2-ancestry mixing, positive discriminant

Percentage Accurate: 44.2% → 95.9%

Time: 8.0s

Alternatives: 8

Speedup: 3.9×

Specification

Math

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

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

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

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, 2.0× speedup

Math

\[\sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{-2 \cdot g} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.2%

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

3. Applied rewrites 50.1%

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

4. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-cbrt.f64 95.3%

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

6. Applied rewrites 95.3%

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

   1. lift-/.f64 N/A

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

   2. mult-flip-rev N/A

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

   3. metadata-eval N/A

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

   4. lift-cbrt.f64 N/A

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

   5. cbrt-undiv N/A

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

   6. lift-+.f64 N/A

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

   7. count-2-rev N/A

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

   8. lift-*.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lower-*.f64 N/A

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

8. Applied rewrites 95.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-cbrt.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. metadata-eval N/A

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

   6. associate-/r* N/A

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lower-*.f64 N/A

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

10. Applied rewrites 95.8%

    \[\leadsto \color{blue}{\sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{-2 \cdot g}} \]
11. Add Preprocessing

Alternative 2: 95.9% accurate, 2.0× speedup

Math

\[\sqrt[3]{-0.5 \cdot g} \cdot \sqrt[3]{\frac{2}{a}} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.2%

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

2. Taylor expanded in g around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-cbrt.f64 95.2%

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

4. Applied rewrites 95.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-cbrt.f64 N/A

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

   9. lift-cbrt.f64 N/A

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

   10. cbrt-unprod N/A

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

   11. lower-cbrt.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lift-cbrt.f64 N/A

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

   14. lift-cbrt.f64 N/A

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

   15. cbrt-undiv N/A

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

   16. lower-cbrt.f64 N/A

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

   17. lower-/.f64 95.9%

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

6. Applied rewrites 95.9%

   \[\leadsto \sqrt[3]{-0.5 \cdot g} \cdot \color{blue}{\sqrt[3]{\frac{2}{a}}} \]
7. Add Preprocessing

Alternative 3: 95.9% accurate, 2.0× speedup

Math

\[\frac{\sqrt[3]{-2 \cdot g}}{\sqrt[3]{a + a}} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.2%

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

3. Applied rewrites 50.1%

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

4. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-cbrt.f64 95.3%

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

6. Applied rewrites 95.3%

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

   1. lift-*.f64 N/A

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

   2. mul-1-neg N/A

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

   3. lower-neg.f64 95.3%

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

   4. lift-*.f64 N/A

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

   5. lift-cbrt.f64 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. cbrt-unprod N/A

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

   8. lower-cbrt.f64 N/A

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

   9. metadata-eval N/A

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

   10. distribute-rgt-in N/A

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

   11. *-lft-identity N/A

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

   12. *-lft-identity N/A

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

   13. lower-+.f64 95.8%

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

8. Applied rewrites 95.8%

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

   1. lift-neg.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. cbrt-neg-rev N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. count-2 N/A

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

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

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

   8. lower-*.f64 N/A

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

   9. metadata-eval 95.8%

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

10. Applied rewrites 95.8%

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

Alternative 4: 95.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.2%

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

2. Taylor expanded in g around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-cbrt.f64 95.2%

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

4. Applied rewrites 95.2%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. distribute-frac-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-cbrt.f64 N/A

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

   8. lift-cbrt.f64 N/A

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

   9. lift-cbrt.f64 N/A

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

   10. cbrt-unprod N/A

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

   11. metadata-eval N/A

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

   12. cbrt-unprod N/A

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

   13. *-commutative N/A

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

   14. mul-1-neg N/A

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

   15. lift-neg.f64 N/A

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

   16. lift-cbrt.f64 N/A

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

   17. cbrt-neg-rev N/A

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

   18. cbrt-undiv N/A

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

   19. lift-neg.f64 N/A

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

   20. frac-2neg N/A

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

   21. lift-/.f64 N/A

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

6. Applied rewrites 73.6%

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

   1. lift-neg.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. cbrt-neg-rev N/A

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

   4. lift-/.f64 N/A

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

   5. mult-flip N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. cbrt-prod N/A

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

   8. lift-cbrt.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cbrt.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-/.f64 95.9%

