2-ancestry mixing, positive discriminant

Percentage Accurate: 44.3% → 95.9%

Time: 39.5s

Alternatives: 6

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: 44.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 95.9% 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 45.4%

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

3. Simplified 45.4%

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

5. Taylor expanded in g around -inf 25.3%

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

   1. *-commutative 25.3%

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

7. Simplified 25.3%

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

8. Taylor expanded in g around -inf 74.9%

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

   1. neg-mul-1 74.9%

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

10. Simplified 74.9%

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

    1. associate-*l/ 74.9%

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

    2. cbrt-div 95.9%

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

    3. *-commutative 95.9%

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

    4. associate-*r* 95.9%

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

    5. metadata-eval 95.9%

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

    6. neg-mul-1 95.9%

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

12. Applied egg-rr 95.9%

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

13. Final simplification 95.9%

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

Alternative 2: 89.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.2 \cdot 10^{-32} \lor \neg \left(a \leq 1.2 \cdot 10^{-30}\right):\\ \;\;\;\;\sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (or (<= a -5.2e-32) (not (<= a 1.2e-30)))
   (+ (cbrt (* (- g g) (/ -0.5 a))) (cbrt (* (/ 0.5 a) (* g -2.0))))
   (+ (/ (cbrt (- g)) (cbrt a)) (cbrt -2.0))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((a <= -5.2e-32) || !(a <= 1.2e-30)) {
		tmp = cbrt(((g - g) * (-0.5 / a))) + cbrt(((0.5 / a) * (g * -2.0)));
	} else {
		tmp = (cbrt(-g) / cbrt(a)) + cbrt(-2.0);
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if ((a <= -5.2e-32) || !(a <= 1.2e-30)) {
		tmp = Math.cbrt(((g - g) * (-0.5 / a))) + Math.cbrt(((0.5 / a) * (g * -2.0)));
	} else {
		tmp = (Math.cbrt(-g) / Math.cbrt(a)) + Math.cbrt(-2.0);
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if ((a <= -5.2e-32) || !(a <= 1.2e-30))
		tmp = Float64(cbrt(Float64(Float64(g - g) * Float64(-0.5 / a))) + cbrt(Float64(Float64(0.5 / a) * Float64(g * -2.0))));
	else
		tmp = Float64(Float64(cbrt(Float64(-g)) / cbrt(a)) + cbrt(-2.0));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[Or[LessEqual[a, -5.2e-32], N[Not[LessEqual[a, 1.2e-30]], $MachinePrecision]], N[(N[Power[N[(N[(g - g), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(g * -2.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[(-g), 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[-2.0, 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.1999999999999995e-32 or 1.19999999999999992e-30 < a

   1. Initial program 46.0%

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

   3. Simplified 46.0%

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

   5. Taylor expanded in g around -inf 22.3%

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

      1. *-commutative 22.3%

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

   7. Simplified 22.3%

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

   8. Taylor expanded in g around -inf 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%

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

   if -5.1999999999999995e-32 < a < 1.19999999999999992e-30

   1. Initial program 44.6%

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

   3. Simplified 44.6%

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

   5. Taylor expanded in g around -inf 29.5%

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

      1. *-commutative 29.5%

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

   7. Simplified 29.5%

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

   8. Taylor expanded in g around inf 12.1%

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

   10. Applied egg-rr 0.0%

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

   12. Simplified 46.9%

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

       1. cbrt-prod 90.3%

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

       2. *-commutative 90.3%

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

       3. add-sqr-sqrt 40.6%

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

       4. sqrt-unprod 22.0%

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

       5. frac-times 22.0%

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

       6. metadata-eval 22.0%

          \[\leadsto \sqrt[3]{-2} + \sqrt[3]{\sqrt{\frac{\color{blue}{0.25}}{a \cdot a}}} \cdot \sqrt[3]{g + g} \]

       7. metadata-eval 22.0%

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

       8. frac-times 22.0%

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

       9. sqrt-unprod 0.7%

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

       10. add-sqr-sqrt 1.2%

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

       11. add-sqr-sqrt 0.6%

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

       12. sqrt-unprod 24.8%

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

       13. count-2 24.8%

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

       14. count-2 24.8%

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

       15. swap-sqr 24.8%

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

       16. metadata-eval 24.8%

           \[\leadsto \sqrt[3]{-2} + \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\sqrt{\color{blue}{4} \cdot \left(g \cdot g\right)}} \]

