2-ancestry mixing, positive discriminant

Percentage Accurate: 44.8% → 97.6%

Time: 14.8s

Alternatives: 15

Speedup: 1.4×

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: 44.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 97.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}} \cdot \sqrt[3]{-0.5}, {2}^{0.3333333333333333}, \frac{\sqrt[3]{\frac{h}{g} \cdot h}}{\sqrt[3]{a}} \cdot \left(\sqrt[3]{0.5} \cdot \sqrt[3]{-0.5}\right)\right) \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (fma
  (* (/ (cbrt g) (cbrt a)) (cbrt -0.5))
  (pow 2.0 0.3333333333333333)
  (* (/ (cbrt (* (/ h g) h)) (cbrt a)) (* (cbrt 0.5) (cbrt -0.5)))))

C

double code(double g, double h, double a) {
	return fma(((cbrt(g) / cbrt(a)) * cbrt(-0.5)), pow(2.0, 0.3333333333333333), ((cbrt(((h / g) * h)) / cbrt(a)) * (cbrt(0.5) * cbrt(-0.5))));
}

Julia

function code(g, h, a)
	return fma(Float64(Float64(cbrt(g) / cbrt(a)) * cbrt(-0.5)), (2.0 ^ 0.3333333333333333), Float64(Float64(cbrt(Float64(Float64(h / g) * h)) / cbrt(a)) * Float64(cbrt(0.5) * cbrt(-0.5))))
end

Wolfram

code[g_, h_, a_] := N[(N[(N[(N[Power[g, 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[-0.5, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[2.0, 0.3333333333333333], $MachinePrecision] + N[(N[(N[Power[N[(N[(h / g), $MachinePrecision] * h), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[Power[0.5, 1/3], $MachinePrecision] * N[Power[-0.5, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.1%

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

3. Taylor expanded in h around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cbrt.f64 N/A

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

   10. unpow2 N/A

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

   11. *-commutative N/A

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

   12. times-frac N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lower-cbrt.f64 N/A

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

   19. lower-cbrt.f64 72.9

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

5. Applied rewrites 72.9%

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

7. Applied rewrites 90.9%

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

9. Applied rewrites 96.3%

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

11. Applied rewrites 97.1%

    \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}} \cdot \sqrt[3]{-0.5}, {2}^{\color{blue}{0.3333333333333333}}, \frac{\sqrt[3]{\frac{h}{g} \cdot h}}{\sqrt[3]{a}} \cdot \left(\sqrt[3]{0.5} \cdot \sqrt[3]{-0.5}\right)\right) \]
12. Add Preprocessing

Alternative 2: 77.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ t_1 := \sqrt[3]{{\left(2 \cdot a\right)}^{-1} \cdot \left(\left(-g\right) + t\_0\right)} + \sqrt[3]{\frac{-1}{2 \cdot a} \cdot \left(g + t\_0\right)}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-93} \lor \neg \left(t\_1 \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{g}{a}}, \sqrt[3]{-1}, \sqrt[3]{-0.25 \cdot \left(\frac{h}{a} \cdot \frac{h}{g}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(-g\right) - g}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{-g}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (sqrt (- (* g g) (* h h))))
        (t_1
         (+
          (cbrt (* (pow (* 2.0 a) -1.0) (+ (- g) t_0)))
          (cbrt (* (/ -1.0 (* 2.0 a)) (+ g t_0))))))
   (if (or (<= t_1 -1e-93) (not (<= t_1 0.0)))
     (fma (cbrt (/ g a)) (cbrt -1.0) (cbrt (* -0.25 (* (/ h a) (/ h g)))))
     (+ (/ (cbrt (- (- g) g)) (cbrt (* 2.0 a))) (cbrt (/ (- g) a))))))

C

double code(double g, double h, double a) {
	double t_0 = sqrt(((g * g) - (h * h)));
	double t_1 = cbrt((pow((2.0 * a), -1.0) * (-g + t_0))) + cbrt(((-1.0 / (2.0 * a)) * (g + t_0)));
	double tmp;
	if ((t_1 <= -1e-93) || !(t_1 <= 0.0)) {
		tmp = fma(cbrt((g / a)), cbrt(-1.0), cbrt((-0.25 * ((h / a) * (h / g)))));
	} else {
		tmp = (cbrt((-g - g)) / cbrt((2.0 * a))) + cbrt((-g / a));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	t_1 = Float64(cbrt(Float64((Float64(2.0 * a) ^ -1.0) * Float64(Float64(-g) + t_0))) + cbrt(Float64(Float64(-1.0 / Float64(2.0 * a)) * Float64(g + t_0))))
	tmp = 0.0
	if ((t_1 <= -1e-93) || !(t_1 <= 0.0))
		tmp = fma(cbrt(Float64(g / a)), cbrt(-1.0), cbrt(Float64(-0.25 * Float64(Float64(h / a) * Float64(h / g)))));
	else
		tmp = Float64(Float64(cbrt(Float64(Float64(-g) - g)) / cbrt(Float64(2.0 * a))) + cbrt(Float64(Float64(-g) / a)));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(N[Power[N[(2.0 * a), $MachinePrecision], -1.0], $MachinePrecision] * N[((-g) + t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(-1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * N[(g + t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e-93], N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision]], N[(N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[-1.0, 1/3], $MachinePrecision] + N[Power[N[(-0.25 * N[(N[(h / a), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[((-g) - g), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[N[(2.0 * a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < -9.999999999999999e-94 or 0.0 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))))

   1. Initial program 51.4%

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

   3. Taylor expanded in h around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cbrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. unpow2 N/A

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

      11. *-commutative N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cbrt.f64 N/A

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

      19. lower-cbrt.f64 76.5

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

   5. Applied rewrites 76.5%

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

   7. Applied rewrites 77.4%

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

   if -9.999999999999999e-94 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < 0.0

   1. Initial program 4.4%

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

   3. Taylor expanded in g around inf

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
   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 4.4

