2-ancestry mixing, positive discriminant

Percentage Accurate: 45.2% → 97.6%

Time: 12.5s

Alternatives: 8

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 45.2% 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

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

FPCore

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

Julia

function code(g, h, a)
	return fma(Float64(Float64(cbrt(g) * (2.0 ^ 0.3333333333333333)) / cbrt(a)), cbrt(-0.5), 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[2.0, 0.3333333333333333], $MachinePrecision]), $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[-0.5, 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 44.8%

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

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. lower-cbrt.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cbrt.f64 N/A

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

   11. unpow2 N/A

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

   12. *-commutative N/A

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

   13. times-frac N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

5. Applied rewrites 73.1%

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

7. Applied rewrites 92.5%

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

9. Applied rewrites 96.3%

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

11. Applied rewrites 96.9%

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

Alternative 2: 78.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ t_1 := \sqrt[3]{{\left(2 \cdot a\right)}^{-1} \cdot \left(\left(-g\right) + t\_0\right)}\\ t_2 := t\_1 + \sqrt[3]{\frac{-1}{2 \cdot a} \cdot \left(g + t\_0\right)}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-104} \lor \neg \left(t\_2 \leq 2 \cdot 10^{-104}\right):\\ \;\;\;\;\sqrt[3]{\left(\frac{g}{a} \cdot 2\right) \cdot -0.5} + \sqrt[3]{-0.25 \cdot \left(\frac{h}{a} \cdot \frac{h}{g}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \frac{\sqrt[3]{\left(-2 \cdot g\right) \cdot 0.5}}{\sqrt[3]{a}}\\ \end{array} \end{array} \]

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))))
        (t_2 (+ t_1 (cbrt (* (/ -1.0 (* 2.0 a)) (+ g t_0))))))
   (if (or (<= t_2 -2e-104) (not (<= t_2 2e-104)))
     (+ (cbrt (* (* (/ g a) 2.0) -0.5)) (cbrt (* -0.25 (* (/ h a) (/ h g)))))
     (+ t_1 (/ (cbrt (* (* -2.0 g) 0.5)) (cbrt 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)));
	double t_2 = t_1 + cbrt(((-1.0 / (2.0 * a)) * (g + t_0)));
	double tmp;
	if ((t_2 <= -2e-104) || !(t_2 <= 2e-104)) {
		tmp = cbrt((((g / a) * 2.0) * -0.5)) + cbrt((-0.25 * ((h / a) * (h / g))));
	} else {
		tmp = t_1 + (cbrt(((-2.0 * g) * 0.5)) / cbrt(a));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.sqrt(((g * g) - (h * h)));
	double t_1 = Math.cbrt((Math.pow((2.0 * a), -1.0) * (-g + t_0)));
	double t_2 = t_1 + Math.cbrt(((-1.0 / (2.0 * a)) * (g + t_0)));
	double tmp;
	if ((t_2 <= -2e-104) || !(t_2 <= 2e-104)) {
		tmp = Math.cbrt((((g / a) * 2.0) * -0.5)) + Math.cbrt((-0.25 * ((h / a) * (h / g))));
	} else {
		tmp = t_1 + (Math.cbrt(((-2.0 * g) * 0.5)) / Math.cbrt(a));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	t_1 = cbrt(Float64((Float64(2.0 * a) ^ -1.0) * Float64(Float64(-g) + t_0)))
	t_2 = Float64(t_1 + cbrt(Float64(Float64(-1.0 / Float64(2.0 * a)) * Float64(g + t_0))))
	tmp = 0.0
	if ((t_2 <= -2e-104) || !(t_2 <= 2e-104))
		tmp = Float64(cbrt(Float64(Float64(Float64(g / a) * 2.0) * -0.5)) + cbrt(Float64(-0.25 * Float64(Float64(h / a) * Float64(h / g)))));
	else
		tmp = Float64(t_1 + Float64(cbrt(Float64(Float64(-2.0 * g) * 0.5)) / cbrt(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[Power[N[(N[Power[N[(2.0 * a), $MachinePrecision], -1.0], $MachinePrecision] * N[((-g) + t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + 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$2, -2e-104], N[Not[LessEqual[t$95$2, 2e-104]], $MachinePrecision]], N[(N[Power[N[(N[(N[(g / a), $MachinePrecision] * 2.0), $MachinePrecision] * -0.5), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(-0.25 * N[(N[(h / a), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[Power[N[(N[(-2.0 * g), $MachinePrecision] * 0.5), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $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))))))) < -1.99999999999999985e-104 or 1.99999999999999985e-104 < (+.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 46.1%

