2-ancestry mixing, positive discriminant

Percentage Accurate: 44.3% → 93.9%

Time: 12.1s

Alternatives: 9

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: 44.3% 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: 93.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{0.5}{a}}\\ \mathbf{if}\;g \leq 4 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -\sqrt[3]{2 \cdot g}, \sqrt[3]{\left(\mathsf{fma}\left(0, h, \frac{\mathsf{fma}\left(-1, h \cdot h, {\left(0 \cdot h\right)}^{2} \cdot -0.25\right)}{g}\right) \cdot 0.5\right) \cdot \frac{0.5}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{0}{a}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (/ 0.5 a))))
   (if (<= g 4e+139)
     (fma
      t_0
      (- (cbrt (* 2.0 g)))
      (cbrt
       (*
        (*
         (fma 0.0 h (/ (fma -1.0 (* h h) (* (pow (* 0.0 h) 2.0) -0.25)) g))
         0.5)
        (/ 0.5 a))))
     (fma
      t_0
      (cbrt (- (fma (sqrt (+ h g)) (sqrt (- g h)) g)))
      (cbrt (/ 0.0 a))))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt((0.5 / a));
	double tmp;
	if (g <= 4e+139) {
		tmp = fma(t_0, -cbrt((2.0 * g)), cbrt(((fma(0.0, h, (fma(-1.0, (h * h), (pow((0.0 * h), 2.0) * -0.25)) / g)) * 0.5) * (0.5 / a))));
	} else {
		tmp = fma(t_0, cbrt(-fma(sqrt((h + g)), sqrt((g - h)), g)), cbrt((0.0 / a)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(0.5 / a))
	tmp = 0.0
	if (g <= 4e+139)
		tmp = fma(t_0, Float64(-cbrt(Float64(2.0 * g))), cbrt(Float64(Float64(fma(0.0, h, Float64(fma(-1.0, Float64(h * h), Float64((Float64(0.0 * h) ^ 2.0) * -0.25)) / g)) * 0.5) * Float64(0.5 / a))));
	else
		tmp = fma(t_0, cbrt(Float64(-fma(sqrt(Float64(h + g)), sqrt(Float64(g - h)), g))), cbrt(Float64(0.0 / a)));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[g, 4e+139], N[(t$95$0 * (-N[Power[N[(2.0 * g), $MachinePrecision], 1/3], $MachinePrecision]) + N[Power[N[(N[(N[(0.0 * h + N[(N[(-1.0 * N[(h * h), $MachinePrecision] + N[(N[Power[N[(0.0 * h), $MachinePrecision], 2.0], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] / g), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[(-N[(N[Sqrt[N[(h + g), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(g - h), $MachinePrecision]], $MachinePrecision] + g), $MachinePrecision]), 1/3], $MachinePrecision] + N[Power[N[(0.0 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if g < 4.00000000000000013e139

   1. Initial program 56.9%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   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. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. cbrt-div N/A

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

      6. *-lft-identity N/A

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

      7. pow1/3 N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 60.6%

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

   6. Applied rewrites 31.0%

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

   7. Taylor expanded in g around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. rem-square-sqrt N/A

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

      5. unpow2 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. rem-square-sqrt N/A

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

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\left(\frac{1}{2} \cdot \mathsf{fma}\left(\color{blue}{0}, h, \frac{-1 \cdot {h}^{2} - \frac{1}{4} \cdot {\left(h + -1 \cdot h\right)}^{2}}{g}\right)\right) \cdot \frac{\frac{1}{2}}{a}}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\left(\frac{1}{2} \cdot \mathsf{fma}\left(0, h, \color{blue}{\frac{-1 \cdot {h}^{2} - \frac{1}{4} \cdot {\left(h + -1 \cdot h\right)}^{2}}{g}}\right)\right) \cdot \frac{\frac{1}{2}}{a}}\right) \]