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

   5. Applied rewrites 4.4%

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

   6. Taylor expanded in g around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 4.4

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

   8. Applied rewrites 4.4%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. cbrt-div N/A

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

      6. *-lft-identity N/A

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

      7. lower-/.f64 N/A

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

   10. Applied rewrites 81.3%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{{\left(2 \cdot a\right)}^{-1} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{-1}{2 \cdot a} \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)} \leq -1 \cdot 10^{-93} \lor \neg \left(\sqrt[3]{{\left(2 \cdot a\right)}^{-1} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{-1}{2 \cdot a} \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)} \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{g}{a}}, \sqrt[3]{-1}, \sqrt[3]{-0.25 \cdot \left(\frac{h}{a} \cdot \frac{h}{g}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(-g\right) - g}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{-g}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ t_1 := \left(-g\right) + t\_0\\ t_2 := \sqrt[3]{{\left(2 \cdot a\right)}^{-1} \cdot t\_1} + \sqrt[3]{\frac{-1}{2 \cdot a} \cdot \left(g + t\_0\right)}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-93}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot t\_1} + \frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}}{\sqrt[3]{a \cdot 2}}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\sqrt[3]{\left(-g\right) - g}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{g}{a}}, \sqrt[3]{-1}, \sqrt[3]{-0.25 \cdot \left(\frac{h}{a} \cdot \frac{h}{g}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (sqrt (- (* g g) (* h h))))
        (t_1 (+ (- g) t_0))
        (t_2
         (+
          (cbrt (* (pow (* 2.0 a) -1.0) t_1))
          (cbrt (* (/ -1.0 (* 2.0 a)) (+ g t_0))))))
   (if (<= t_2 -1e-93)
     (+
      (cbrt (* (/ 0.5 a) t_1))
      (/ (cbrt (- (- g) (sqrt (* (- g h) (+ h g))))) (cbrt (* a 2.0))))
     (if (<= t_2 0.0)
       (+ (/ (cbrt (- (- g) g)) (cbrt (* 2.0 a))) (cbrt (/ (- g) a)))
       (fma
        (cbrt (/ g a))
        (cbrt -1.0)
        (cbrt (* -0.25 (* (/ h a) (/ h g)))))))))

C

double code(double g, double h, double a) {
	double t_0 = sqrt(((g * g) - (h * h)));
	double t_1 = -g + t_0;
	double t_2 = cbrt((pow((2.0 * a), -1.0) * t_1)) + cbrt(((-1.0 / (2.0 * a)) * (g + t_0)));
	double tmp;
	if (t_2 <= -1e-93) {
		tmp = cbrt(((0.5 / a) * t_1)) + (cbrt((-g - sqrt(((g - h) * (h + g))))) / cbrt((a * 2.0)));
	} else if (t_2 <= 0.0) {
		tmp = (cbrt((-g - g)) / cbrt((2.0 * a))) + cbrt((-g / a));
	} else {
		tmp = fma(cbrt((g / a)), cbrt(-1.0), cbrt((-0.25 * ((h / a) * (h / g)))));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	t_1 = Float64(Float64(-g) + t_0)
	t_2 = Float64(cbrt(Float64((Float64(2.0 * a) ^ -1.0) * t_1)) + cbrt(Float64(Float64(-1.0 / Float64(2.0 * a)) * Float64(g + t_0))))
	tmp = 0.0
	if (t_2 <= -1e-93)
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * t_1)) + Float64(cbrt(Float64(Float64(-g) - sqrt(Float64(Float64(g - h) * Float64(h + g))))) / cbrt(Float64(a * 2.0))));
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(cbrt(Float64(Float64(-g) - g)) / cbrt(Float64(2.0 * a))) + cbrt(Float64(Float64(-g) / a)));
	else
		tmp = fma(cbrt(Float64(g / a)), cbrt(-1.0), cbrt(Float64(-0.25 * Float64(Float64(h / a) * Float64(h / g)))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[((-g) + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[(N[Power[N[(2.0 * a), $MachinePrecision], -1.0], $MachinePrecision] * t$95$1), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(-1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * N[(g + t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-93], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * t$95$1), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[((-g) - N[Sqrt[N[(N[(g - h), $MachinePrecision] * N[(h + g), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[N[(a * 2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(N[Power[N[((-g) - g), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[N[(2.0 * a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[-1.0, 1/3], $MachinePrecision] + N[Power[N[(-0.25 * N[(N[(h / a), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < -9.999999999999999e-94

   1. Initial program 90.5%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. cbrt-div N/A

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

      6. *-lft-identity N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 93.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. metadata-eval N/A

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

      5. lift-/.f64 93.0

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

   6. Applied rewrites 93.0%

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

   if -9.999999999999999e-94 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < 0.0

   1. Initial program 4.4%

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

   3. Taylor expanded in g around inf

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
   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 4.4

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

   5. Applied rewrites 4.4%

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

   6. Taylor expanded in g around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 4.4

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

   8. Applied rewrites 4.4%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. cbrt-div N/A

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

      6. *-lft-identity N/A

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

      7. lower-/.f64 N/A

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

   10. Applied rewrites 81.3%

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

   if 0.0 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))))

   1. Initial program 35.0%

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

   3. Taylor expanded in h around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cbrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. unpow2 N/A

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

      11. *-commutative N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cbrt.f64 N/A

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

      19. lower-cbrt.f64 70.6

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

   5. Applied rewrites 70.6%

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

   7. Applied rewrites 71.4%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{g}{a}}, \color{blue}{\sqrt[3]{-1}}, \sqrt[3]{-0.25 \cdot \left(\frac{h}{a} \cdot \frac{h}{g}\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{{\left(2 \cdot a\right)}^{-1} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{-1}{2 \cdot a} \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)} \leq -1 \cdot 10^{-93}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}}{\sqrt[3]{a \cdot 2}}\\ \mathbf{elif}\;\sqrt[3]{{\left(2 \cdot a\right)}^{-1} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{-1}{2 \cdot a} \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)} \leq 0:\\ \;\;\;\;\frac{\sqrt[3]{\left(-g\right) - g}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{g}{a}}, \sqrt[3]{-1}, \sqrt[3]{-0.25 \cdot \left(\frac{h}{a} \cdot \frac{h}{g}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(2 \cdot a\right)}^{-1}\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ t_2 := \sqrt[3]{t\_0 \cdot \left(\left(-g\right) + t\_1\right)} + \sqrt[3]{\frac{-1}{2 \cdot a} \cdot \left(g + t\_1\right)}\\ t_3 := \sqrt[3]{\frac{-g}{a}}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-93}:\\ \;\;\;\;\sqrt[3]{t\_0 \cdot \left(\frac{h \cdot h}{g} \cdot -0.5\right)} + t\_3\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\sqrt[3]{\left(-g\right) - g}}{\sqrt[3]{2 \cdot a}} + t\_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (pow (* 2.0 a) -1.0))
        (t_1 (sqrt (- (* g g) (* h h))))
        (t_2
         (+
          (cbrt (* t_0 (+ (- g) t_1)))
          (cbrt (* (/ -1.0 (* 2.0 a)) (+ g t_1)))))
        (t_3 (cbrt (/ (- g) a))))
   (if (<= t_2 -1e-93)
     (+ (cbrt (* t_0 (* (/ (* h h) g) -0.5))) t_3)
     (if (<= t_2 0.0)
       (+ (/ (cbrt (- (- g) g)) (cbrt (* 2.0 a))) t_3)
       (* (cbrt (/ g a)) (cbrt -1.0))))))

