2-ancestry mixing, positive discriminant

Percentage Accurate: 42.8% → 96.6%

Time: 12.7s

Alternatives: 7

Speedup: 1.2×

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: 42.8% 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: 96.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{2 \cdot a}\\ t_1 := \frac{h}{g} \cdot \left(-0.5 \cdot h\right)\\ \mathbf{if}\;g \leq 1.6 \cdot 10^{+168}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, -\sqrt[3]{2 \cdot g}, \sqrt[3]{t\_1 \cdot \frac{0.5}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{-\mathsf{fma}\left(\sqrt{g - h}, \sqrt{h + g}, g\right)}, t\_0, \sqrt[3]{t\_1} \cdot t\_0\right)}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (* 2.0 a))) (t_1 (* (/ h g) (* -0.5 h))))
   (if (<= g 1.6e+168)
     (fma (cbrt (/ 0.5 a)) (- (cbrt (* 2.0 g))) (cbrt (* t_1 (/ 0.5 a))))
     (/
      (fma
       (cbrt (- (fma (sqrt (- g h)) (sqrt (+ h g)) g)))
       t_0
       (* (cbrt t_1) t_0))
      (* t_0 t_0)))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt((2.0 * a));
	double t_1 = (h / g) * (-0.5 * h);
	double tmp;
	if (g <= 1.6e+168) {
		tmp = fma(cbrt((0.5 / a)), -cbrt((2.0 * g)), cbrt((t_1 * (0.5 / a))));
	} else {
		tmp = fma(cbrt(-fma(sqrt((g - h)), sqrt((h + g)), g)), t_0, (cbrt(t_1) * t_0)) / (t_0 * t_0);
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(2.0 * a))
	t_1 = Float64(Float64(h / g) * Float64(-0.5 * h))
	tmp = 0.0
	if (g <= 1.6e+168)
		tmp = fma(cbrt(Float64(0.5 / a)), Float64(-cbrt(Float64(2.0 * g))), cbrt(Float64(t_1 * Float64(0.5 / a))));
	else
		tmp = Float64(fma(cbrt(Float64(-fma(sqrt(Float64(g - h)), sqrt(Float64(h + g)), g))), t_0, Float64(cbrt(t_1) * t_0)) / Float64(t_0 * t_0));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[(2.0 * a), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[(N[(h / g), $MachinePrecision] * N[(-0.5 * h), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[g, 1.6e+168], N[(N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * (-N[Power[N[(2.0 * g), $MachinePrecision], 1/3], $MachinePrecision]) + N[Power[N[(t$95$1 * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[(-N[(N[Sqrt[N[(g - h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(h + g), $MachinePrecision]], $MachinePrecision] + g), $MachinePrecision]), 1/3], $MachinePrecision] * t$95$0 + N[(N[Power[t$95$1, 1/3], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if g < 1.6000000000000001e168

   1. Initial program 60.4%

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

   3. Taylor expanded in g around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 35.9

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

   5. Applied rewrites 35.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. div-inv N/A

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

      10. *-lft-identity N/A

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

      11. lower-/.f64 N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 35.8

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

      15. lift--.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. difference-of-squares N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 N/A

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

      21. lift--.f64 N/A

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

      22. lower-*.f64 35.8

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

   7. Applied rewrites 35.8%

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

   9. Applied rewrites 39.5%

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

   10. Taylor expanded in g around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. lower-cbrt.f64 N/A

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

       4. unpow2 N/A

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

       5. rem-square-sqrt N/A

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

       6. metadata-eval N/A

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

       7. lower-*.f64 96.2

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

   12. Applied rewrites 96.2%

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

   if 1.6000000000000001e168 < g

   1. Initial program 0.0%

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

   3. Taylor expanded in g around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 2.8

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

   5. Applied rewrites 2.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. div-inv N/A

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

      10. *-lft-identity N/A

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

      11. lower-/.f64 N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 2.8

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

      15. lift--.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. difference-of-squares N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 N/A