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

   3. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lower-cbrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cbrt.f64 N/A

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

      11. unpow2 N/A

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

      12. *-commutative N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   5. Applied rewrites 75.9%

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

   7. Applied rewrites 92.8%

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

   9. Applied rewrites 76.7%

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

   if -1.99999999999999985e-104 < (+.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))))))) < 1.99999999999999985e-104

   1. Initial program 16.8%

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

      1. lift-cbrt.f64 N/A

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

      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. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. cbrt-div N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 35.7%

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

   5. Taylor expanded in g around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. rem-square-sqrt N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. rem-square-sqrt N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval 87.1

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

   7. Applied rewrites 87.1%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{{\left(2 \cdot a\right)}^{-1} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{-1}{2 \cdot a} \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)} \leq -2 \cdot 10^{-104} \lor \neg \left(\sqrt[3]{{\left(2 \cdot a\right)}^{-1} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{-1}{2 \cdot a} \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)} \leq 2 \cdot 10^{-104}\right):\\ \;\;\;\;\sqrt[3]{\left(\frac{g}{a} \cdot 2\right) \cdot -0.5} + \sqrt[3]{-0.25 \cdot \left(\frac{h}{a} \cdot \frac{h}{g}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot a\right)}^{-1} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \frac{\sqrt[3]{\left(-2 \cdot g\right) \cdot 0.5}}{\sqrt[3]{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.8%

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

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. lower-cbrt.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cbrt.f64 N/A

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

   11. unpow2 N/A

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

   12. *-commutative N/A

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

   13. times-frac N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

5. Applied rewrites 73.1%

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

7. Applied rewrites 92.5%

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

9. Applied rewrites 96.3%

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

11. Applied rewrites 96.8%

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

Alternative 4: 87.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;g \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\left(\frac{g}{a} \cdot 2\right) \cdot -0.5} + \sqrt[3]{-0.25 \cdot \left(\frac{h}{a} \cdot \frac{h}{g}\right)}\\ \mathbf{elif}\;g \leq -1 \cdot 10^{-307}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{\mathsf{fma}\left(-h, h, g \cdot g\right)} - g}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{-0.25}{a} \cdot \frac{h \cdot h}{g}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a}} \cdot \left(\sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)} + \sqrt[3]{\left(\frac{h}{g} \cdot h\right) \cdot -0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= g -1.35e+154)
   (+ (cbrt (* (* (/ g a) 2.0) -0.5)) (cbrt (* -0.25 (* (/ h a) (/ h g)))))
   (if (<= g -1e-307)
     (+
      (/ (cbrt (- (sqrt (fma (- h) h (* g g))) g)) (cbrt (* 2.0 a)))
      (cbrt (* (/ -0.25 a) (/ (* h h) g))))
     (*
      (cbrt (/ 0.5 a))
      (+
       (cbrt (- (fma (sqrt (+ h g)) (sqrt (- g h)) g)))
       (cbrt (* (* (/ h g) h) -0.5)))))))

C

double code(double g, double h, double a) {
	double tmp;
	if (g <= -1.35e+154) {
		tmp = cbrt((((g / a) * 2.0) * -0.5)) + cbrt((-0.25 * ((h / a) * (h / g))));
	} else if (g <= -1e-307) {
		tmp = (cbrt((sqrt(fma(-h, h, (g * g))) - g)) / cbrt((2.0 * a))) + cbrt(((-0.25 / a) * ((h * h) / g)));
	} else {
		tmp = cbrt((0.5 / a)) * (cbrt(-fma(sqrt((h + g)), sqrt((g - h)), g)) + cbrt((((h / g) * h) * -0.5)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (g <= -1.35e+154)
		tmp = Float64(cbrt(Float64(Float64(Float64(g / a) * 2.0) * -0.5)) + cbrt(Float64(-0.25 * Float64(Float64(h / a) * Float64(h / g)))));
	elseif (g <= -1e-307)
		tmp = Float64(Float64(cbrt(Float64(sqrt(fma(Float64(-h), h, Float64(g * g))) - g)) / cbrt(Float64(2.0 * a))) + cbrt(Float64(Float64(-0.25 / a) * Float64(Float64(h * h) / g))));
	else
		tmp = Float64(cbrt(Float64(0.5 / a)) * Float64(cbrt(Float64(-fma(sqrt(Float64(h + g)), sqrt(Float64(g - h)), g))) + cbrt(Float64(Float64(Float64(h / g) * h) * -0.5))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[g, -1.35e+154], N[(N[Power[N[(N[(N[(g / a), $MachinePrecision] * 2.0), $MachinePrecision] * -0.5), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(-0.25 * N[(N[(h / a), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[g, -1e-307], N[(N[(N[Power[N[(N[Sqrt[N[((-h) * h + N[(g * g), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - g), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[N[(2.0 * a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(-0.25 / a), $MachinePrecision] * N[(N[(h * h), $MachinePrecision] / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[(N[Power[(-N[(N[Sqrt[N[(h + g), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(g - h), $MachinePrecision]], $MachinePrecision] + g), $MachinePrecision]), 1/3], $MachinePrecision] + N[Power[N[(N[(N[(h / g), $MachinePrecision] * h), $MachinePrecision] * -0.5), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if g < -1.35000000000000003e154