C

double code(double g, double h, double a) {
	double t_0 = pow((2.0 * a), -1.0);
	double t_1 = sqrt(((g * g) - (h * h)));
	double t_2 = cbrt((t_0 * (-g + t_1))) + cbrt(((-1.0 / (2.0 * a)) * (g + t_1)));
	double t_3 = cbrt((-g / a));
	double tmp;
	if (t_2 <= -1e-93) {
		tmp = cbrt((t_0 * (((h * h) / g) * -0.5))) + t_3;
	} else if (t_2 <= 0.0) {
		tmp = (cbrt((-g - g)) / cbrt((2.0 * a))) + t_3;
	} else {
		tmp = cbrt((g / a)) * cbrt(-1.0);
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.pow((2.0 * a), -1.0);
	double t_1 = Math.sqrt(((g * g) - (h * h)));
	double t_2 = Math.cbrt((t_0 * (-g + t_1))) + Math.cbrt(((-1.0 / (2.0 * a)) * (g + t_1)));
	double t_3 = Math.cbrt((-g / a));
	double tmp;
	if (t_2 <= -1e-93) {
		tmp = Math.cbrt((t_0 * (((h * h) / g) * -0.5))) + t_3;
	} else if (t_2 <= 0.0) {
		tmp = (Math.cbrt((-g - g)) / Math.cbrt((2.0 * a))) + t_3;
	} else {
		tmp = Math.cbrt((g / a)) * Math.cbrt(-1.0);
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64(2.0 * a) ^ -1.0
	t_1 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	t_2 = Float64(cbrt(Float64(t_0 * Float64(Float64(-g) + t_1))) + cbrt(Float64(Float64(-1.0 / Float64(2.0 * a)) * Float64(g + t_1))))
	t_3 = cbrt(Float64(Float64(-g) / a))
	tmp = 0.0
	if (t_2 <= -1e-93)
		tmp = Float64(cbrt(Float64(t_0 * Float64(Float64(Float64(h * h) / g) * -0.5))) + t_3);
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(cbrt(Float64(Float64(-g) - g)) / cbrt(Float64(2.0 * a))) + t_3);
	else
		tmp = Float64(cbrt(Float64(g / a)) * cbrt(-1.0));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[(2.0 * a), $MachinePrecision], -1.0], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[(t$95$0 * N[((-g) + t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(-1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * N[(g + t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[t$95$2, -1e-93], N[(N[Power[N[(t$95$0 * N[(N[(N[(h * h), $MachinePrecision] / g), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + t$95$3), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(N[Power[N[((-g) - g), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[N[(2.0 * a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision], N[(N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[-1.0, 1/3], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < -9.999999999999999e-94

   1. Initial program 90.5%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   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 50.4

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

   5. Applied rewrites 50.4%

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

   6. Taylor expanded in g around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 90.9

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

   8. Applied rewrites 90.9%

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

   if -9.999999999999999e-94 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < 0.0

   1. Initial program 4.4%

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

   3. Taylor expanded in g around inf

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
   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 4.4

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

   5. Applied rewrites 4.4%

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

   6. Taylor expanded in g around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 4.4

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

   8. Applied rewrites 4.4%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. cbrt-div N/A

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

      6. *-lft-identity N/A

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

      7. lower-/.f64 N/A

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

   10. Applied rewrites 81.3%

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

   if 0.0 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))))

   1. Initial program 35.0%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. cbrt-div N/A

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

      6. *-lft-identity N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 34.9%

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

   5. Taylor expanded in g around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cbrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-cbrt.f64 71.1

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

   7. Applied rewrites 71.1%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{{\left(2 \cdot a\right)}^{-1} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{-1}{2 \cdot a} \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)} \leq -1 \cdot 10^{-93}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot a\right)}^{-1} \cdot \left(\frac{h \cdot h}{g} \cdot -0.5\right)} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{elif}\;\sqrt[3]{{\left(2 \cdot a\right)}^{-1} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{-1}{2 \cdot a} \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)} \leq 0:\\ \;\;\;\;\frac{\sqrt[3]{\left(-g\right) - g}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(2 \cdot a\right)}^{-1}\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ t_2 := \sqrt[3]{t\_0 \cdot \left(\left(-g\right) + t\_1\right)} + \sqrt[3]{\frac{-1}{2 \cdot a} \cdot \left(g + t\_1\right)}\\ t_3 := \sqrt[3]{\frac{-g}{a}}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-93}:\\ \;\;\;\;\sqrt[3]{t\_0 \cdot \left(\frac{h \cdot h}{g} \cdot -0.5\right)} + t\_3\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(-g\right) - g\right) \cdot 0.5}}{\sqrt[3]{a}} + t\_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (pow (* 2.0 a) -1.0))
        (t_1 (sqrt (- (* g g) (* h h))))
        (t_2
         (+
          (cbrt (* t_0 (+ (- g) t_1)))
          (cbrt (* (/ -1.0 (* 2.0 a)) (+ g t_1)))))
        (t_3 (cbrt (/ (- g) a))))
   (if (<= t_2 -1e-93)
     (+ (cbrt (* t_0 (* (/ (* h h) g) -0.5))) t_3)
     (if (<= t_2 0.0)
       (+ (/ (cbrt (* (- (- g) g) 0.5)) (cbrt a)) t_3)
       (* (cbrt (/ g a)) (cbrt -1.0))))))

C

double code(double g, double h, double a) {
	double t_0 = pow((2.0 * a), -1.0);
	double t_1 = sqrt(((g * g) - (h * h)));
	double t_2 = cbrt((t_0 * (-g + t_1))) + cbrt(((-1.0 / (2.0 * a)) * (g + t_1)));
	double t_3 = cbrt((-g / a));
	double tmp;
	if (t_2 <= -1e-93) {
		tmp = cbrt((t_0 * (((h * h) / g) * -0.5))) + t_3;
	} else if (t_2 <= 0.0) {
		tmp = (cbrt(((-g - g) * 0.5)) / cbrt(a)) + t_3;
	} else {
		tmp = cbrt((g / a)) * cbrt(-1.0);
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.pow((2.0 * a), -1.0);
	double t_1 = Math.sqrt(((g * g) - (h * h)));
	double t_2 = Math.cbrt((t_0 * (-g + t_1))) + Math.cbrt(((-1.0 / (2.0 * a)) * (g + t_1)));
	double t_3 = Math.cbrt((-g / a));
	double tmp;
	if (t_2 <= -1e-93) {
		tmp = Math.cbrt((t_0 * (((h * h) / g) * -0.5))) + t_3;
	} else if (t_2 <= 0.0) {
		tmp = (Math.cbrt(((-g - g) * 0.5)) / Math.cbrt(a)) + t_3;
	} else {
		tmp = Math.cbrt((g / a)) * Math.cbrt(-1.0);
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64(2.0 * a) ^ -1.0
	t_1 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	t_2 = Float64(cbrt(Float64(t_0 * Float64(Float64(-g) + t_1))) + cbrt(Float64(Float64(-1.0 / Float64(2.0 * a)) * Float64(g + t_1))))
	t_3 = cbrt(Float64(Float64(-g) / a))
	tmp = 0.0
	if (t_2 <= -1e-93)
		tmp = Float64(cbrt(Float64(t_0 * Float64(Float64(Float64(h * h) / g) * -0.5))) + t_3);
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(cbrt(Float64(Float64(Float64(-g) - g) * 0.5)) / cbrt(a)) + t_3);
	else
		tmp = Float64(cbrt(Float64(g / a)) * cbrt(-1.0));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[(2.0 * a), $MachinePrecision], -1.0], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[(t$95$0 * N[((-g) + t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(-1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * N[(g + t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[t$95$2, -1e-93], N[(N[Power[N[(t$95$0 * N[(N[(N[(h * h), $MachinePrecision] / g), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + t$95$3), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(N[Power[N[(N[((-g) - g), $MachinePrecision] * 0.5), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision], N[(N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[-1.0, 1/3], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < -9.999999999999999e-94