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

      21. lift--.f64 N/A

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

      22. lower-*.f64 3.9

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

   7. Applied rewrites 3.9%

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

   9. Applied rewrites 96.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{-\mathsf{fma}\left(\sqrt{g - h}, \sqrt{h + g}, g\right)}, \sqrt[3]{a \cdot 2}, \sqrt[3]{a \cdot 2} \cdot \sqrt[3]{\left(-0.5 \cdot h\right) \cdot \frac{h}{g}}\right)}{\sqrt[3]{a \cdot 2} \cdot \sqrt[3]{a \cdot 2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq 1.6 \cdot 10^{+168}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, -\sqrt[3]{2 \cdot g}, \sqrt[3]{\left(\frac{h}{g} \cdot \left(-0.5 \cdot h\right)\right) \cdot \frac{0.5}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{-\mathsf{fma}\left(\sqrt{g - h}, \sqrt{h + g}, g\right)}, \sqrt[3]{2 \cdot a}, \sqrt[3]{\frac{h}{g} \cdot \left(-0.5 \cdot h\right)} \cdot \sqrt[3]{2 \cdot a}\right)}{\sqrt[3]{2 \cdot a} \cdot \sqrt[3]{2 \cdot a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{h}{g} \cdot \left(-0.5 \cdot h\right)\\ t_1 := \sqrt[3]{\frac{0.5}{a}}\\ \mathbf{if}\;g \leq 3 \cdot 10^{+186}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, -\sqrt[3]{2 \cdot g}, \sqrt[3]{t\_0 \cdot \frac{0.5}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{t\_0} + \sqrt[3]{-\mathsf{fma}\left(\sqrt{g - h}, \sqrt{h + g}, g\right)}\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (* (/ h g) (* -0.5 h))) (t_1 (cbrt (/ 0.5 a))))
   (if (<= g 3e+186)
     (fma t_1 (- (cbrt (* 2.0 g))) (cbrt (* t_0 (/ 0.5 a))))
     (* (+ (cbrt t_0) (cbrt (- (fma (sqrt (- g h)) (sqrt (+ h g)) g)))) t_1))))

C

double code(double g, double h, double a) {
	double t_0 = (h / g) * (-0.5 * h);
	double t_1 = cbrt((0.5 / a));
	double tmp;
	if (g <= 3e+186) {
		tmp = fma(t_1, -cbrt((2.0 * g)), cbrt((t_0 * (0.5 / a))));
	} else {
		tmp = (cbrt(t_0) + cbrt(-fma(sqrt((g - h)), sqrt((h + g)), g))) * t_1;
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64(Float64(h / g) * Float64(-0.5 * h))
	t_1 = cbrt(Float64(0.5 / a))
	tmp = 0.0
	if (g <= 3e+186)
		tmp = fma(t_1, Float64(-cbrt(Float64(2.0 * g))), cbrt(Float64(t_0 * Float64(0.5 / a))));
	else
		tmp = Float64(Float64(cbrt(t_0) + cbrt(Float64(-fma(sqrt(Float64(g - h)), sqrt(Float64(h + g)), g)))) * t_1);
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(N[(h / g), $MachinePrecision] * N[(-0.5 * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[g, 3e+186], N[(t$95$1 * (-N[Power[N[(2.0 * g), $MachinePrecision], 1/3], $MachinePrecision]) + N[Power[N[(t$95$0 * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[t$95$0, 1/3], $MachinePrecision] + N[Power[(-N[(N[Sqrt[N[(g - h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(h + g), $MachinePrecision]], $MachinePrecision] + g), $MachinePrecision]), 1/3], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if g < 2.99999999999999982e186

   1. Initial program 57.4%

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

   3. Taylor expanded in g around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 34.2

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

   5. Applied rewrites 34.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. div-inv N/A

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

      10. *-lft-identity N/A

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

      11. lower-/.f64 N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 34.1

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

      15. lift--.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. difference-of-squares N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 N/A