       17. metadata-eval 24.8%

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

       18. swap-sqr 24.8%

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

       19. *-commutative 24.8%

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

       20. *-commutative 24.8%

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

       21. sqrt-unprod 42.1%

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

       22. add-sqr-sqrt 90.3%

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

   14. Applied egg-rr 90.3%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.2 \cdot 10^{-32} \lor \neg \left(a \leq 1.2 \cdot 10^{-30}\right):\\ \;\;\;\;\sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.6% 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 45.4%

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

3. Simplified 45.4%

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

5. Taylor expanded in g around -inf 25.3%

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

   1. *-commutative 25.3%

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

7. Simplified 25.3%

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

8. Taylor expanded in g around -inf 74.9%

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

   1. neg-mul-1 74.9%

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

10. Simplified 74.9%

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

11. Taylor expanded in g around -inf 74.9%

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

    1. mul-1-neg 74.9%

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

13. Simplified 74.9%

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

14. Final simplification 74.9%

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

Alternative 4: 43.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 45.4%

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

3. Simplified 45.4%

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

5. Taylor expanded in g around -inf 25.3%

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

   1. *-commutative 25.3%

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

7. Simplified 25.3%

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

8. Taylor expanded in g around inf 15.6%

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

10. Applied egg-rr 0.0%

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

12. Simplified 46.7%

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

    1. +-commutative 46.7%

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

    2. *-un-lft-identity 46.7%

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

    3. fma-define 46.7%

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

    4. count-2 46.7%

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

14. Applied egg-rr 46.7%

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

    1. fma-undefine 46.7%

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

    2. *-lft-identity 46.7%

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

    3. associate-*r/ 46.7%

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

    4. *-commutative 46.7%

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

    5. associate-*r* 46.7%

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

    6. metadata-eval 46.7%

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

    7. neg-mul-1 46.7%

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

16. Simplified 46.7%

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

17. Final simplification 46.7%

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

Alternative 5: 4.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{-2} + \sqrt[3]{0} \end{array} \]

FPCore

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

C

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

Java

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

Julia

function code(g, h, a)
	return Float64(cbrt(-2.0) + cbrt(0.0))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[-2.0, 1/3], $MachinePrecision] + N[Power[0.0, 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 45.4%

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

3. Simplified 45.4%

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

5. Taylor expanded in g around -inf 25.3%

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

   1. *-commutative 25.3%

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

7. Simplified 25.3%

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

8. Taylor expanded in g around inf 15.6%

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

10. Applied egg-rr 0.0%

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

12. Simplified 46.7%

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

    1. flip-+ 0.0%

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

    2. frac-times 0.0%

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

    3. pow2 0.0%

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

    4. pow2 0.0%

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

    5. *-un-lft-identity 0.0%

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

    6. fma-neg 0.0%

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

    7. add-sqr-sqrt 0.5%

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

    8. sqrt-unprod 1.5%

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

    9. sqr-neg 1.5%

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

    10. sqrt-prod 1.5%

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

    11. add-sqr-sqrt 2.3%

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

    12. fma-define 2.3%

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

    13. *-un-lft-identity 2.3%

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

    14. count-2 2.3%

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

14. Applied egg-rr 2.3%

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

    1. +-inverses 4.3%

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

    2. metadata-eval 4.3%

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

    3. div0 4.5%

       \[\leadsto \sqrt[3]{-2} + \sqrt[3]{\color{blue}{0}} \]

16. Simplified 4.5%

    \[\leadsto \sqrt[3]{-2} + \sqrt[3]{\color{blue}{0}} \]

17. Final simplification 4.5%

    \[\leadsto \sqrt[3]{-2} + \sqrt[3]{0} \]
18. Add Preprocessing

Alternative 6: 4.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 45.4%

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

3. Simplified 45.4%

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

5. Taylor expanded in g around -inf 25.3%

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

   1. *-commutative 25.3%

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

7. Simplified 25.3%

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

8. Taylor expanded in g around inf 15.6%

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

10. Applied egg-rr 0.0%

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

12. Simplified 46.7%

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

13. Taylor expanded in g around 0 46.7%

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

15. Simplified 4.9%

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

16. Final simplification 4.9%

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

Reproduce

herbie shell --seed 2024076 
(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))))))))