   1. Initial program 90.5%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   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 50.4

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

   5. Applied rewrites 50.4%

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

   6. Taylor expanded in g around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 90.9

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

   8. Applied rewrites 90.9%

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

   if -9.999999999999999e-94 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < 0.0

   1. Initial program 4.4%

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

   3. Taylor expanded in g around inf

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
   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 4.4

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

   5. Applied rewrites 4.4%

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

   6. Taylor expanded in g around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 4.4

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

   8. Applied rewrites 4.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 4.4

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

   10. Applied rewrites 4.4%

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

       1. lift-cbrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. associate-*l/ N/A

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

       5. cbrt-div N/A

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

       6. lift-cbrt.f64 N/A

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

       7. lower-/.f64 N/A

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

   12. Applied rewrites 81.2%

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

   if 0.0 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))))

   1. Initial program 35.0%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. cbrt-div N/A

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

      6. *-lft-identity N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 34.9%

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

   5. Taylor expanded in g around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cbrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-cbrt.f64 71.1

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

   7. Applied rewrites 71.1%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{{\left(2 \cdot a\right)}^{-1} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{-1}{2 \cdot a} \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)} \leq -1 \cdot 10^{-93}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot a\right)}^{-1} \cdot \left(\frac{h \cdot h}{g} \cdot -0.5\right)} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{elif}\;\sqrt[3]{{\left(2 \cdot a\right)}^{-1} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{-1}{2 \cdot a} \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)} \leq 0:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(-g\right) - g\right) \cdot 0.5}}{\sqrt[3]{a}} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(2 \cdot a\right)}^{-1}\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ t_2 := \sqrt[3]{t\_0 \cdot \left(\left(-g\right) + t\_1\right)} + \sqrt[3]{\frac{-1}{2 \cdot a} \cdot \left(g + t\_1\right)}\\ t_3 := \sqrt[3]{\frac{-g}{a}}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-93}:\\ \;\;\;\;\sqrt[3]{t\_0 \cdot \left(\frac{h \cdot h}{g} \cdot -0.5\right)} + t\_3\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{\left(-g\right) - g}, t\_3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (pow (* 2.0 a) -1.0))
        (t_1 (sqrt (- (* g g) (* h h))))
        (t_2
         (+
          (cbrt (* t_0 (+ (- g) t_1)))
          (cbrt (* (/ -1.0 (* 2.0 a)) (+ g t_1)))))
        (t_3 (cbrt (/ (- g) a))))
   (if (<= t_2 -1e-93)
     (+ (cbrt (* t_0 (* (/ (* h h) g) -0.5))) t_3)
     (if (<= t_2 0.0)
       (fma (cbrt (/ 0.5 a)) (cbrt (- (- g) g)) t_3)
       (* (cbrt (/ g a)) (cbrt -1.0))))))

C

double code(double g, double h, double a) {
	double t_0 = pow((2.0 * a), -1.0);
	double t_1 = sqrt(((g * g) - (h * h)));
	double t_2 = cbrt((t_0 * (-g + t_1))) + cbrt(((-1.0 / (2.0 * a)) * (g + t_1)));
	double t_3 = cbrt((-g / a));
	double tmp;
	if (t_2 <= -1e-93) {
		tmp = cbrt((t_0 * (((h * h) / g) * -0.5))) + t_3;
	} else if (t_2 <= 0.0) {
		tmp = fma(cbrt((0.5 / a)), cbrt((-g - g)), t_3);
	} else {
		tmp = cbrt((g / a)) * cbrt(-1.0);
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64(2.0 * a) ^ -1.0
	t_1 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	t_2 = Float64(cbrt(Float64(t_0 * Float64(Float64(-g) + t_1))) + cbrt(Float64(Float64(-1.0 / Float64(2.0 * a)) * Float64(g + t_1))))
	t_3 = cbrt(Float64(Float64(-g) / a))
	tmp = 0.0
	if (t_2 <= -1e-93)
		tmp = Float64(cbrt(Float64(t_0 * Float64(Float64(Float64(h * h) / g) * -0.5))) + t_3);
	elseif (t_2 <= 0.0)
		tmp = fma(cbrt(Float64(0.5 / a)), cbrt(Float64(Float64(-g) - g)), t_3);
	else
		tmp = Float64(cbrt(Float64(g / a)) * cbrt(-1.0));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[(2.0 * a), $MachinePrecision], -1.0], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[(t$95$0 * N[((-g) + t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(-1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * N[(g + t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[t$95$2, -1e-93], N[(N[Power[N[(t$95$0 * N[(N[(N[(h * h), $MachinePrecision] / g), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + t$95$3), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[((-g) - g), $MachinePrecision], 1/3], $MachinePrecision] + t$95$3), $MachinePrecision], N[(N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[-1.0, 1/3], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < -9.999999999999999e-94

   1. Initial program 90.5%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   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 50.4

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

   5. Applied rewrites 50.4%

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

   6. Taylor expanded in g around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 90.9

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

   8. Applied rewrites 90.9%

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

   if -9.999999999999999e-94 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < 0.0

   1. Initial program 4.4%

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

   3. Taylor expanded in g around inf

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
   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 4.4

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

   5. Applied rewrites 4.4%

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

   6. Taylor expanded in g around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 4.4

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

   8. Applied rewrites 4.4%

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

      1. lift-+.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. cbrt-prod N/A

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

      5. lower-fma.f64 N/A

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

   10. Applied rewrites 80.8%

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

   if 0.0 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))))

   1. Initial program 35.0%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. cbrt-div N/A

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

      6. *-lft-identity N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 34.9%

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

   5. Taylor expanded in g around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cbrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-cbrt.f64 71.1

