frac-times 18.0%

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

   18. swap-sqr 15.4%

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

   19. sqrt-unprod 8.0%

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

   20. add-sqr-sqrt 15.5%

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

   21. associate-*r/ 15.5%

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

10. Applied egg-rr 0.0%

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

12. Simplified 42.1%

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

    1. associate-*r/ 42.1%

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

    2. cbrt-div 60.5%

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

14. Applied egg-rr 0.0%

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

16. Simplified 4.9%

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

17. Final simplification 4.9%

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

Alternative 9: 4.4% accurate, 433.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 45.9%

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

3. Simplified 45.9%

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

5. Taylor expanded in g around -inf 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 15.5%

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

   1. add-sqr-sqrt 8.0%

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

   2. sqrt-unprod 15.4%

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

   3. *-commutative 15.4%

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

   4. *-commutative 15.4%

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

   5. swap-sqr 18.0%

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

   6. *-commutative 18.0%

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

   7. *-commutative 18.0%

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

   8. swap-sqr 18.0%

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

   9. metadata-eval 18.0%

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

   10. metadata-eval 18.0%

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

   11. swap-sqr 18.0%

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

   12. count-2 18.0%

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

   13. count-2 18.0%

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

   14. frac-times 18.0%

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

   15. metadata-eval 18.0%

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

   16. metadata-eval 18.0%

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

   17. frac-times 18.0%

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

   18. swap-sqr 15.4%

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

   19. sqrt-unprod 8.0%

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

   20. add-sqr-sqrt 15.5%

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

   21. associate-*r/ 15.5%

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

10. Applied egg-rr 0.0%

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

12. Simplified 42.1%

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

14. Applied egg-rr 4.3%

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

15. Final simplification 4.3%

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

Alternative 10: 4.4% accurate, 433.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[g_, h_, a_] := 1.0

Derivation

1. Initial program 45.9%

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

3. Simplified 45.9%

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

5. Taylor expanded in g around -inf 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 15.5%

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

   1. add-sqr-sqrt 8.0%

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

   2. sqrt-unprod 15.4%

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

   3. *-commutative 15.4%

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

   4. *-commutative 15.4%

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

   5. swap-sqr 18.0%

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

   6. *-commutative 18.0%

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

   7. *-commutative 18.0%

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

   8. swap-sqr 18.0%

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

   9. metadata-eval 18.0%

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

   10. metadata-eval 18.0%

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

   11. swap-sqr 18.0%

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

   12. count-2 18.0%

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

   13. count-2 18.0%

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

   14. frac-times 18.0%

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

   15. metadata-eval 18.0%

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

   16. metadata-eval 18.0%

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

   17. frac-times 18.0%

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

   18. swap-sqr 15.4%

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

   19. sqrt-unprod 8.0%

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

   20. add-sqr-sqrt 15.5%

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

   21. associate-*r/ 15.5%

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

10. Applied egg-rr 0.0%

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

12. Simplified 42.1%

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

14. Applied egg-rr 4.7%

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

15. Final simplification 4.7%

    \[\leadsto 1 \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024080 
(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))))))))