2-ancestry mixing, positive discriminant

Percentage Accurate: 43.2% → 96.0%

Time: 8.6s

Alternatives: 4

Speedup: 3.8×

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: 43.2% 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: 96.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 43.2%

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

3. Applied rewrites 48.9%

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

   1. lift-cbrt.f64 N/A

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

   2. metadata-eval N/A

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

   3. cbrt-div N/A

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

   4. metadata-eval N/A

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

   5. lift-cbrt.f64 N/A

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

   6. lower-/.f64 48.8%

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

5. Applied rewrites 48.8%

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

6. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-cbrt.f64 96.0%

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

8. Applied rewrites 96.0%

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

Alternative 2: 96.0% accurate, 2.2× 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(Float64(-cbrt(g)) / 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 43.2%

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

3. Applied rewrites 49.0%

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

4. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-cbrt.f64 96.0%

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

6. Applied rewrites 96.0%

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

   1. lift-*.f64 N/A

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

   2. mul-1-neg N/A

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

   3. lift-/.f64 N/A

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

   4. distribute-neg-frac N/A

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

   5. lower-/.f64 N/A

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

   6. lower-neg.f64 96.0%

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

8. Applied rewrites 96.0%

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

Alternative 3: 73.8% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 43.2%

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

3. Applied rewrites 49.0%

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

4. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-cbrt.f64 96.0%

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

6. Applied rewrites 96.0%

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

   1. lift-*.f64 N/A

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

   2. mul-1-neg N/A

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

   3. lower-neg.f64 96.0%

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

   4. lift-/.f64 N/A

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

   5. lift-cbrt.f64 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. cbrt-undiv N/A

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

   8. lower-cbrt.f64 N/A

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

   9. lower-/.f64 73.1%

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

8. Applied rewrites 73.1%

   \[\leadsto -\sqrt[3]{\frac{g}{a}} \]
9. 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. lift-cbrt.f64 N/A

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

   7. distribute-neg-frac N/A

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

   8. lift-neg.f64 N/A

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

   9. div-flip-rev N/A

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

   10. lift-/.f64 N/A

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

   11. frac-2neg N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. distribute-neg-frac N/A

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

   16. lift-neg.f64 N/A

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

   17. frac-2neg N/A

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

   18. lift-cbrt.f64 N/A

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

   19. lift-cbrt.f64 N/A

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

   20. cbrt-undiv N/A

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

   21. lower-cbrt.f64 N/A

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

   22. lower-/.f64 73.8%

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

10. Applied rewrites 73.8%

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

Alternative 4: 73.1% accurate, 3.8× 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 43.2%

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

3. Applied rewrites 49.0%

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

4. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-cbrt.f64 96.0%

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

6. Applied rewrites 96.0%

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

   1. lift-*.f64 N/A

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

   2. mul-1-neg N/A

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

   3. lower-neg.f64 96.0%

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

   4. lift-/.f64 N/A

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

   5. lift-cbrt.f64 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. cbrt-undiv N/A

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

   8. lower-cbrt.f64 N/A

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

   9. lower-/.f64 73.1%

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

8. Applied rewrites 73.1%

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

Reproduce

herbie shell --seed 2025212 
(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))))))))