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

   7. Applied rewrites 71.1%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{{\left(2 \cdot a\right)}^{-1} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{-1}{2 \cdot a} \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)} \leq -1 \cdot 10^{-93}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot a\right)}^{-1} \cdot \left(\frac{h \cdot h}{g} \cdot -0.5\right)} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{elif}\;\sqrt[3]{{\left(2 \cdot a\right)}^{-1} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{-1}{2 \cdot a} \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{\left(-g\right) - g}, \sqrt[3]{\frac{-g}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot g\right) \cdot g\\ t_1 := {\left(2 \cdot a\right)}^{-1}\\ t_2 := \left(g \cdot g\right) \cdot a\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+215}:\\ \;\;\;\;\left(-g\right) \cdot \mathsf{fma}\left(\sqrt[3]{{t\_0}^{-1}} \cdot \sqrt[3]{0.5}, \sqrt[3]{2}, \sqrt[3]{\frac{2}{t\_2}} \cdot \sqrt[3]{0.5}\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+171}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{g}{a}}, \sqrt[3]{-1}, \sqrt[3]{-0.25 \cdot \left(\frac{h}{a} \cdot \frac{h}{g}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-g\right) \cdot \mathsf{fma}\left({t\_2}^{-0.3333333333333333} \cdot \sqrt[3]{0.5}, \sqrt[3]{2}, \sqrt[3]{\frac{2}{t\_0}} \cdot \sqrt[3]{0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (* (* a g) g)) (t_1 (pow (* 2.0 a) -1.0)) (t_2 (* (* g g) a)))
   (if (<= t_1 -2e+215)
     (*
      (- g)
      (fma
       (* (cbrt (pow t_0 -1.0)) (cbrt 0.5))
       (cbrt 2.0)
       (* (cbrt (/ 2.0 t_2)) (cbrt 0.5))))
     (if (<= t_1 5e+171)
       (fma (cbrt (/ g a)) (cbrt -1.0) (cbrt (* -0.25 (* (/ h a) (/ h g)))))
       (*
        (- g)
        (fma
         (* (pow t_2 -0.3333333333333333) (cbrt 0.5))
         (cbrt 2.0)
         (* (cbrt (/ 2.0 t_0)) (cbrt 0.5))))))))

C

double code(double g, double h, double a) {
	double t_0 = (a * g) * g;
	double t_1 = pow((2.0 * a), -1.0);
	double t_2 = (g * g) * a;
	double tmp;
	if (t_1 <= -2e+215) {
		tmp = -g * fma((cbrt(pow(t_0, -1.0)) * cbrt(0.5)), cbrt(2.0), (cbrt((2.0 / t_2)) * cbrt(0.5)));
	} else if (t_1 <= 5e+171) {
		tmp = fma(cbrt((g / a)), cbrt(-1.0), cbrt((-0.25 * ((h / a) * (h / g)))));
	} else {
		tmp = -g * fma((pow(t_2, -0.3333333333333333) * cbrt(0.5)), cbrt(2.0), (cbrt((2.0 / t_0)) * cbrt(0.5)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64(Float64(a * g) * g)
	t_1 = Float64(2.0 * a) ^ -1.0
	t_2 = Float64(Float64(g * g) * a)
	tmp = 0.0
	if (t_1 <= -2e+215)
		tmp = Float64(Float64(-g) * fma(Float64(cbrt((t_0 ^ -1.0)) * cbrt(0.5)), cbrt(2.0), Float64(cbrt(Float64(2.0 / t_2)) * cbrt(0.5))));
	elseif (t_1 <= 5e+171)
		tmp = fma(cbrt(Float64(g / a)), cbrt(-1.0), cbrt(Float64(-0.25 * Float64(Float64(h / a) * Float64(h / g)))));
	else
		tmp = Float64(Float64(-g) * fma(Float64((t_2 ^ -0.3333333333333333) * cbrt(0.5)), cbrt(2.0), Float64(cbrt(Float64(2.0 / t_0)) * cbrt(0.5))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(N[(a * g), $MachinePrecision] * g), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(2.0 * a), $MachinePrecision], -1.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(g * g), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+215], N[((-g) * N[(N[(N[Power[N[Power[t$95$0, -1.0], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[0.5, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision] + N[(N[Power[N[(2.0 / t$95$2), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[0.5, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+171], N[(N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[-1.0, 1/3], $MachinePrecision] + N[Power[N[(-0.25 * N[(N[(h / a), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[((-g) * N[(N[(N[Power[t$95$2, -0.3333333333333333], $MachinePrecision] * N[Power[0.5, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision] + N[(N[Power[N[(2.0 / t$95$0), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[0.5, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) < -1.99999999999999981e215

   1. Initial program 30.5%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. cbrt-prod N/A

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

      4. pow1/3 N/A

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

      5. lower-*.f64 N/A

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

      6. pow1/3 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cbrt.f64 30.5

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

      14. lift--.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. difference-of-squares N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower--.f64 N/A

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

   4. Applied rewrites 30.5%

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

   5. Taylor expanded in g around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

   7. Applied rewrites 5.4%

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

   9. Applied rewrites 55.4%

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

   if -1.99999999999999981e215 < (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) < 5.0000000000000004e171

   1. Initial program 54.2%

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

   3. Taylor expanded in h around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cbrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. unpow2 N/A

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

      11. *-commutative N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cbrt.f64 N/A

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

      19. lower-cbrt.f64 82.7

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

   5. Applied rewrites 82.7%

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

   7. Applied rewrites 83.7%

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

   if 5.0000000000000004e171 < (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 24.1%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. cbrt-prod N/A

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

      4. pow1/3 N/A

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

      5. lower-*.f64 N/A

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

      6. pow1/3 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cbrt.f64 31.3

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

      14. lift--.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. difference-of-squares N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower--.f64 N/A