   9. Applied rewrites 33.5%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\color{blue}{\left(0.5 \cdot \mathsf{fma}\left(0, h, \frac{\mathsf{fma}\left(-1, h \cdot h, -0.25 \cdot {\left(0 \cdot h\right)}^{2}\right)}{g}\right)\right)} \cdot \frac{0.5}{a}}\right) \]

   10. Taylor expanded in g around -inf

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \color{blue}{-1 \cdot \sqrt[3]{g \cdot \left(1 - {\left(\sqrt{-1}\right)}^{2}\right)}}, \sqrt[3]{\left(\frac{1}{2} \cdot \mathsf{fma}\left(0, h, \frac{\mathsf{fma}\left(-1, h \cdot h, \frac{-1}{4} \cdot {\left(0 \cdot h\right)}^{2}\right)}{g}\right)\right) \cdot \frac{\frac{1}{2}}{a}}\right) \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \color{blue}{\mathsf{neg}\left(\sqrt[3]{g \cdot \left(1 - {\left(\sqrt{-1}\right)}^{2}\right)}\right)}, \sqrt[3]{\left(\frac{1}{2} \cdot \mathsf{fma}\left(0, h, \frac{\mathsf{fma}\left(-1, h \cdot h, \frac{-1}{4} \cdot {\left(0 \cdot h\right)}^{2}\right)}{g}\right)\right) \cdot \frac{\frac{1}{2}}{a}}\right) \]

       2. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \color{blue}{-\sqrt[3]{g \cdot \left(1 - {\left(\sqrt{-1}\right)}^{2}\right)}}, \sqrt[3]{\left(\frac{1}{2} \cdot \mathsf{fma}\left(0, h, \frac{\mathsf{fma}\left(-1, h \cdot h, \frac{-1}{4} \cdot {\left(0 \cdot h\right)}^{2}\right)}{g}\right)\right) \cdot \frac{\frac{1}{2}}{a}}\right) \]

       3. lower-cbrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, -\color{blue}{\sqrt[3]{g \cdot \left(1 - {\left(\sqrt{-1}\right)}^{2}\right)}}, \sqrt[3]{\left(\frac{1}{2} \cdot \mathsf{fma}\left(0, h, \frac{\mathsf{fma}\left(-1, h \cdot h, \frac{-1}{4} \cdot {\left(0 \cdot h\right)}^{2}\right)}{g}\right)\right) \cdot \frac{\frac{1}{2}}{a}}\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, -\sqrt[3]{g \cdot \left(1 - \color{blue}{\sqrt{-1} \cdot \sqrt{-1}}\right)}, \sqrt[3]{\left(\frac{1}{2} \cdot \mathsf{fma}\left(0, h, \frac{\mathsf{fma}\left(-1, h \cdot h, \frac{-1}{4} \cdot {\left(0 \cdot h\right)}^{2}\right)}{g}\right)\right) \cdot \frac{\frac{1}{2}}{a}}\right) \]

       5. rem-square-sqrt N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, -\sqrt[3]{g \cdot \left(1 - \color{blue}{-1}\right)}, \sqrt[3]{\left(\frac{1}{2} \cdot \mathsf{fma}\left(0, h, \frac{\mathsf{fma}\left(-1, h \cdot h, \frac{-1}{4} \cdot {\left(0 \cdot h\right)}^{2}\right)}{g}\right)\right) \cdot \frac{\frac{1}{2}}{a}}\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, -\sqrt[3]{g \cdot \color{blue}{2}}, \sqrt[3]{\left(\frac{1}{2} \cdot \mathsf{fma}\left(0, h, \frac{\mathsf{fma}\left(-1, h \cdot h, \frac{-1}{4} \cdot {\left(0 \cdot h\right)}^{2}\right)}{g}\right)\right) \cdot \frac{\frac{1}{2}}{a}}\right) \]

