2-ancestry mixing, positive discriminant

Percentage Accurate: 43.7% → 97.9%

Time: 10.2s

Alternatives: 5

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: 43.7% 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: 97.9% accurate, 0.7× speedup

Math

\[-\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}} + \frac{{\left(\left|h\right|\right)}^{0.6666666666666666} \cdot \sqrt[3]{0.25}}{\sqrt[3]{a} \cdot \sqrt[3]{g}}\right) \]

FPCore

(FPCore (g h a)
 :precision binary64
 (-
  (+
   (/ (cbrt g) (cbrt a))
   (/
    (* (pow (fabs h) 0.6666666666666666) (cbrt 0.25))
    (* (cbrt a) (cbrt g))))))

C

double code(double g, double h, double a) {
	return -((cbrt(g) / cbrt(a)) + ((pow(fabs(h), 0.6666666666666666) * cbrt(0.25)) / (cbrt(a) * cbrt(g))));
}

Java

public static double code(double g, double h, double a) {
	return -((Math.cbrt(g) / Math.cbrt(a)) + ((Math.pow(Math.abs(h), 0.6666666666666666) * Math.cbrt(0.25)) / (Math.cbrt(a) * Math.cbrt(g))));
}

Julia

function code(g, h, a)
	return Float64(-Float64(Float64(cbrt(g) / cbrt(a)) + Float64(Float64((abs(h) ^ 0.6666666666666666) * cbrt(0.25)) / Float64(cbrt(a) * cbrt(g)))))
end

Wolfram

code[g_, h_, a_] := (-N[(N[(N[Power[g, 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[N[Abs[h], $MachinePrecision], 0.6666666666666666], $MachinePrecision] * N[Power[0.25, 1/3], $MachinePrecision]), $MachinePrecision] / N[(N[Power[a, 1/3], $MachinePrecision] * N[Power[g, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])

Derivation

1. Initial program 43.7%

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

3. Applied rewrites 43.3%

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

4. Taylor expanded in g around -inf

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

   1. lower-fma.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. lower-cbrt.f64 N/A

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

   9. lower-*.f64 N/A

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

6. Applied rewrites 48.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{\sqrt[3]{g} \cdot \left(\sqrt[3]{0.5} \cdot \sqrt[3]{2}\right)}{\sqrt[3]{a}}, -1 \cdot \frac{{h}^{0.6666666666666666} \cdot {\left(\sqrt[3]{0.5}\right)}^{2}}{\sqrt[3]{a} \cdot \sqrt[3]{g}}\right)} \]

8. Applied rewrites 35.9%

   \[\leadsto \color{blue}{-\mathsf{fma}\left(\sqrt[3]{\frac{0.25}{a \cdot g}}, {h}^{0.6666666666666666}, \sqrt[3]{\frac{g}{a}}\right)} \]

9. Taylor expanded in g around 0

   \[\leadsto -\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}} + \frac{{h}^{\frac{2}{3}} \cdot \sqrt[3]{\frac{1}{4}}}{\sqrt[3]{a} \cdot \sqrt[3]{g}}\right) \]
10. Step-by-step derivation

    1. lower-+.f64 N/A

       \[\leadsto -\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}} + \frac{{h}^{\frac{2}{3}} \cdot \sqrt[3]{\frac{1}{4}}}{\sqrt[3]{a} \cdot \sqrt[3]{g}}\right) \]

    2. lower-/.f64 N/A

       \[\leadsto -\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}} + \frac{{h}^{\frac{2}{3}} \cdot \sqrt[3]{\frac{1}{4}}}{\sqrt[3]{a} \cdot \sqrt[3]{g}}\right) \]

    3. lower-cbrt.f64 N/A

       \[\leadsto -\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}} + \frac{{h}^{\frac{2}{3}} \cdot \sqrt[3]{\frac{1}{4}}}{\sqrt[3]{a} \cdot \sqrt[3]{g}}\right) \]

    4. lower-cbrt.f64 N/A

       \[\leadsto -\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}} + \frac{{h}^{\frac{2}{3}} \cdot \sqrt[3]{\frac{1}{4}}}{\sqrt[3]{a} \cdot \sqrt[3]{g}}\right) \]

    5. lower-/.f64 N/A

       \[\leadsto -\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}} + \frac{{h}^{\frac{2}{3}} \cdot \sqrt[3]{\frac{1}{4}}}{\sqrt[3]{a} \cdot \sqrt[3]{g}}\right) \]

    6. lower-*.f64 N/A

       \[\leadsto -\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}} + \frac{{h}^{\frac{2}{3}} \cdot \sqrt[3]{\frac{1}{4}}}{\sqrt[3]{a} \cdot \sqrt[3]{g}}\right) \]