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

   4. Applied rewrites 31.3%

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

   5. Taylor expanded in g around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

   7. Applied rewrites 7.4%

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

   9. Applied rewrites 57.2%

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

   11. Applied rewrites 57.2%

       \[\leadsto \left(-g\right) \cdot \mathsf{fma}\left({\left(\left(g \cdot g\right) \cdot a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{0.5}, \sqrt[3]{2}, \sqrt[3]{\frac{2}{\left(a \cdot g\right) \cdot g}} \cdot \sqrt[3]{0.5}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(2 \cdot a\right)}^{-1} \leq -2 \cdot 10^{+215}:\\ \;\;\;\;\left(-g\right) \cdot \mathsf{fma}\left(\sqrt[3]{{\left(\left(a \cdot g\right) \cdot g\right)}^{-1}} \cdot \sqrt[3]{0.5}, \sqrt[3]{2}, \sqrt[3]{\frac{2}{\left(g \cdot g\right) \cdot a}} \cdot \sqrt[3]{0.5}\right)\\ \mathbf{elif}\;{\left(2 \cdot a\right)}^{-1} \leq 5 \cdot 10^{+171}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{g}{a}}, \sqrt[3]{-1}, \sqrt[3]{-0.25 \cdot \left(\frac{h}{a} \cdot \frac{h}{g}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-g\right) \cdot \mathsf{fma}\left({\left(\left(g \cdot g\right) \cdot a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{0.5}, \sqrt[3]{2}, \sqrt[3]{\frac{2}{\left(a \cdot g\right) \cdot g}} \cdot \sqrt[3]{0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \cdot h \leq 2 \cdot 10^{+239}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{g \cdot -0.5}}{\sqrt[3]{a}}, \sqrt[3]{2}, \sqrt[3]{\frac{h}{g} \cdot \frac{h}{a}} \cdot \left(\sqrt[3]{0.5} \cdot \sqrt[3]{-0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-g\right) \cdot \mathsf{fma}\left(\sqrt[3]{{\left(\left(g \cdot g\right) \cdot a\right)}^{-1}} \cdot \sqrt[3]{0.5}, \sqrt[3]{2}, \frac{\sqrt[3]{\frac{2}{a \cdot g}}}{\sqrt[3]{g}} \cdot \sqrt[3]{0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= (* h h) 2e+239)
   (fma
    (/ (cbrt (* g -0.5)) (cbrt a))
    (cbrt 2.0)
    (* (cbrt (* (/ h g) (/ h a))) (* (cbrt 0.5) (cbrt -0.5))))
   (*
    (- g)
    (fma
     (* (cbrt (pow (* (* g g) a) -1.0)) (cbrt 0.5))
     (cbrt 2.0)
     (* (/ (cbrt (/ 2.0 (* a g))) (cbrt g)) (cbrt 0.5))))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((h * h) <= 2e+239) {
		tmp = fma((cbrt((g * -0.5)) / cbrt(a)), cbrt(2.0), (cbrt(((h / g) * (h / a))) * (cbrt(0.5) * cbrt(-0.5))));
	} else {
		tmp = -g * fma((cbrt(pow(((g * g) * a), -1.0)) * cbrt(0.5)), cbrt(2.0), ((cbrt((2.0 / (a * g))) / cbrt(g)) * cbrt(0.5)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (Float64(h * h) <= 2e+239)
		tmp = fma(Float64(cbrt(Float64(g * -0.5)) / cbrt(a)), cbrt(2.0), Float64(cbrt(Float64(Float64(h / g) * Float64(h / a))) * Float64(cbrt(0.5) * cbrt(-0.5))));
	else
		tmp = Float64(Float64(-g) * fma(Float64(cbrt((Float64(Float64(g * g) * a) ^ -1.0)) * cbrt(0.5)), cbrt(2.0), Float64(Float64(cbrt(Float64(2.0 / Float64(a * g))) / cbrt(g)) * cbrt(0.5))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[N[(h * h), $MachinePrecision], 2e+239], N[(N[(N[Power[N[(g * -0.5), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision] + N[(N[Power[N[(N[(h / g), $MachinePrecision] * N[(h / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(N[Power[0.5, 1/3], $MachinePrecision] * N[Power[-0.5, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-g) * N[(N[(N[Power[N[Power[N[(N[(g * g), $MachinePrecision] * a), $MachinePrecision], -1.0], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[0.5, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision] + N[(N[(N[Power[N[(2.0 / N[(a * g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[g, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[0.5, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 h h) < 1.99999999999999998e239

   1. Initial program 53.0%

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

   3. Taylor expanded in h around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cbrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. unpow2 N/A

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

      11. *-commutative N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cbrt.f64 N/A

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

      19. lower-cbrt.f64 76.0

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

   5. Applied rewrites 76.0%

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

   7. Applied rewrites 95.0%

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

   9. Applied rewrites 95.2%

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

   if 1.99999999999999998e239 < (*.f64 h h)

   1. Initial program 0.0%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. cbrt-prod N/A

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

      4. pow1/3 N/A

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

      5. lower-*.f64 N/A

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

      6. pow1/3 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cbrt.f64 0.0

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

      14. lift--.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. difference-of-squares N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower--.f64 N/A

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in g around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

   7. Applied rewrites 2.5%

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

   9. Applied rewrites 40.1%

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

   11. Applied rewrites 65.6%

       \[\leadsto \left(-g\right) \cdot \mathsf{fma}\left(\sqrt[3]{\frac{1}{\left(g \cdot g\right) \cdot a}} \cdot \sqrt[3]{0.5}, \sqrt[3]{2}, \frac{\sqrt[3]{\frac{2}{a \cdot g}}}{\sqrt[3]{g}} \cdot \sqrt[3]{0.5}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \cdot h \leq 2 \cdot 10^{+239}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{g \cdot -0.5}}{\sqrt[3]{a}}, \sqrt[3]{2}, \sqrt[3]{\frac{h}{g} \cdot \frac{h}{a}} \cdot \left(\sqrt[3]{0.5} \cdot \sqrt[3]{-0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-g\right) \cdot \mathsf{fma}\left(\sqrt[3]{{\left(\left(g \cdot g\right) \cdot a\right)}^{-1}} \cdot \sqrt[3]{0.5}, \sqrt[3]{2}, \frac{\sqrt[3]{\frac{2}{a \cdot g}}}{\sqrt[3]{g}} \cdot \sqrt[3]{0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 94.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{0.5} \cdot \sqrt[3]{-0.5}\\ \mathbf{if}\;h \leq 9.6 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{g \cdot -0.5}}{\sqrt[3]{a}}, \sqrt[3]{2}, \sqrt[3]{\frac{h}{g} \cdot \frac{h}{a}} \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}} \cdot \sqrt[3]{-0.5}, \sqrt[3]{2}, \sqrt[3]{h \cdot \frac{h}{a \cdot g}} \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (* (cbrt 0.5) (cbrt -0.5))))
   (if (<= h 9.6e+14)
     (fma
      (/ (cbrt (* g -0.5)) (cbrt a))
      (cbrt 2.0)
      (* (cbrt (* (/ h g) (/ h a))) t_0))
     (fma
      (* (/ (cbrt g) (cbrt a)) (cbrt -0.5))
      (cbrt 2.0)
      (* (cbrt (* h (/ h (* a g)))) t_0)))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt(0.5) * cbrt(-0.5);
	double tmp;
	if (h <= 9.6e+14) {
		tmp = fma((cbrt((g * -0.5)) / cbrt(a)), cbrt(2.0), (cbrt(((h / g) * (h / a))) * t_0));
	} else {
		tmp = fma(((cbrt(g) / cbrt(a)) * cbrt(-0.5)), cbrt(2.0), (cbrt((h * (h / (a * g)))) * t_0));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64(cbrt(0.5) * cbrt(-0.5))
	tmp = 0.0
	if (h <= 9.6e+14)
		tmp = fma(Float64(cbrt(Float64(g * -0.5)) / cbrt(a)), cbrt(2.0), Float64(cbrt(Float64(Float64(h / g) * Float64(h / a))) * t_0));
	else
		tmp = fma(Float64(Float64(cbrt(g) / cbrt(a)) * cbrt(-0.5)), cbrt(2.0), Float64(cbrt(Float64(h * Float64(h / Float64(a * g)))) * t_0));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(N[Power[0.5, 1/3], $MachinePrecision] * N[Power[-0.5, 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, 9.6e+14], N[(N[(N[Power[N[(g * -0.5), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision] + N[(N[Power[N[(N[(h / g), $MachinePrecision] * N[(h / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[g, 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[-0.5, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision] + N[(N[Power[N[(h * N[(h / N[(a * g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < 9.6e14