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

      21. lift--.f64 N/A

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

      22. lower-*.f64 34.2

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

   7. Applied rewrites 34.2%

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

   9. Applied rewrites 42.4%

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

   10. Taylor expanded in g around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. lower-cbrt.f64 N/A

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

       4. unpow2 N/A

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

       5. rem-square-sqrt N/A

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

       6. metadata-eval N/A

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

       7. lower-*.f64 96.3

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

   12. Applied rewrites 96.3%

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

   if 2.99999999999999982e186 < g

   1. Initial program 0.0%

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

   3. Taylor expanded in g around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 2.9

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

   5. Applied rewrites 2.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. div-inv N/A

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

      10. *-lft-identity N/A

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

      11. lower-/.f64 N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 2.9

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

      15. lift--.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. difference-of-squares N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 N/A

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

      21. lift--.f64 N/A

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

      22. lower-*.f64 3.9

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

   7. Applied rewrites 3.9%

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

   9. Applied rewrites 95.6%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{0.5}{a}} \cdot \left(\sqrt[3]{-\mathsf{fma}\left(\sqrt{g - h}, \sqrt{h + g}, g\right)} + \sqrt[3]{\left(-0.5 \cdot h\right) \cdot \frac{h}{g}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq 3 \cdot 10^{+186}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, -\sqrt[3]{2 \cdot g}, \sqrt[3]{\left(\frac{h}{g} \cdot \left(-0.5 \cdot h\right)\right) \cdot \frac{0.5}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{h}{g} \cdot \left(-0.5 \cdot h\right)} + \sqrt[3]{-\mathsf{fma}\left(\sqrt{g - h}, \sqrt{h + g}, g\right)}\right) \cdot \sqrt[3]{\frac{0.5}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{\left(\left(\frac{h}{g} \cdot h\right) \cdot -0.5\right) \cdot \frac{1}{2 \cdot a}} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (+
  (/ (cbrt (- g)) (cbrt a))
  (cbrt (* (* (* (/ h g) h) -0.5) (/ 1.0 (* 2.0 a))))))

C

double code(double g, double h, double a) {
	return (cbrt(-g) / cbrt(a)) + cbrt(((((h / g) * h) * -0.5) * (1.0 / (2.0 * a))));
}

Java

public static double code(double g, double h, double a) {
	return (Math.cbrt(-g) / Math.cbrt(a)) + Math.cbrt(((((h / g) * h) * -0.5) * (1.0 / (2.0 * a))));
}

Julia

function code(g, h, a)
	return Float64(Float64(cbrt(Float64(-g)) / cbrt(a)) + cbrt(Float64(Float64(Float64(Float64(h / g) * h) * -0.5) * Float64(1.0 / Float64(2.0 * a)))))
end

Wolfram

code[g_, h_, a_] := N[(N[(N[Power[(-g), 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(N[(N[(h / g), $MachinePrecision] * h), $MachinePrecision] * -0.5), $MachinePrecision] * N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 45.3%

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

3. Taylor expanded in g around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 27.6

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

5. Applied rewrites 27.6%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. metadata-eval N/A

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

   8. associate-*r/ N/A

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

   9. cbrt-div N/A

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

   10. lower-/.f64 N/A

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

7. Applied rewrites 29.2%

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

8. Taylor expanded in g around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 86.1

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

10. Applied rewrites 86.1%

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

12. Applied rewrites 94.0%

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

13. Final simplification 94.0%

    \[\leadsto \frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{\left(\left(\frac{h}{g} \cdot h\right) \cdot -0.5\right) \cdot \frac{1}{2 \cdot a}} \]
14. Add Preprocessing

Alternative 4: 92.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{h \cdot h}{g} \cdot \frac{-0.25}{a}} + \frac{\sqrt[3]{-g}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\frac{h}{a} \cdot \frac{h}{g}\right) \cdot -0.25} + \sqrt[3]{\frac{-g}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= h 1.35e+154)
   (+ (cbrt (* (/ (* h h) g) (/ -0.25 a))) (/ (cbrt (- g)) (cbrt a)))
   (+ (cbrt (* (* (/ h a) (/ h g)) -0.25)) (cbrt (/ (- g) a)))))