       7. lower-*.f64 95.2

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, -\sqrt[3]{\color{blue}{g \cdot 2}}, \sqrt[3]{\left(0.5 \cdot \mathsf{fma}\left(0, h, \frac{\mathsf{fma}\left(-1, h \cdot h, -0.25 \cdot {\left(0 \cdot h\right)}^{2}\right)}{g}\right)\right) \cdot \frac{0.5}{a}}\right) \]

   12. Applied rewrites 95.2%

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \color{blue}{-\sqrt[3]{g \cdot 2}}, \sqrt[3]{\left(0.5 \cdot \mathsf{fma}\left(0, h, \frac{\mathsf{fma}\left(-1, h \cdot h, -0.25 \cdot {\left(0 \cdot h\right)}^{2}\right)}{g}\right)\right) \cdot \frac{0.5}{a}}\right) \]

   if 4.00000000000000013e139 < g

   1. Initial program 5.8%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   2. Add Preprocessing
   3. 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. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. cbrt-div N/A

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

      6. *-lft-identity N/A

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

      7. pow1/3 N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 5.8%

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

   6. Applied rewrites 6.1%

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

   7. Taylor expanded in g around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. rem-square-sqrt N/A

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

      5. unpow2 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. rem-square-sqrt N/A

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

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\left(\frac{1}{2} \cdot \mathsf{fma}\left(\color{blue}{0}, h, \frac{-1 \cdot {h}^{2} - \frac{1}{4} \cdot {\left(h + -1 \cdot h\right)}^{2}}{g}\right)\right) \cdot \frac{\frac{1}{2}}{a}}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\left(\frac{1}{2} \cdot \mathsf{fma}\left(0, h, \color{blue}{\frac{-1 \cdot {h}^{2} - \frac{1}{4} \cdot {\left(h + -1 \cdot h\right)}^{2}}{g}}\right)\right) \cdot \frac{\frac{1}{2}}{a}}\right) \]

   9. Applied rewrites 89.8%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\color{blue}{\left(0.5 \cdot \mathsf{fma}\left(0, h, \frac{\mathsf{fma}\left(-1, h \cdot h, -0.25 \cdot {\left(0 \cdot h\right)}^{2}\right)}{g}\right)\right)} \cdot \frac{0.5}{a}}\right) \]

   10. Taylor expanded in g around inf

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

       1. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\color{blue}{\frac{\frac{1}{4} \cdot \left(h + -1 \cdot h\right)}{a}}}\right) \]

       2. distribute-rgt1-in N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{1}{4} \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot h\right)}}{a}}\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{1}{4} \cdot \left(\color{blue}{0} \cdot h\right)}{a}}\right) \]

       4. mul0-lft N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{1}{4} \cdot \color{blue}{0}}{a}}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{1}{4} \cdot \color{blue}{\left(0 \cdot 0\right)}}{a}}\right) \]

       6. mul0-lft N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{1}{4} \cdot \left(\color{blue}{\left(0 \cdot h\right)} \cdot 0\right)}{a}}\right) \]

       7. mul0-lft N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{1}{4} \cdot \left(\left(0 \cdot h\right) \cdot \color{blue}{\left(0 \cdot h\right)}\right)}{a}}\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{1}{4} \cdot \left(\left(\color{blue}{\left(-1 + 1\right)} \cdot h\right) \cdot \left(0 \cdot h\right)\right)}{a}}\right) \]

       9. distribute-rgt1-in N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{1}{4} \cdot \left(\color{blue}{\left(h + -1 \cdot h\right)} \cdot \left(0 \cdot h\right)\right)}{a}}\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{1}{4} \cdot \left(\left(h + -1 \cdot h\right) \cdot \left(\color{blue}{\left(-1 + 1\right)} \cdot h\right)\right)}{a}}\right) \]

       11. distribute-rgt1-in N/A

           \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{1}{4} \cdot \left(\left(h + -1 \cdot h\right) \cdot \color{blue}{\left(h + -1 \cdot h\right)}\right)}{a}}\right) \]