    7. lower-pow.f64 N/A

       \[\leadsto -\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}} + \frac{{h}^{\frac{2}{3}} \cdot \sqrt[3]{\frac{1}{4}}}{\sqrt[3]{a} \cdot \sqrt[3]{g}}\right) \]

    8. lower-cbrt.f64 N/A

       \[\leadsto -\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}} + \frac{{h}^{\frac{2}{3}} \cdot \sqrt[3]{\frac{1}{4}}}{\sqrt[3]{a} \cdot \sqrt[3]{g}}\right) \]

    9. lower-*.f64 N/A

       \[\leadsto -\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}} + \frac{{h}^{\frac{2}{3}} \cdot \sqrt[3]{\frac{1}{4}}}{\sqrt[3]{a} \cdot \sqrt[3]{g}}\right) \]

    10. lower-cbrt.f64 N/A

        \[\leadsto -\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}} + \frac{{h}^{\frac{2}{3}} \cdot \sqrt[3]{\frac{1}{4}}}{\sqrt[3]{a} \cdot \sqrt[3]{g}}\right) \]

    11. lower-cbrt.f64 48.8%

        \[\leadsto -\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}} + \frac{{h}^{0.6666666666666666} \cdot \sqrt[3]{0.25}}{\sqrt[3]{a} \cdot \sqrt[3]{g}}\right) \]

11. Applied rewrites 48.8%

    \[\leadsto -\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}} + \frac{{h}^{0.6666666666666666} \cdot \sqrt[3]{0.25}}{\sqrt[3]{a} \cdot \sqrt[3]{g}}\right) \]
12. Add Preprocessing

Alternative 2: 95.8% 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 43.7%

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

3. Applied rewrites 49.6%

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

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 95.8%

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

6. Applied rewrites 95.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. cbrt-neg-rev N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. cbrt-unprod N/A

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

   11. lift-cbrt.f64 N/A

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

   12. lift-cbrt.f64 N/A

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

   13. associate-*r* N/A

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

   14. associate-/l* N/A

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

   15. lower-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lift-cbrt.f64 N/A

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

   18. lift-cbrt.f64 N/A

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

   19. cbrt-unprod N/A

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

   20. lower-cbrt.f64 N/A

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

   21. lower-*.f64 N/A

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

   22. lift-cbrt.f64 N/A

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

   23. lift-cbrt.f64 N/A

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

   24. cbrt-undiv N/A

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

   25. lower-cbrt.f64 N/A

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

   26. lower-/.f64 95.8%

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

8. Applied rewrites 95.8%

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

Alternative 3: 95.8% 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 43.7%

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

3. Applied rewrites 49.6%

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

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 95.8%

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

6. Applied rewrites 95.8%

   \[\leadsto \color{blue}{-1 \cdot \frac{\sqrt[3]{g}}{\sqrt[3]{a}}} \]
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. mult-flip N/A

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

   5. associate-/l* N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. cbrt-unprod N/A

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

   9. lift-cbrt.f64 N/A

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

   10. lift-cbrt.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. lower-neg.f64 95.1%

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

   15. lift-/.f64 N/A

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

8. Applied rewrites 73.6%

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

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

   9. lower-neg.f64 95.8%

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

10. Applied rewrites 95.8%

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

Alternative 4: 73.6% accurate, 3.5× 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 43.7%

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

3. Applied rewrites 49.6%

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

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 95.8%

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

6. Applied rewrites 95.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. cbrt-neg-rev N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. cbrt-unprod N/A

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

   11. lift-cbrt.f64 N/A

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

   12. lift-cbrt.f64 N/A

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

   13. associate-*r* N/A

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

   14. associate-*r* N/A

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

   15. lift-*.f64 N/A

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

   16. associate-/l* N/A

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

   17. *-commutative N/A

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

8. Applied rewrites 73.6%

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

Alternative 5: 73.6% 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 43.7%

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

3. Applied rewrites 49.6%

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

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 95.8%

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

6. Applied rewrites 95.8%

   \[\leadsto \color{blue}{-1 \cdot \frac{\sqrt[3]{g}}{\sqrt[3]{a}}} \]
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. mult-flip N/A

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

   5. associate-/l* N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. cbrt-unprod N/A

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

   9. lift-cbrt.f64 N/A

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

   10. lift-cbrt.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. lower-neg.f64 95.1%

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

   15. lift-/.f64 N/A

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

8. Applied rewrites 73.6%

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

Reproduce

herbie shell --seed 2025209 
(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))))))))