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

8. Applied rewrites 95.9%

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

Alternative 5: 95.8% accurate, 2.3× speedup

Math

\[\frac{\sqrt[3]{g}}{-\sqrt[3]{a}} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.2%

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

2. Taylor expanded in g around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-cbrt.f64 95.2%

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

4. Applied rewrites 95.2%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. distribute-frac-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-cbrt.f64 N/A

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

   8. lift-cbrt.f64 N/A

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

   9. lift-cbrt.f64 N/A

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

   10. cbrt-unprod N/A

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

   11. metadata-eval N/A

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

   12. cbrt-unprod N/A

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

   13. *-commutative N/A

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

   14. mul-1-neg N/A

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

   15. lift-neg.f64 N/A

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

   16. lift-cbrt.f64 N/A

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

   17. cbrt-neg-rev N/A

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

   18. cbrt-undiv N/A

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

   19. lift-neg.f64 N/A

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

   20. frac-2neg N/A

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

   21. lift-/.f64 N/A

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

6. Applied rewrites 73.6%

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

   1. lift-neg.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. cbrt-div N/A

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

   5. lift-cbrt.f64 N/A

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

   6. distribute-neg-frac2 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-cbrt.f64 95.9%

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

8. Applied rewrites 95.9%

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

Alternative 6: 73.6% accurate, 3.4× speedup

Math

\[-\sqrt[3]{\frac{1}{\frac{a}{g}}} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.2%

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

2. Taylor expanded in g around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-cbrt.f64 95.2%

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

4. Applied rewrites 95.2%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. distribute-frac-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-cbrt.f64 N/A

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

   8. lift-cbrt.f64 N/A

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

   9. lift-cbrt.f64 N/A

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

   10. cbrt-unprod N/A

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

   11. metadata-eval N/A

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

   12. cbrt-unprod N/A

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

   13. *-commutative N/A

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

   14. mul-1-neg N/A

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

   15. lift-neg.f64 N/A

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

   16. lift-cbrt.f64 N/A

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

   17. cbrt-neg-rev N/A

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

   18. cbrt-undiv N/A

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

   19. lift-neg.f64 N/A

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

   20. frac-2neg N/A

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

   21. lift-/.f64 N/A

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

6. Applied rewrites 73.6%

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

   1. lift-/.f64 N/A

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

   2. div-flip N/A

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

   3. lower-unsound-/.f64 N/A

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

   4. lower-unsound-/.f64 73.2%

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

8. Applied rewrites 73.2%

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

Alternative 7: 73.6% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.2%

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

2. Taylor expanded in g around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-cbrt.f64 95.2%

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

4. Applied rewrites 95.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. lift-cbrt.f64 N/A

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

   8. cbrt-unprod N/A

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

   9. metadata-eval N/A

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

   10. lift-cbrt.f64 N/A

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

   11. cbrt-undiv N/A

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

   12. lift-cbrt.f64 N/A

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

   13. cbrt-unprod N/A

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

   14. lower-cbrt.f64 N/A

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

   15. frac-2neg-rev N/A

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

   16. metadata-eval N/A

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

   17. lower-*.f64 N/A

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

   18. metadata-eval N/A

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

   19. frac-2neg-rev N/A

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

   20. lower-/.f64 73.6%

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

6. Applied rewrites 73.6%

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

Alternative 8: 73.2% accurate, 3.9× speedup

Math

\[-\sqrt[3]{\frac{g}{a}} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.2%

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

2. Taylor expanded in g around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-cbrt.f64 95.2%

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

4. Applied rewrites 95.2%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. distribute-frac-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-cbrt.f64 N/A

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

   8. lift-cbrt.f64 N/A

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

   9. lift-cbrt.f64 N/A

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

   10. cbrt-unprod N/A

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

   11. metadata-eval N/A

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

   12. cbrt-unprod N/A

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

   13. *-commutative N/A

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

   14. mul-1-neg N/A

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

   15. lift-neg.f64 N/A

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

   16. lift-cbrt.f64 N/A

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

   17. cbrt-neg-rev N/A

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

   18. cbrt-undiv N/A

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

   19. lift-neg.f64 N/A

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

   20. frac-2neg N/A

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

   21. lift-/.f64 N/A

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

6. Applied rewrites 73.6%

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

Reproduce

herbie shell --seed 2025196 
(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))))))))