2-ancestry mixing, positive discriminant

Percentage Accurate: 44.3% → 96.0%

Time: 8.1s

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: 44.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 96.0% accurate, 1.2× speedup

Math

\[\mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{g + g}, \sqrt[3]{\frac{g - g}{a + a}}\right) \]

FPCore

(FPCore (g h a)
 :precision binary64
 (fma (cbrt (/ -0.5 a)) (cbrt (+ g g)) (cbrt (/ (- g g) (+ a a)))))

C

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

Julia

function code(g, h, a)
	return fma(cbrt(Float64(-0.5 / a)), cbrt(Float64(g + g)), cbrt(Float64(Float64(g - g) / Float64(a + a))))
end

Wolfram

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

Derivation

1. Initial program 44.3%

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

3. Applied rewrites 47.0%

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

4. Taylor expanded in g around inf

   \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{\color{blue}{g \cdot \left(1 + \frac{1}{2} \cdot \frac{h + -1 \cdot h}{g}\right)} + g}, \sqrt[3]{\frac{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}{a + a}}\right) \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-*.f64 32.4%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right) + g}, \sqrt[3]{\frac{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}{a + a}}\right) \]

6. Applied rewrites 32.4%

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

7. Taylor expanded in g around inf

   \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right) + g}, \sqrt[3]{\frac{\color{blue}{g \cdot \left(1 + \frac{1}{2} \cdot \frac{h + -1 \cdot h}{g}\right)} - g}{a + a}}\right) \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-*.f64 95.9%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right) + g}, \sqrt[3]{\frac{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right) - g}{a + a}}\right) \]

9. Applied rewrites 95.9%

   \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right) + g}, \sqrt[3]{\frac{\color{blue}{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right)} - g}{a + a}}\right) \]
10. Step-by-step derivation

11. Applied rewrites 95.9%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + 0\right) - g}{a + a}}\right)} \]

12. Taylor expanded in g around 0

    \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{g + g}, \sqrt[3]{\frac{\left(g + 0\right) - g}{a + a}}\right) \]
13. Step-by-step derivation

14. Applied rewrites 95.9%

    \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{g + g}, \sqrt[3]{\frac{\left(g + 0\right) - g}{a + a}}\right) \]

15. Taylor expanded in g around 0

    \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{g + g}, \sqrt[3]{\frac{g - g}{a + a}}\right) \]
16. Step-by-step derivation

17. Applied rewrites 95.9%

    \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{g + g}, \sqrt[3]{\frac{g - g}{a + a}}\right) \]
18. Add Preprocessing

Alternative 2: 96.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.3%

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

2. Taylor expanded in g around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-cbrt.f64 95.4%

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

4. Applied rewrites 95.4%

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

   1. lift-/.f64 N/A

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

   2. div-flip N/A

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

   3. lower-unsound-/.f64 N/A

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

   4. lower-unsound-/.f64 95.3%

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

   5. lift-*.f64 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. lift-cbrt.f64 N/A

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

   8. cbrt-unprod N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. cbrt-neg N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval 96.0%

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

   14. lower-*.f64 N/A

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

   15. lift-cbrt.f64 N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. cbrt-neg N/A

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

   19. metadata-eval N/A

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

   20. cbrt-unprod N/A

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

   21. *-commutative N/A

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

   22. mul-1-neg N/A

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

   23. cbrt-neg-rev N/A

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

   24. lift-cbrt.f64 N/A

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

   25. lower-neg.f64 96.0%

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

6. Applied rewrites 96.0%

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

Alternative 3: 95.9% 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 44.3%

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

2. Taylor expanded in g around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-cbrt.f64 95.4%

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

4. Applied rewrites 95.4%

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

   1. lift-*.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. cbrt-unprod N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. cbrt-neg N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval 96.0%

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

   10. lower-*.f64 N/A

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

   11. lift-cbrt.f64 N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. cbrt-neg N/A

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

   15. metadata-eval N/A

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

   16. cbrt-unprod N/A

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

   17. *-commutative N/A

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

   18. mul-1-neg N/A

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

   19. cbrt-neg-rev N/A

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

   20. lift-cbrt.f64 N/A

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

   21. lower-neg.f64 96.0%

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

6. Applied rewrites 96.0%

   \[\leadsto \frac{-\sqrt[3]{g}}{\color{blue}{\sqrt[3]{a}}} \]
7. 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 44.3%

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

2. Taylor expanded in g around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-cbrt.f64 95.4%

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

4. Applied rewrites 95.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. lift-cbrt.f64 N/A

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

   8. cbrt-unprod N/A

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

   9. metadata-eval N/A

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

   10. lift-cbrt.f64 N/A

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

   11. cbrt-undiv N/A

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

   12. lift-cbrt.f64 N/A

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

   13. cbrt-unprod N/A

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

   14. lower-cbrt.f64 N/A

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

   15. frac-2neg-rev N/A

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

   16. metadata-eval N/A

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

   17. lower-*.f64 N/A

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

   18. metadata-eval N/A

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

   19. frac-2neg-rev N/A

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

   20. lower-/.f64 73.6%

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

6. Applied rewrites 73.6%

   \[\leadsto \sqrt[3]{\frac{-1}{a} \cdot g} \]
7. 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 44.3%

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

2. Taylor expanded in g around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-cbrt.f64 95.4%

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

4. Applied rewrites 95.4%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. distribute-frac-neg N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cbrt.f64 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. cbrt-unprod N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. cbrt-neg N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower-*.f64 N/A

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

   14. lift-cbrt.f64 N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. cbrt-neg N/A

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

   18. metadata-eval N/A

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

   19. cbrt-unprod N/A

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

   20. *-commutative N/A

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

   21. mul-1-neg N/A

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

   22. cbrt-neg-rev N/A

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

   23. lift-cbrt.f64 N/A

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

6. Applied rewrites 73.6%

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

Reproduce

herbie shell --seed 2025201 
(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))))))))