   1. Initial program 0.0%

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

   3. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lower-cbrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cbrt.f64 N/A

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

      11. unpow2 N/A

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

      12. *-commutative N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   5. Applied rewrites 57.5%

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

   7. Applied rewrites 92.6%

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

   9. Applied rewrites 58.1%

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

   if -1.35000000000000003e154 < g < -9.99999999999999909e-308

   1. Initial program 73.0%

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

   3. Taylor expanded in g around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 16.0

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

   5. Applied rewrites 16.0%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. cbrt-div N/A

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

      6. *-lft-identity N/A

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

      7. lower-/.f64 N/A

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

   7. Applied rewrites 0.0%

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

   8. Taylor expanded in g around -inf

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 0.0

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

   10. Applied rewrites 0.0%

       \[\leadsto \frac{\sqrt[3]{\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, -g\right)}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\color{blue}{\frac{-0.25}{a} \cdot \frac{h \cdot h}{g}}} \]
   11. Step-by-step derivation

       1. lift-neg.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. lift-+.f64 N/A

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

       5. lift-sqrt.f64 N/A

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

       6. lift--.f64 N/A

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

       7. lift-+.f64 N/A

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

       8. lift--.f64 N/A

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

       9. sqrt-prod N/A

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

       10. lift-*.f64 N/A

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

       11. lift-sqrt.f64 N/A

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

       12. lower-+.f64 N/A

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

   12. Applied rewrites 86.9%

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

   if -9.99999999999999909e-308 < g

   1. Initial program 46.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sqr-neg-rev N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. difference-of-squares N/A

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

      9. sqrt-prod N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower--.f64 0.0

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in g around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 0.0

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

   7. Applied rewrites 0.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. *-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lift--.f64 N/A

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

      10. difference-of-squares-rev N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. sqr-neg-rev N/A

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

      14. difference-of-squares N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift--.f64 N/A

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

      18. *-commutative N/A

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

      19. sqrt-unprod N/A

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

   9. Applied rewrites 76.0%

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

   11. Applied rewrites 97.3%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\left(\frac{g}{a} \cdot 2\right) \cdot -0.5} + \sqrt[3]{-0.25 \cdot \left(\frac{h}{a} \cdot \frac{h}{g}\right)}\\ \mathbf{elif}\;g \leq -1 \cdot 10^{-307}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{\mathsf{fma}\left(-h, h, g \cdot g\right)} - g}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{-0.25}{a} \cdot \frac{h \cdot h}{g}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a}} \cdot \left(\sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)} + \sqrt[3]{\left(\frac{h}{g} \cdot h\right) \cdot -0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;g \leq -1 \cdot 10^{-307}:\\ \;\;\;\;\sqrt[3]{\left(\frac{g}{a} \cdot 2\right) \cdot -0.5} + \sqrt[3]{-0.25 \cdot \left(\frac{h}{a} \cdot \frac{h}{g}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a}} \cdot \left(\sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)} + \sqrt[3]{\left(\frac{h}{g} \cdot h\right) \cdot -0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= g -1e-307)
   (+ (cbrt (* (* (/ g a) 2.0) -0.5)) (cbrt (* -0.25 (* (/ h a) (/ h g)))))
   (*
    (cbrt (/ 0.5 a))
    (+
     (cbrt (- (fma (sqrt (+ h g)) (sqrt (- g h)) g)))
     (cbrt (* (* (/ h g) h) -0.5))))))