       12. unpow2 N/A

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

       13. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{1}{4} \cdot \color{blue}{\left(\left(h + -1 \cdot h\right) \cdot \left(h + -1 \cdot h\right)\right)}}{a}}\right) \]

       14. distribute-rgt1-in N/A

           \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{1}{4} \cdot \left(\color{blue}{\left(\left(-1 + 1\right) \cdot h\right)} \cdot \left(h + -1 \cdot h\right)\right)}{a}}\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{1}{4} \cdot \left(\left(\color{blue}{0} \cdot h\right) \cdot \left(h + -1 \cdot h\right)\right)}{a}}\right) \]

       16. distribute-rgt1-in N/A

           \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{1}{4} \cdot \left(\left(0 \cdot h\right) \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot h\right)}\right)}{a}}\right) \]

       17. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{1}{4} \cdot \left(\left(0 \cdot h\right) \cdot \left(\color{blue}{0} \cdot h\right)\right)}{a}}\right) \]

       18. mul0-lft N/A

           \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{1}{4} \cdot \left(\color{blue}{0} \cdot \left(0 \cdot h\right)\right)}{a}}\right) \]

       19. mul0-lft N/A

           \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{1}{4} \cdot \left(0 \cdot \color{blue}{0}\right)}{a}}\right) \]

       20. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{1}{4} \cdot \color{blue}{0}}{a}}\right) \]

       21. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\color{blue}{0}}{a}}\right) \]

       22. lower-/.f64 98.8

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

   12. Applied rewrites 98.8%

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\color{blue}{\frac{0}{a}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq 4 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, -\sqrt[3]{2 \cdot g}, \sqrt[3]{\left(\mathsf{fma}\left(0, h, \frac{\mathsf{fma}\left(-1, h \cdot h, {\left(0 \cdot h\right)}^{2} \cdot -0.25\right)}{g}\right) \cdot 0.5\right) \cdot \frac{0.5}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{0}{a}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{0.5}{a}}\\ \mathbf{if}\;g \leq -6.1 \cdot 10^{+137}:\\ \;\;\;\;\sqrt[3]{\left(\frac{h}{a} \cdot \frac{h}{g}\right) \cdot -0.25} + \sqrt[3]{\frac{-1}{\frac{a}{g}}}\\ \mathbf{elif}\;g \leq -2 \cdot 10^{-293}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}, \sqrt[3]{\frac{h \cdot h}{g} \cdot \frac{-0.25}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\left(\left(-0.5\right) \cdot \left(\frac{h}{g} \cdot h\right)\right) \cdot \frac{0.5}{a}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (/ 0.5 a))))
   (if (<= g -6.1e+137)
     (+ (cbrt (* (* (/ h a) (/ h g)) -0.25)) (cbrt (/ -1.0 (/ a g))))
     (if (<= g -2e-293)
       (fma
        t_0
        (cbrt (- (sqrt (* (- g h) (+ h g))) g))
        (cbrt (* (/ (* h h) g) (/ -0.25 a))))
       (fma
        t_0
        (cbrt (- (fma (sqrt (+ h g)) (sqrt (- g h)) g)))
        (cbrt (* (* (- 0.5) (* (/ h g) h)) (/ 0.5 a))))))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt((0.5 / a));
	double tmp;
	if (g <= -6.1e+137) {
		tmp = cbrt((((h / a) * (h / g)) * -0.25)) + cbrt((-1.0 / (a / g)));
	} else if (g <= -2e-293) {
		tmp = fma(t_0, cbrt((sqrt(((g - h) * (h + g))) - g)), cbrt((((h * h) / g) * (-0.25 / a))));
	} else {
		tmp = fma(t_0, cbrt(-fma(sqrt((h + g)), sqrt((g - h)), g)), cbrt(((-0.5 * ((h / g) * h)) * (0.5 / a))));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(0.5 / a))
	tmp = 0.0
	if (g <= -6.1e+137)
		tmp = Float64(cbrt(Float64(Float64(Float64(h / a) * Float64(h / g)) * -0.25)) + cbrt(Float64(-1.0 / Float64(a / g))));
	elseif (g <= -2e-293)
		tmp = fma(t_0, cbrt(Float64(sqrt(Float64(Float64(g - h) * Float64(h + g))) - g)), cbrt(Float64(Float64(Float64(h * h) / g) * Float64(-0.25 / a))));
	else
		tmp = fma(t_0, cbrt(Float64(-fma(sqrt(Float64(h + g)), sqrt(Float64(g - h)), g))), cbrt(Float64(Float64(Float64(-0.5) * Float64(Float64(h / g) * h)) * Float64(0.5 / a))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[g, -6.1e+137], N[(N[Power[N[(N[(N[(h / a), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(-1.0 / N[(a / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[g, -2e-293], N[(t$95$0 * N[Power[N[(N[Sqrt[N[(N[(g - h), $MachinePrecision] * N[(h + g), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - g), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(N[(h * h), $MachinePrecision] / g), $MachinePrecision] * N[(-0.25 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[(-N[(N[Sqrt[N[(h + g), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(g - h), $MachinePrecision]], $MachinePrecision] + g), $MachinePrecision]), 1/3], $MachinePrecision] + N[Power[N[(N[((-0.5) * N[(N[(h / g), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if g < -6.10000000000000004e137