   1. Initial program 51.8%

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

   3. Taylor expanded in h around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cbrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. unpow2 N/A

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

      11. *-commutative N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cbrt.f64 N/A

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

      19. lower-cbrt.f64 73.4

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

   5. Applied rewrites 73.4%

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

   7. Applied rewrites 92.1%

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

   9. Applied rewrites 92.4%

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

   if 9.6e14 < h

   1. Initial program 23.5%

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

   3. Taylor expanded in h around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cbrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. unpow2 N/A

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

      11. *-commutative N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cbrt.f64 N/A

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

      19. lower-cbrt.f64 68.2

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

   5. Applied rewrites 68.2%

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

   7. Applied rewrites 79.2%

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

   9. Applied rewrites 86.7%

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

Alternative 10: 97.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.1%

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

3. Taylor expanded in h around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cbrt.f64 N/A

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

   10. unpow2 N/A

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

   11. *-commutative N/A

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

   12. times-frac N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lower-cbrt.f64 N/A

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

   19. lower-cbrt.f64 72.9

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

5. Applied rewrites 72.9%

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

7. Applied rewrites 90.9%

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

9. Applied rewrites 96.3%

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

11. Applied rewrites 96.6%

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

Alternative 11: 96.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}} \cdot \sqrt[3]{-0.5}, \sqrt[3]{2}, \frac{\sqrt[3]{\frac{h}{g} \cdot h}}{\sqrt[3]{a}} \cdot \sqrt[3]{-0.25}\right) \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (fma
  (* (/ (cbrt g) (cbrt a)) (cbrt -0.5))
  (cbrt 2.0)
  (* (/ (cbrt (* (/ h g) h)) (cbrt a)) (cbrt -0.25))))

C

double code(double g, double h, double a) {
	return fma(((cbrt(g) / cbrt(a)) * cbrt(-0.5)), cbrt(2.0), ((cbrt(((h / g) * h)) / cbrt(a)) * cbrt(-0.25)));
}

Julia

function code(g, h, a)
	return fma(Float64(Float64(cbrt(g) / cbrt(a)) * cbrt(-0.5)), cbrt(2.0), Float64(Float64(cbrt(Float64(Float64(h / g) * h)) / cbrt(a)) * cbrt(-0.25)))
end

Wolfram

code[g_, h_, a_] := N[(N[(N[(N[Power[g, 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[-0.5, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision] + N[(N[(N[Power[N[(N[(h / g), $MachinePrecision] * h), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[-0.25, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.1%

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

3. Taylor expanded in h around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cbrt.f64 N/A

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

   10. unpow2 N/A

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

   11. *-commutative N/A

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

   12. times-frac N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lower-cbrt.f64 N/A

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

   19. lower-cbrt.f64 72.9

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

5. Applied rewrites 72.9%

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

7. Applied rewrites 90.9%

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

9. Applied rewrites 96.3%

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

11. Applied rewrites 96.3%

    \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}} \cdot \sqrt[3]{-0.5}, \sqrt[3]{2}, \frac{\sqrt[3]{\frac{h}{g} \cdot h}}{\sqrt[3]{a}} \cdot \sqrt[3]{-0.25}\right) \]
12. Add Preprocessing

Alternative 12: 75.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{-215} \lor \neg \left(a \leq 6 \cdot 10^{-174}\right):\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{g}{a}}, \sqrt[3]{-1}, \sqrt[3]{-0.25 \cdot \left(\frac{h}{a} \cdot \frac{h}{g}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-g\right) \cdot \mathsf{fma}\left(\sqrt[3]{{\left(\left(g \cdot g\right) \cdot a\right)}^{-1}} \cdot \sqrt[3]{0.5}, \sqrt[3]{2}, \sqrt[3]{\frac{2}{\left(a \cdot g\right) \cdot g}} \cdot \sqrt[3]{0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (or (<= a -2.4e-215) (not (<= a 6e-174)))
   (fma (cbrt (/ g a)) (cbrt -1.0) (cbrt (* -0.25 (* (/ h a) (/ h g)))))
   (*
    (- g)
    (fma
     (* (cbrt (pow (* (* g g) a) -1.0)) (cbrt 0.5))
     (cbrt 2.0)
     (* (cbrt (/ 2.0 (* (* a g) g))) (cbrt 0.5))))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((a <= -2.4e-215) || !(a <= 6e-174)) {
		tmp = fma(cbrt((g / a)), cbrt(-1.0), cbrt((-0.25 * ((h / a) * (h / g)))));
	} else {
		tmp = -g * fma((cbrt(pow(((g * g) * a), -1.0)) * cbrt(0.5)), cbrt(2.0), (cbrt((2.0 / ((a * g) * g))) * cbrt(0.5)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if ((a <= -2.4e-215) || !(a <= 6e-174))
		tmp = fma(cbrt(Float64(g / a)), cbrt(-1.0), cbrt(Float64(-0.25 * Float64(Float64(h / a) * Float64(h / g)))));
	else
		tmp = Float64(Float64(-g) * fma(Float64(cbrt((Float64(Float64(g * g) * a) ^ -1.0)) * cbrt(0.5)), cbrt(2.0), Float64(cbrt(Float64(2.0 / Float64(Float64(a * g) * g))) * cbrt(0.5))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[Or[LessEqual[a, -2.4e-215], N[Not[LessEqual[a, 6e-174]], $MachinePrecision]], N[(N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[-1.0, 1/3], $MachinePrecision] + N[Power[N[(-0.25 * N[(N[(h / a), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[((-g) * N[(N[(N[Power[N[Power[N[(N[(g * g), $MachinePrecision] * a), $MachinePrecision], -1.0], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[0.5, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision] + N[(N[Power[N[(2.0 / N[(N[(a * g), $MachinePrecision] * g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[0.5, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.4000000000000001e-215 or 6.00000000000000042e-174 < a

   1. Initial program 54.2%

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

   3. Taylor expanded in h around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cbrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. unpow2 N/A