C

double code(double g, double h, double a) {
	double tmp;
	if (g <= -1e-307) {
		tmp = cbrt((((g / a) * 2.0) * -0.5)) + cbrt((-0.25 * ((h / a) * (h / g))));
	} else {
		tmp = cbrt((0.5 / a)) * (cbrt(-fma(sqrt((h + g)), sqrt((g - h)), g)) + cbrt((((h / g) * h) * -0.5)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (g <= -1e-307)
		tmp = Float64(cbrt(Float64(Float64(Float64(g / a) * 2.0) * -0.5)) + cbrt(Float64(-0.25 * Float64(Float64(h / a) * Float64(h / g)))));
	else
		tmp = Float64(cbrt(Float64(0.5 / a)) * Float64(cbrt(Float64(-fma(sqrt(Float64(h + g)), sqrt(Float64(g - h)), g))) + cbrt(Float64(Float64(Float64(h / g) * h) * -0.5))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[g, -1e-307], N[(N[Power[N[(N[(N[(g / a), $MachinePrecision] * 2.0), $MachinePrecision] * -0.5), $MachinePrecision], 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[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[(N[Power[(-N[(N[Sqrt[N[(h + g), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(g - h), $MachinePrecision]], $MachinePrecision] + g), $MachinePrecision]), 1/3], $MachinePrecision] + N[Power[N[(N[(N[(h / g), $MachinePrecision] * h), $MachinePrecision] * -0.5), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if g < -9.99999999999999909e-308

   1. Initial program 43.5%

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

   3. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lower-cbrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cbrt.f64 N/A

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

      11. unpow2 N/A

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

      12. *-commutative N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   5. Applied rewrites 70.0%

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

   7. Applied rewrites 94.4%

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

   9. Applied rewrites 70.6%

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

   if -9.99999999999999909e-308 < g

   1. Initial program 46.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sqr-neg-rev N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. difference-of-squares N/A

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

      9. sqrt-prod N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower--.f64 0.0

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in g around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 0.0

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

   7. Applied rewrites 0.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. *-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lift--.f64 N/A

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

      10. difference-of-squares-rev N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. sqr-neg-rev N/A

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

      14. difference-of-squares N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift--.f64 N/A

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

      18. *-commutative N/A

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

      19. sqrt-unprod N/A

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

   9. Applied rewrites 76.0%

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

   11. Applied rewrites 97.3%

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

Alternative 6: 75.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double g, double h, double a) {
	return Math.cbrt((((g / a) * 2.0) * -0.5)) + Math.cbrt((-0.25 * ((h / a) * (h / g))));
}

Julia

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

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(N[(N[(g / a), $MachinePrecision] * 2.0), $MachinePrecision] * -0.5), $MachinePrecision], 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. Initial program 44.8%

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

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. lower-cbrt.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cbrt.f64 N/A

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

   11. unpow2 N/A

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

   12. *-commutative N/A

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

   13. times-frac N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

5. Applied rewrites 73.1%

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

7. Applied rewrites 92.5%

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

9. Applied rewrites 73.9%

   \[\leadsto \sqrt[3]{\left(\frac{g}{a} \cdot 2\right) \cdot -0.5} + \color{blue}{\sqrt[3]{-0.25 \cdot \left(\frac{h}{a} \cdot \frac{h}{g}\right)}} \]
10. Add Preprocessing

Alternative 7: 15.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.8%

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

3. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 27.9

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

5. Applied rewrites 27.9%

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

6. Taylor expanded in g around -inf

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \color{blue}{-1 \cdot g}\right)} + \sqrt[3]{\frac{-g}{a}} \]
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 15.2

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

8. Applied rewrites 15.2%

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \color{blue}{\left(-g\right)}\right)} + \sqrt[3]{\frac{-g}{a}} \]
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 15.2

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

10. Applied rewrites 15.2%

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

Alternative 8: 3.0% accurate, 100.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(g, h, a)
	return Float64(-0.0)
end

MATLAB

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

Wolfram

code[g_, h_, a_] := (-0.0)

Derivation

1. Initial program 44.8%

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

3. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 27.9

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

5. Applied rewrites 27.9%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. cbrt-div N/A

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

   6. *-lft-identity N/A

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

   7. lower-/.f64 N/A

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

7. Applied rewrites 18.6%

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

8. Taylor expanded in g around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-*.f64 N/A

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

10. Applied rewrites 3.0%

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

11. Taylor expanded in g around 0

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

13. Applied rewrites 3.0%

    \[\leadsto -0 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024333 
(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))))))))