   1. Initial program 7.9%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   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 4.8

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

   5. Applied rewrites 4.8%

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

   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. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. lower-cbrt.f64 61.9

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

   8. Applied rewrites 61.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 61.9

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

   10. Applied rewrites 61.9%

       \[\leadsto \color{blue}{\sqrt[3]{\frac{-g}{a}} + \sqrt[3]{-0.25 \cdot \left(\frac{h}{g} \cdot \frac{h}{a}\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 61.9%

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

   if -6.10000000000000004e137 < g < -2.0000000000000001e-293

   1. Initial program 80.2%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   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 16.8

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

   5. Applied rewrites 16.8%

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

   7. Applied rewrites 20.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, {\left({\left(\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g\right)}^{-1}\right)}^{-0.3333333333333333}, \sqrt[3]{\frac{-g}{a}}\right)} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-pow N/A

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

      4. metadata-eval N/A

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

      5. pow1/3 N/A

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

      6. lower-cbrt.f64 20.5

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 20.5

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

   9. Applied rewrites 20.5%

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

   10. Taylor expanded in g around -inf

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

       1. associate-*r/ N/A

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

       2. times-frac N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}, \sqrt[3]{\frac{\frac{-1}{4}}{a} \cdot \frac{\color{blue}{h \cdot h}}{g}}\right) \]

       7. lower-*.f64 92.1

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}, \sqrt[3]{\frac{-0.25}{a} \cdot \frac{\color{blue}{h \cdot h}}{g}}\right) \]

   12. Applied rewrites 92.1%

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}, \sqrt[3]{\color{blue}{\frac{-0.25}{a} \cdot \frac{h \cdot h}{g}}}\right) \]

   if -2.0000000000000001e-293 < g

   1. Initial program 45.4%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   2. Add Preprocessing
   3. 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. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. cbrt-div N/A

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

      6. *-lft-identity N/A

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

      7. pow1/3 N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 45.6%

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

   6. Applied rewrites 54.0%

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

   7. Taylor expanded in g around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. rem-square-sqrt N/A

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

      5. unpow2 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. rem-square-sqrt N/A

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

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\left(\frac{1}{2} \cdot \mathsf{fma}\left(\color{blue}{0}, h, \frac{-1 \cdot {h}^{2} - \frac{1}{4} \cdot {\left(h + -1 \cdot h\right)}^{2}}{g}\right)\right) \cdot \frac{\frac{1}{2}}{a}}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\left(\frac{1}{2} \cdot \mathsf{fma}\left(0, h, \color{blue}{\frac{-1 \cdot {h}^{2} - \frac{1}{4} \cdot {\left(h + -1 \cdot h\right)}^{2}}{g}}\right)\right) \cdot \frac{\frac{1}{2}}{a}}\right) \]

