2-ancestry mixing, positive discriminant

Percentage Accurate: 44.5% → 97.9%

Time: 11.4s

Alternatives: 6

Speedup: 2.5×

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: 44.5% 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: 97.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 46.5%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
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 77.6%

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

7. Applied rewrites 94.4%

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

9. Applied rewrites 97.5%

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

11. Applied rewrites 98.3%

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

Alternative 2: 95.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 46.5%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
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 77.6%

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

7. Applied rewrites 94.4%

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

8. Taylor expanded in g around inf

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

10. Applied rewrites 76.0%

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

12. Applied rewrites 96.4%

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

Alternative 3: 95.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt[3]{g} \cdot \sqrt[3]{-1}}{\sqrt[3]{a}} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 46.5%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
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 77.6%

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

7. Applied rewrites 94.4%

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

8. Taylor expanded in g around inf

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

10. Applied rewrites 76.0%

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

12. Applied rewrites 96.4%

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

Alternative 4: 73.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 46.5%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. cbrt-div N/A

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

   6. *-lft-identity N/A

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

   7. lower-/.f64 N/A

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

4. Applied rewrites 48.7%

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

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 76.8

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

7. Applied rewrites 76.8%

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

Alternative 5: 73.4% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 46.5%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
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 77.6%

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

7. Applied rewrites 94.4%

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

8. Taylor expanded in g around inf

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

10. Applied rewrites 76.0%

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

12. Applied rewrites 76.8%

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

Alternative 6: 3.0% accurate, 302.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (g h a) :precision binary64 0.0)

C

double code(double g, double h, double a) {
	return 0.0;
}

Fortran

real(8) function code(g, h, a)
    real(8), intent (in) :: g
    real(8), intent (in) :: h
    real(8), intent (in) :: a
    code = 0.0d0
end function

Java

public static double code(double g, double h, double a) {
	return 0.0;
}

Python

def code(g, h, a):
	return 0.0

Julia

function code(g, h, a)
	return 0.0
end

MATLAB

function tmp = code(g, h, a)
	tmp = 0.0;
end

Wolfram

code[g_, h_, a_] := 0.0

Derivation

1. Initial program 46.5%

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

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. difference-of-squares N/A

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

   7. *-commutative N/A

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

   8. unpow-prod-down N/A

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

   9. lower-*.f64 N/A

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

   10. pow1/2 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower--.f64 N/A

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

   13. pow1/2 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 19.3

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

4. Applied rewrites 19.3%

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

5. Taylor expanded in g around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. rem-square-sqrt N/A

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

   8. metadata-eval N/A

      \[\leadsto -\sqrt[3]{\frac{g \cdot \color{blue}{0}}{a}} \cdot \sqrt[3]{\frac{1}{2}} \]

   9. lower-*.f64 N/A

      \[\leadsto -\sqrt[3]{\frac{\color{blue}{g \cdot 0}}{a}} \cdot \sqrt[3]{\frac{1}{2}} \]

   10. lower-cbrt.f64 3.0

       \[\leadsto -\sqrt[3]{\frac{g \cdot 0}{a}} \cdot \color{blue}{\sqrt[3]{0.5}} \]

7. Applied rewrites 3.0%

   \[\leadsto \color{blue}{-\sqrt[3]{\frac{g \cdot 0}{a}} \cdot \sqrt[3]{0.5}} \]
8. Step-by-step derivation

9. Applied rewrites 3.0%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024340 
(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))))))))