2-ancestry mixing, positive discriminant

Percentage Accurate: 44.4% → 94.0%

Time: 11.7s

Alternatives: 4

Speedup: 2.6×

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 44.4% 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: 94.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\left(-0.25 \cdot \frac{h}{g}\right) \cdot \frac{h}{a}} + \sqrt[3]{\frac{-1}{a}} \cdot \sqrt[3]{g} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (+ (cbrt (* (* -0.25 (/ h g)) (/ h a))) (* (cbrt (/ -1.0 a)) (cbrt g))))

C

double code(double g, double h, double a) {
	return cbrt(((-0.25 * (h / g)) * (h / a))) + (cbrt((-1.0 / a)) * cbrt(g));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt(((-0.25 * (h / g)) * (h / a))) + (Math.cbrt((-1.0 / a)) * Math.cbrt(g));
}

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(-0.25 * Float64(h / g)) * Float64(h / a))) + Float64(cbrt(Float64(-1.0 / a)) * cbrt(g)))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(N[(-0.25 * N[(h / g), $MachinePrecision]), $MachinePrecision] * N[(h / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(-1.0 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[g, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.8%

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

3. Taylor expanded in h around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. lower-cbrt.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cbrt.f64 N/A

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

   11. unpow2 N/A

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

   12. *-commutative N/A

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

   13. times-frac N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

5. Applied rewrites 74.5%

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

7. Applied rewrites 94.3%

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

9. Applied rewrites 95.2%

   \[\leadsto \mathsf{fma}\left(\sqrt[3]{g}, \color{blue}{\sqrt[3]{\frac{2}{a}} \cdot \sqrt[3]{-0.5}}, \sqrt[3]{-0.25 \cdot \left(\frac{h}{a} \cdot \frac{h}{g}\right)}\right) \]
10. Step-by-step derivation

11. Applied rewrites 95.4%

    \[\leadsto \sqrt[3]{\left(-0.25 \cdot \frac{h}{g}\right) \cdot \frac{h}{a}} - \color{blue}{\left(-\sqrt[3]{\frac{-1}{a}}\right) \cdot \sqrt[3]{g}} \]

12. Final simplification 95.4%

    \[\leadsto \sqrt[3]{\left(-0.25 \cdot \frac{h}{g}\right) \cdot \frac{h}{a}} + \sqrt[3]{\frac{-1}{a}} \cdot \sqrt[3]{g} \]
13. Add Preprocessing

Alternative 2: 94.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt[3]{\frac{-1}{a}}, \sqrt[3]{g}, \sqrt[3]{\left(-0.25 \cdot \frac{h}{g}\right) \cdot \frac{h}{a}}\right) \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (fma (cbrt (/ -1.0 a)) (cbrt g) (cbrt (* (* -0.25 (/ h g)) (/ h a)))))

C

double code(double g, double h, double a) {
	return fma(cbrt((-1.0 / a)), cbrt(g), cbrt(((-0.25 * (h / g)) * (h / a))));
}

Julia

function code(g, h, a)
	return fma(cbrt(Float64(-1.0 / a)), cbrt(g), cbrt(Float64(Float64(-0.25 * Float64(h / g)) * Float64(h / a))))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(-1.0 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[g, 1/3], $MachinePrecision] + N[Power[N[(N[(-0.25 * N[(h / g), $MachinePrecision]), $MachinePrecision] * N[(h / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.8%

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

3. Taylor expanded in h around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. lower-cbrt.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cbrt.f64 N/A

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

   11. unpow2 N/A

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

   12. *-commutative N/A

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

   13. times-frac N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

5. Applied rewrites 74.5%

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

7. Applied rewrites 94.3%

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

9. Applied rewrites 95.2%

   \[\leadsto \mathsf{fma}\left(\sqrt[3]{g}, \color{blue}{\sqrt[3]{\frac{2}{a}} \cdot \sqrt[3]{-0.5}}, \sqrt[3]{-0.25 \cdot \left(\frac{h}{a} \cdot \frac{h}{g}\right)}\right) \]
10. Step-by-step derivation

11. Applied rewrites 95.4%

    \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{-1}{a}}, \color{blue}{\sqrt[3]{g}}, \sqrt[3]{\left(-0.25 \cdot \frac{h}{g}\right) \cdot \frac{h}{a}}\right) \]
12. Add Preprocessing

Alternative 3: 96.0% accurate, 1.4× 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(Float64(-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 46.8%

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

4. Applied rewrites 15.3%

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

5. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-cbrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-cbrt.f64 72.6

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

7. Applied rewrites 72.6%

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

9. Applied rewrites 72.6%

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

11. Applied rewrites 94.5%

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

Alternative 4: 73.6% accurate, 2.6× 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 cbrt(Float64(Float64(-g) / a))
end

Wolfram

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

Derivation

1. Initial program 46.8%

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

4. Applied rewrites 15.3%

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

5. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-cbrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-cbrt.f64 72.6

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

7. Applied rewrites 72.6%

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

9. Applied rewrites 72.6%

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

11. Applied rewrites 72.6%

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

Reproduce

herbie shell --seed 2024358 
(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))))))))