   9. Applied rewrites 93.5%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\color{blue}{\left(0.5 \cdot \mathsf{fma}\left(0, h, \frac{\mathsf{fma}\left(-1, h \cdot h, -0.25 \cdot {\left(0 \cdot h\right)}^{2}\right)}{g}\right)\right)} \cdot \frac{0.5}{a}}\right) \]

   11. Applied rewrites 96.6%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -6.1 \cdot 10^{+137}:\\ \;\;\;\;\sqrt[3]{\left(\frac{h}{a} \cdot \frac{h}{g}\right) \cdot -0.25} + \sqrt[3]{\frac{-1}{\frac{a}{g}}}\\ \mathbf{elif}\;g \leq -2 \cdot 10^{-293}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}, \sqrt[3]{\frac{h \cdot h}{g} \cdot \frac{-0.25}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\left(\left(-0.5\right) \cdot \left(\frac{h}{g} \cdot h\right)\right) \cdot \frac{0.5}{a}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;g \leq 1.15 \cdot 10^{-181}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\frac{h}{g} \cdot h\right) \cdot -0.25}}{\sqrt[3]{a}} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{0}{a}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= g 1.15e-181)
   (+ (/ (cbrt (* (* (/ h g) h) -0.25)) (cbrt a)) (cbrt (/ (- g) a)))
   (fma
    (cbrt (/ 0.5 a))
    (cbrt (- (fma (sqrt (+ h g)) (sqrt (- g h)) g)))
    (cbrt (/ 0.0 a)))))

C

double code(double g, double h, double a) {
	double tmp;
	if (g <= 1.15e-181) {
		tmp = (cbrt((((h / g) * h) * -0.25)) / cbrt(a)) + cbrt((-g / a));
	} else {
		tmp = fma(cbrt((0.5 / a)), cbrt(-fma(sqrt((h + g)), sqrt((g - h)), g)), cbrt((0.0 / a)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (g <= 1.15e-181)
		tmp = Float64(Float64(cbrt(Float64(Float64(Float64(h / g) * h) * -0.25)) / cbrt(a)) + cbrt(Float64(Float64(-g) / a)));
	else
		tmp = fma(cbrt(Float64(0.5 / a)), cbrt(Float64(-fma(sqrt(Float64(h + g)), sqrt(Float64(g - h)), g))), cbrt(Float64(0.0 / a)));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[g, 1.15e-181], N[(N[(N[Power[N[(N[(N[(h / g), $MachinePrecision] * h), $MachinePrecision] * -0.25), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[(-N[(N[Sqrt[N[(h + g), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(g - h), $MachinePrecision]], $MachinePrecision] + g), $MachinePrecision]), 1/3], $MachinePrecision] + N[Power[N[(0.0 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if g < 1.14999999999999995e-181

   1. Initial program 46.9%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   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 11.5

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

   5. Applied rewrites 11.5%

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

   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. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. lower-cbrt.f64 75.7

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

   8. Applied rewrites 75.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 75.7

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

   10. Applied rewrites 75.7%

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

   12. Applied rewrites 75.8%

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

   if 1.14999999999999995e-181 < g

   1. Initial program 46.1%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   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. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. cbrt-div N/A

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

      6. *-lft-identity N/A

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

      7. pow1/3 N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 46.3%

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

   6. Applied rewrites 54.9%

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

   7. Taylor expanded in g around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. rem-square-sqrt N/A

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

      5. unpow2 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. rem-square-sqrt N/A

