2-ancestry mixing, positive discriminant

Percentage Accurate: 45.3% → 95.7%

Time: 6.6s

Alternatives: 3

Speedup: 3.9×

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: 45.3% 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: 95.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{0.5}{a}} \cdot \left(-1 \cdot \left(\sqrt[3]{g} \cdot {2}^{0.3333333333333333}\right)\right) \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (* (cbrt (/ 0.5 a)) (* -1.0 (* (cbrt g) (pow 2.0 0.3333333333333333)))))

C

double code(double g, double h, double a) {
	return cbrt((0.5 / a)) * (-1.0 * (cbrt(g) * pow(2.0, 0.3333333333333333)));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt((0.5 / a)) * (-1.0 * (Math.cbrt(g) * Math.pow(2.0, 0.3333333333333333)));
}

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(0.5 / a)) * Float64(-1.0 * Float64(cbrt(g) * (2.0 ^ 0.3333333333333333))))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[(-1.0 * N[(N[Power[g, 1/3], $MachinePrecision] * N[Power[2.0, 0.3333333333333333], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 45.3%

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

   1. lift-+.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. cbrt-prod N/A

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

   5. lift-cbrt.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. cbrt-prod N/A

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

   8. distribute-lft-out N/A

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

3. Applied rewrites 51.2%

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

4. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-cbrt.f64 95.3

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

6. Applied rewrites 95.3%

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

   1. lift-cbrt.f64 N/A

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

   2. pow1/3 N/A

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

   3. lower-pow.f64 95.7

      \[\leadsto \sqrt[3]{\frac{0.5}{a}} \cdot \left(-1 \cdot \left(\sqrt[3]{g} \cdot {2}^{\color{blue}{0.3333333333333333}}\right)\right) \]

8. Applied rewrites 95.7%

   \[\leadsto \sqrt[3]{\frac{0.5}{a}} \cdot \left(-1 \cdot \left(\sqrt[3]{g} \cdot {2}^{\color{blue}{0.3333333333333333}}\right)\right) \]
9. Add Preprocessing

Alternative 2: 95.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

function code(g, h, a)
	return Float64(cbrt(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 45.3%

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

2. 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 95.0

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

4. Applied rewrites 95.0%

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

   1. lift-*.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. pow1/3 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cbrt.f64 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. cbrt-unprod N/A

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

   8. metadata-eval N/A

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

   9. pow1/3 N/A

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

   10. pow-prod-down N/A

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

   11. *-commutative N/A

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

   12. mul-1-neg N/A

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

   13. lift-neg.f64 N/A

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

   14. pow1/3 N/A

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

   15. lift-neg.f64 N/A

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

   16. mul-1-neg N/A

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

   17. *-commutative N/A

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

   18. *-commutative N/A

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

   19. mul-1-neg N/A

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

   20. cbrt-neg-rev N/A

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

   21. lift-cbrt.f64 N/A

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

   22. lower-neg.f64 95.7

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

6. Applied rewrites 95.7%

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

   1. lift-/.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. distribute-frac-neg N/A

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

   4. distribute-neg-frac2 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-neg.f64 95.7

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

8. Applied rewrites 95.7%

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

Alternative 3: 73.9% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 45.3%

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

2. 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 95.0

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

4. Applied rewrites 95.0%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{g} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right)}{\sqrt[3]{a}}} \]
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-cbrt.f64 N/A

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

   7. pow1/3 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-cbrt.f64 N/A

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

   10. lift-cbrt.f64 N/A

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

   11. cbrt-unprod N/A

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

   12. metadata-eval N/A

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

   13. pow1/3 N/A

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

   14. pow-prod-down N/A

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

   15. *-commutative N/A

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

   16. mul-1-neg N/A

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

   17. lift-neg.f64 N/A

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

   18. pow1/3 N/A

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

   19. lift-cbrt.f64 N/A

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

   20. cbrt-neg-rev N/A

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

   21. cbrt-undiv N/A

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

   22. lift-neg.f64 N/A

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

   23. frac-2neg N/A

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

6. Applied rewrites 73.9%

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

Reproduce

herbie shell --seed 2025142 
(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))))))))