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

      11. *-commutative N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cbrt.f64 N/A

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

      19. lower-cbrt.f64 82.7

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

   5. Applied rewrites 82.7%

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

   7. Applied rewrites 83.7%

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

   if -2.4000000000000001e-215 < a < 6.00000000000000042e-174

   1. Initial program 27.1%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. cbrt-prod N/A

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

      4. pow1/3 N/A

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

      5. lower-*.f64 N/A

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

      6. pow1/3 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cbrt.f64 31.0

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

      14. lift--.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. difference-of-squares N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower--.f64 N/A

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

   4. Applied rewrites 31.0%

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

   5. Taylor expanded in g around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

   7. Applied rewrites 6.5%

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

   9. Applied rewrites 56.6%

      \[\leadsto \left(-g\right) \cdot \mathsf{fma}\left(\sqrt[3]{\frac{1}{\left(g \cdot g\right) \cdot a}} \cdot \sqrt[3]{0.5}, \sqrt[3]{2}, \sqrt[3]{\frac{2}{\left(a \cdot g\right) \cdot g}} \cdot \sqrt[3]{0.5}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{-215} \lor \neg \left(a \leq 6 \cdot 10^{-174}\right):\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{g}{a}}, \sqrt[3]{-1}, \sqrt[3]{-0.25 \cdot \left(\frac{h}{a} \cdot \frac{h}{g}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-g\right) \cdot \mathsf{fma}\left(\sqrt[3]{{\left(\left(g \cdot g\right) \cdot a\right)}^{-1}} \cdot \sqrt[3]{0.5}, \sqrt[3]{2}, \sqrt[3]{\frac{2}{\left(a \cdot g\right) \cdot g}} \cdot \sqrt[3]{0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 93.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.1%

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

3. Taylor expanded in h around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cbrt.f64 N/A

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

   10. unpow2 N/A

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

   11. *-commutative N/A

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

   12. times-frac N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lower-cbrt.f64 N/A

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

   19. lower-cbrt.f64 72.9

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

5. Applied rewrites 72.9%

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

7. Applied rewrites 90.9%

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

9. Applied rewrites 91.1%

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

Alternative 14: 75.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.1 \cdot 10^{-230} \lor \neg \left(a \leq 6 \cdot 10^{-174}\right):\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{g}{a}}, \sqrt[3]{-1}, \sqrt[3]{-0.25 \cdot \left(\frac{h}{a} \cdot \frac{h}{g}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-g\right) \cdot \mathsf{fma}\left({\left(\left(g \cdot g\right) \cdot a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{0.5}, \sqrt[3]{2}, \sqrt[3]{\frac{2}{\left(a \cdot g\right) \cdot g}} \cdot \sqrt[3]{0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (or (<= a -3.1e-230) (not (<= a 6e-174)))
   (fma (cbrt (/ g a)) (cbrt -1.0) (cbrt (* -0.25 (* (/ h a) (/ h g)))))
   (*
    (- g)
    (fma
     (* (pow (* (* g g) a) -0.3333333333333333) (cbrt 0.5))
     (cbrt 2.0)
     (* (cbrt (/ 2.0 (* (* a g) g))) (cbrt 0.5))))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((a <= -3.1e-230) || !(a <= 6e-174)) {
		tmp = fma(cbrt((g / a)), cbrt(-1.0), cbrt((-0.25 * ((h / a) * (h / g)))));
	} else {
		tmp = -g * fma((pow(((g * g) * a), -0.3333333333333333) * cbrt(0.5)), cbrt(2.0), (cbrt((2.0 / ((a * g) * g))) * cbrt(0.5)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if ((a <= -3.1e-230) || !(a <= 6e-174))
		tmp = fma(cbrt(Float64(g / a)), cbrt(-1.0), cbrt(Float64(-0.25 * Float64(Float64(h / a) * Float64(h / g)))));
	else
		tmp = Float64(Float64(-g) * fma(Float64((Float64(Float64(g * g) * a) ^ -0.3333333333333333) * cbrt(0.5)), cbrt(2.0), Float64(cbrt(Float64(2.0 / Float64(Float64(a * g) * g))) * cbrt(0.5))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[Or[LessEqual[a, -3.1e-230], N[Not[LessEqual[a, 6e-174]], $MachinePrecision]], N[(N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[-1.0, 1/3], $MachinePrecision] + N[Power[N[(-0.25 * N[(N[(h / a), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[((-g) * N[(N[(N[Power[N[(N[(g * g), $MachinePrecision] * a), $MachinePrecision], -0.3333333333333333], $MachinePrecision] * N[Power[0.5, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision] + N[(N[Power[N[(2.0 / N[(N[(a * g), $MachinePrecision] * g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[0.5, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.1e-230 or 6.00000000000000042e-174 < a

   1. Initial program 53.7%

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

   3. Taylor expanded in h around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cbrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. unpow2 N/A

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

      11. *-commutative N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cbrt.f64 N/A

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

      19. lower-cbrt.f64 81.4

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

   5. Applied rewrites 81.4%

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

   7. Applied rewrites 82.3%

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

   if -3.1e-230 < a < 6.00000000000000042e-174

   1. Initial program 26.2%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. cbrt-prod N/A

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

      4. pow1/3 N/A

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

      5. lower-*.f64 N/A

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

      6. pow1/3 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cbrt.f64 30.6

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

      14. lift--.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. difference-of-squares N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower--.f64 N/A

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

   4. Applied rewrites 30.6%

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

   5. Taylor expanded in g around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

   7. Applied rewrites 5.9%

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

   9. Applied rewrites 58.5%

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

   11. Applied rewrites 57.6%

       \[\leadsto \left(-g\right) \cdot \mathsf{fma}\left({\left(\left(g \cdot g\right) \cdot a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{0.5}, \sqrt[3]{2}, \sqrt[3]{\frac{2}{\left(a \cdot g\right) \cdot g}} \cdot \sqrt[3]{0.5}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.1 \cdot 10^{-230} \lor \neg \left(a \leq 6 \cdot 10^{-174}\right):\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{g}{a}}, \sqrt[3]{-1}, \sqrt[3]{-0.25 \cdot \left(\frac{h}{a} \cdot \frac{h}{g}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-g\right) \cdot \mathsf{fma}\left({\left(\left(g \cdot g\right) \cdot a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{0.5}, \sqrt[3]{2}, \sqrt[3]{\frac{2}{\left(a \cdot g\right) \cdot g}} \cdot \sqrt[3]{0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 73.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.1%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. cbrt-div N/A

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

   6. *-lft-identity N/A

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

   7. lower-/.f64 N/A

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

4. Applied rewrites 51.3%

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

5. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-cbrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-cbrt.f64 72.8

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

7. Applied rewrites 72.8%

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

Reproduce

herbie shell --seed 2024354 
(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))))))))