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

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\left(\frac{1}{2} \cdot \mathsf{fma}\left(\color{blue}{0}, h, \frac{-1 \cdot {h}^{2} - \frac{1}{4} \cdot {\left(h + -1 \cdot h\right)}^{2}}{g}\right)\right) \cdot \frac{\frac{1}{2}}{a}}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\left(\frac{1}{2} \cdot \mathsf{fma}\left(0, h, \color{blue}{\frac{-1 \cdot {h}^{2} - \frac{1}{4} \cdot {\left(h + -1 \cdot h\right)}^{2}}{g}}\right)\right) \cdot \frac{\frac{1}{2}}{a}}\right) \]

   9. Applied rewrites 94.2%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\color{blue}{\left(0.5 \cdot \mathsf{fma}\left(0, h, \frac{\mathsf{fma}\left(-1, h \cdot h, -0.25 \cdot {\left(0 \cdot h\right)}^{2}\right)}{g}\right)\right)} \cdot \frac{0.5}{a}}\right) \]

   10. Taylor expanded in g around inf

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

       1. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\color{blue}{\frac{\frac{1}{4} \cdot \left(h + -1 \cdot h\right)}{a}}}\right) \]

       2. distribute-rgt1-in N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{1}{4} \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot h\right)}}{a}}\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{1}{4} \cdot \left(\color{blue}{0} \cdot h\right)}{a}}\right) \]

       4. mul0-lft N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{1}{4} \cdot \color{blue}{0}}{a}}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{1}{4} \cdot \color{blue}{\left(0 \cdot 0\right)}}{a}}\right) \]

       6. mul0-lft N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{1}{4} \cdot \left(\color{blue}{\left(0 \cdot h\right)} \cdot 0\right)}{a}}\right) \]

       7. mul0-lft N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{1}{4} \cdot \left(\left(0 \cdot h\right) \cdot \color{blue}{\left(0 \cdot h\right)}\right)}{a}}\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{1}{4} \cdot \left(\left(\color{blue}{\left(-1 + 1\right)} \cdot h\right) \cdot \left(0 \cdot h\right)\right)}{a}}\right) \]

       9. distribute-rgt1-in N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{1}{4} \cdot \left(\color{blue}{\left(h + -1 \cdot h\right)} \cdot \left(0 \cdot h\right)\right)}{a}}\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{1}{4} \cdot \left(\left(h + -1 \cdot h\right) \cdot \left(\color{blue}{\left(-1 + 1\right)} \cdot h\right)\right)}{a}}\right) \]

       11. distribute-rgt1-in N/A

           \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{1}{4} \cdot \left(\left(h + -1 \cdot h\right) \cdot \color{blue}{\left(h + -1 \cdot h\right)}\right)}{a}}\right) \]

       12. unpow2 N/A

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

       13. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{1}{4} \cdot \color{blue}{\left(\left(h + -1 \cdot h\right) \cdot \left(h + -1 \cdot h\right)\right)}}{a}}\right) \]

       14. distribute-rgt1-in N/A

           \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{1}{4} \cdot \left(\color{blue}{\left(\left(-1 + 1\right) \cdot h\right)} \cdot \left(h + -1 \cdot h\right)\right)}{a}}\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{1}{4} \cdot \left(\left(\color{blue}{0} \cdot h\right) \cdot \left(h + -1 \cdot h\right)\right)}{a}}\right) \]

       16. distribute-rgt1-in N/A

           \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{1}{4} \cdot \left(\left(0 \cdot h\right) \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot h\right)}\right)}{a}}\right) \]

       17. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{1}{4} \cdot \left(\left(0 \cdot h\right) \cdot \left(\color{blue}{0} \cdot h\right)\right)}{a}}\right) \]

       18. mul0-lft N/A

           \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{1}{4} \cdot \left(\color{blue}{0} \cdot \left(0 \cdot h\right)\right)}{a}}\right) \]

       19. mul0-lft N/A

           \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{1}{4} \cdot \left(0 \cdot \color{blue}{0}\right)}{a}}\right) \]

       20. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{1}{4} \cdot \color{blue}{0}}{a}}\right) \]

       21. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\color{blue}{0}}{a}}\right) \]