C

double code(double g, double h, double a) {
	double tmp;
	if (h <= 1.35e+154) {
		tmp = cbrt((((h * h) / g) * (-0.25 / a))) + (cbrt(-g) / cbrt(a));
	} else {
		tmp = cbrt((((h / a) * (h / g)) * -0.25)) + cbrt((-g / a));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if (h <= 1.35e+154) {
		tmp = Math.cbrt((((h * h) / g) * (-0.25 / a))) + (Math.cbrt(-g) / Math.cbrt(a));
	} else {
		tmp = Math.cbrt((((h / a) * (h / g)) * -0.25)) + Math.cbrt((-g / a));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (h <= 1.35e+154)
		tmp = Float64(cbrt(Float64(Float64(Float64(h * h) / g) * Float64(-0.25 / a))) + Float64(cbrt(Float64(-g)) / cbrt(a)));
	else
		tmp = Float64(cbrt(Float64(Float64(Float64(h / a) * Float64(h / g)) * -0.25)) + cbrt(Float64(Float64(-g) / a)));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[h, 1.35e+154], N[(N[Power[N[(N[(N[(h * h), $MachinePrecision] / g), $MachinePrecision] * N[(-0.25 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[(-g), 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(N[(h / a), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 1.35000000000000003e154

   1. Initial program 49.6%

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

   3. Taylor expanded in g around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 30.2

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

   5. Applied rewrites 30.2%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. cbrt-div N/A

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

      10. lower-/.f64 N/A

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

   7. Applied rewrites 31.7%

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

   8. Taylor expanded in g around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 93.8

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

   10. Applied rewrites 93.8%

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

   11. Taylor expanded in g around inf

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

       1. associate-*r/ N/A

          \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{-1}{4} \cdot {h}^{2}}{a \cdot g}}} + \frac{\sqrt[3]{-g}}{\sqrt[3]{a}} \]

       2. times-frac N/A

          \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{-1}{4}}{a} \cdot \frac{{h}^{2}}{g}}} + \frac{\sqrt[3]{-g}}{\sqrt[3]{a}} \]

       3. lower-*.f64 N/A

          \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{-1}{4}}{a} \cdot \frac{{h}^{2}}{g}}} + \frac{\sqrt[3]{-g}}{\sqrt[3]{a}} \]

       4. lower-/.f64 N/A

          \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{-1}{4}}{a}} \cdot \frac{{h}^{2}}{g}} + \frac{\sqrt[3]{-g}}{\sqrt[3]{a}} \]

       5. lower-/.f64 N/A

          \[\leadsto \sqrt[3]{\frac{\frac{-1}{4}}{a} \cdot \color{blue}{\frac{{h}^{2}}{g}}} + \frac{\sqrt[3]{-g}}{\sqrt[3]{a}} \]

       6. unpow2 N/A

          \[\leadsto \sqrt[3]{\frac{\frac{-1}{4}}{a} \cdot \frac{\color{blue}{h \cdot h}}{g}} + \frac{\sqrt[3]{-g}}{\sqrt[3]{a}} \]

       7. lower-*.f64 93.8

          \[\leadsto \sqrt[3]{\frac{-0.25}{a} \cdot \frac{\color{blue}{h \cdot h}}{g}} + \frac{\sqrt[3]{-g}}{\sqrt[3]{a}} \]

   13. Applied rewrites 93.8%

       \[\leadsto \sqrt[3]{\color{blue}{\frac{-0.25}{a} \cdot \frac{h \cdot h}{g}}} + \frac{\sqrt[3]{-g}}{\sqrt[3]{a}} \]

   if 1.35000000000000003e154 < h

   1. Initial program 0.0%

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

   3. Taylor expanded in g around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 0.0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in g around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cbrt.f64 N/A