       22. lower-/.f64 96.6

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

   12. Applied rewrites 96.6%

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\color{blue}{\frac{0}{a}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq 1.15 \cdot 10^{-181}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\frac{h}{g} \cdot h\right) \cdot -0.25}}{\sqrt[3]{a}} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{0}{a}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 46.5%

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

3. Taylor expanded in 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 27.8

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

5. Applied rewrites 27.8%

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

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. times-frac N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cbrt.f64 N/A

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

   10. lower-cbrt.f64 75.5

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

8. Applied rewrites 75.5%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 75.5

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

10. Applied rewrites 75.5%

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

12. Applied rewrites 75.6%

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

13. Final simplification 75.6%

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

Alternative 5: 75.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 46.5%

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

3. Taylor expanded in 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 27.8

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

5. Applied rewrites 27.8%

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

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. times-frac N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cbrt.f64 N/A

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

   10. lower-cbrt.f64 75.5

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

8. Applied rewrites 75.5%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 75.5

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

10. Applied rewrites 75.5%

    \[\leadsto \color{blue}{\sqrt[3]{\frac{-g}{a}} + \sqrt[3]{-0.25 \cdot \left(\frac{h}{g} \cdot \frac{h}{a}\right)}} \]
11. Step-by-step derivation

12. Applied rewrites 75.5%

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

13. Final simplification 75.5%

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

Alternative 6: 74.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(Float64(h / a) * Float64(h / g)) * -0.25)) + cbrt(Float64(-1.0 / Float64(a / g))))
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[(-1.0 / N[(a / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.5%

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

3. Taylor expanded in 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 27.8

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

5. Applied rewrites 27.8%

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

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. times-frac N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cbrt.f64 N/A

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

   10. lower-cbrt.f64 75.5

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

8. Applied rewrites 75.5%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 75.5

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

10. Applied rewrites 75.5%

    \[\leadsto \color{blue}{\sqrt[3]{\frac{-g}{a}} + \sqrt[3]{-0.25 \cdot \left(\frac{h}{g} \cdot \frac{h}{a}\right)}} \]
11. Step-by-step derivation

12. Applied rewrites 75.5%

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

13. Final simplification 75.5%

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

Alternative 7: 75.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 46.5%

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

3. Taylor expanded in 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 27.8

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

5. Applied rewrites 27.8%

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

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. times-frac N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cbrt.f64 N/A

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

   10. lower-cbrt.f64 75.5

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

8. Applied rewrites 75.5%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 75.5

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

10. Applied rewrites 75.5%

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

11. Final simplification 75.5%

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

Alternative 8: 71.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 46.5%

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

3. Taylor expanded in 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 27.8

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

5. Applied rewrites 27.8%

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

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. times-frac N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cbrt.f64 N/A

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

   10. lower-cbrt.f64 75.5

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

8. Applied rewrites 75.5%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 75.5

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

10. Applied rewrites 75.5%

    \[\leadsto \color{blue}{\sqrt[3]{\frac{-g}{a}} + \sqrt[3]{-0.25 \cdot \left(\frac{h}{g} \cdot \frac{h}{a}\right)}} \]
11. Step-by-step derivation

12. Applied rewrites 72.4%

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

13. Final simplification 72.4%

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

Alternative 9: 3.0% accurate, 302.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[g_, h_, a_] := 0.0

Derivation

1. Initial program 46.5%

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

   1. lift-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. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. cbrt-prod N/A

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

   5. pow1/3 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/r* N/A

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

   9. metadata-eval N/A

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

   10. cbrt-div N/A

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

   11. pow1/3 N/A

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

   12. associate-*r/ N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 49.5%

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

5. Taylor expanded in g around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. unpow2 N/A

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

   8. rem-square-sqrt N/A

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

   9. metadata-eval N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cbrt.f64 3.0

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

7. Applied rewrites 3.0%

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

9. Applied rewrites 3.0%

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

Reproduce

herbie shell --seed 2024331 
(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))))))))