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

      3. unpow2 N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cbrt.f64 N/A

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

      12. lower-cbrt.f64 51.7

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

   8. Applied rewrites 51.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 51.7

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

   10. Applied rewrites 51.7%

       \[\leadsto \color{blue}{\sqrt[3]{\frac{-g}{a}} + \sqrt[3]{-0.25 \cdot \left(\frac{h}{a} \cdot \frac{h}{g}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{h \cdot h}{g} \cdot \frac{-0.25}{a}} + \frac{\sqrt[3]{-g}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\frac{h}{a} \cdot \frac{h}{g}\right) \cdot -0.25} + \sqrt[3]{\frac{-g}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 45.3%

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

3. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 29.9

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

5. Applied rewrites 29.9%

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

6. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-cbrt.f64 N/A

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

   3. unpow2 N/A

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

   4. *-commutative N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cbrt.f64 N/A

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

   12. lower-cbrt.f64 76.6

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

8. Applied rewrites 76.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 76.6

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

10. Applied rewrites 76.6%

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

11. Final simplification 76.6%

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

Alternative 6: 15.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\left(-2 \cdot g\right) \cdot \frac{1}{2 \cdot a}} + \sqrt[3]{\frac{-g}{a}} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 45.3%

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

3. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 29.9

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

5. Applied rewrites 29.9%

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

6. Taylor expanded in g around -inf

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

   1. lower-*.f64 15.5

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

8. Applied rewrites 15.5%

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

9. Final simplification 15.5%

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

Alternative 7: 2.9% 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 45.3%

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

   1. lift-cbrt.f64 N/A

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

   2. pow1/3 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-*l/ N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. times-frac N/A

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

   9. unpow-prod-down N/A

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

   10. lower-*.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. inv-pow N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-pow.f64 N/A

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

4. Applied rewrites 25.6%

   \[\leadsto \color{blue}{{\left({a}^{-1}\right)}^{0.3333333333333333} \cdot {\left(\left(\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g\right) \cdot 0.5\right)}^{0.3333333333333333}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

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. *-commutative N/A

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

   3. distribute-lft-neg-in N/A

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. *-commutative N/A

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

   8. +-commutative N/A

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

   9. unpow2 N/A

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

   10. rem-square-sqrt N/A

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

   11. metadata-eval N/A

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

   12. mul0-lft N/A

       \[\leadsto \left(-\sqrt[3]{\frac{1}{2}}\right) \cdot \sqrt[3]{\frac{\color{blue}{0}}{a}} \]

   13. mul0-lft N/A

       \[\leadsto \left(-\sqrt[3]{\frac{1}{2}}\right) \cdot \sqrt[3]{\frac{\color{blue}{0 \cdot h}}{a}} \]

   14. metadata-eval N/A

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

   15. distribute-rgt1-in N/A

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

   16. lower-cbrt.f64 N/A

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

   17. distribute-rgt1-in N/A

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

   18. metadata-eval N/A

       \[\leadsto \left(-\sqrt[3]{\frac{1}{2}}\right) \cdot \sqrt[3]{\frac{\color{blue}{0} \cdot h}{a}} \]

   19. mul0-lft N/A

       \[\leadsto \left(-\sqrt[3]{\frac{1}{2}}\right) \cdot \sqrt[3]{\frac{\color{blue}{0}}{a}} \]

   20. lower-/.f64 3.0

       \[\leadsto \left(-\sqrt[3]{0.5}\right) \cdot \sqrt[3]{\color{blue}{\frac{0}{a}}} \]

7. Applied rewrites 3.0%

   \[\leadsto \color{blue}{\left(-\sqrt[3]{0.5}\right) \cdot \sqrt[3]{\frac{0}{a}}} \]

8. Taylor expanded in a around 0

   \[\leadsto 0 \]
9. Step-by-step derivation

10. Applied rewrites 3.0%

    \[\leadsto 0 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024248 
(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))))))))