2-ancestry mixing, positive discriminant

Percentage Accurate: 43.9% → 78.0%

Time: 14.1s

Alternatives: 7

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 43.9% 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: 78.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\left(\frac{h}{a} \cdot \frac{h}{g}\right) \cdot -0.25}\\ \mathbf{if}\;g \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t\_0 + \sqrt[3]{{a}^{-1} \cdot \left(-g\right)}\\ \mathbf{elif}\;g \leq -3.4 \cdot 10^{-160}:\\ \;\;\;\;\sqrt[3]{\frac{h \cdot h}{g} \cdot \frac{-0.25}{a}} + \frac{\sqrt[3]{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}}{\sqrt[3]{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{-g}{a}} + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (* (* (/ h a) (/ h g)) -0.25))))
   (if (<= g -1.35e+154)
     (+ t_0 (cbrt (* (pow a -1.0) (- g))))
     (if (<= g -3.4e-160)
       (+
        (cbrt (* (/ (* h h) g) (/ -0.25 a)))
        (/ (cbrt (- (sqrt (* (+ h g) (- g h))) g)) (cbrt (* a 2.0))))
       (+ (cbrt (/ (- g) a)) t_0)))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt((((h / a) * (h / g)) * -0.25));
	double tmp;
	if (g <= -1.35e+154) {
		tmp = t_0 + cbrt((pow(a, -1.0) * -g));
	} else if (g <= -3.4e-160) {
		tmp = cbrt((((h * h) / g) * (-0.25 / a))) + (cbrt((sqrt(((h + g) * (g - h))) - g)) / cbrt((a * 2.0)));
	} else {
		tmp = cbrt((-g / a)) + t_0;
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.cbrt((((h / a) * (h / g)) * -0.25));
	double tmp;
	if (g <= -1.35e+154) {
		tmp = t_0 + Math.cbrt((Math.pow(a, -1.0) * -g));
	} else if (g <= -3.4e-160) {
		tmp = Math.cbrt((((h * h) / g) * (-0.25 / a))) + (Math.cbrt((Math.sqrt(((h + g) * (g - h))) - g)) / Math.cbrt((a * 2.0)));
	} else {
		tmp = Math.cbrt((-g / a)) + t_0;
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(Float64(Float64(h / a) * Float64(h / g)) * -0.25))
	tmp = 0.0
	if (g <= -1.35e+154)
		tmp = Float64(t_0 + cbrt(Float64((a ^ -1.0) * Float64(-g))));
	elseif (g <= -3.4e-160)
		tmp = Float64(cbrt(Float64(Float64(Float64(h * h) / g) * Float64(-0.25 / a))) + Float64(cbrt(Float64(sqrt(Float64(Float64(h + g) * Float64(g - h))) - g)) / cbrt(Float64(a * 2.0))));
	else
		tmp = Float64(cbrt(Float64(Float64(-g) / a)) + t_0);
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[(N[(N[(h / a), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[g, -1.35e+154], N[(t$95$0 + N[Power[N[(N[Power[a, -1.0], $MachinePrecision] * (-g)), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[g, -3.4e-160], N[(N[Power[N[(N[(N[(h * h), $MachinePrecision] / g), $MachinePrecision] * N[(-0.25 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(N[Sqrt[N[(N[(h + g), $MachinePrecision] * N[(g - h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - g), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[N[(a * 2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision] + t$95$0), $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 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 3.2

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

   5. Applied rewrites 3.2%

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

   6. Taylor expanded in g around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cbrt.f64 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. lower-cbrt.f64 63.6

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

   8. Applied rewrites 63.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 63.6

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

   10. Applied rewrites 63.6%

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

   12. Applied rewrites 63.7%

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

   if -1.35000000000000003e154 < g < -3.40000000000000021e-160

   1. Initial program 83.9%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. cbrt-div N/A

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

      6. *-lft-identity N/A

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

      7. pow1/3 N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 94.5%

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

   5. Taylor expanded in g around -inf

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 97.2

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

   7. Applied rewrites 97.2%

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

   if -3.40000000000000021e-160 < g

   1. Initial program 48.0%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   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 48.1

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

   5. Applied rewrites 48.1%

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

   6. Taylor expanded in g around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cbrt.f64 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. lower-cbrt.f64 78.9

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

   8. Applied rewrites 78.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 78.9

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

   10. Applied rewrites 78.9%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\left(\frac{h}{a} \cdot \frac{h}{g}\right) \cdot -0.25} + \sqrt[3]{{a}^{-1} \cdot \left(-g\right)}\\ \mathbf{elif}\;g \leq -3.4 \cdot 10^{-160}:\\ \;\;\;\;\sqrt[3]{\frac{h \cdot h}{g} \cdot \frac{-0.25}{a}} + \frac{\sqrt[3]{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}}{\sqrt[3]{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{-g}{a}} + \sqrt[3]{\left(\frac{h}{a} \cdot \frac{h}{g}\right) \cdot -0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left(h + g\right) \cdot \left(g - h\right)}\\ t_1 := \sqrt[3]{a \cdot 2}\\ \mathbf{if}\;\frac{1}{a \cdot 2} \leq -4 \cdot 10^{+273}:\\ \;\;\;\;\frac{\sqrt[3]{\left(-g\right) - t\_0}}{t\_1} + \frac{\sqrt[3]{t\_0 - g}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\frac{h}{a} \cdot \frac{h}{g}\right) \cdot -0.25} + \sqrt[3]{{a}^{-1} \cdot \left(-g\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (sqrt (* (+ h g) (- g h)))) (t_1 (cbrt (* a 2.0))))
   (if (<= (/ 1.0 (* a 2.0)) -4e+273)
     (+ (/ (cbrt (- (- g) t_0)) t_1) (/ (cbrt (- t_0 g)) t_1))
     (+ (cbrt (* (* (/ h a) (/ h g)) -0.25)) (cbrt (* (pow a -1.0) (- g)))))))

C

double code(double g, double h, double a) {
	double t_0 = sqrt(((h + g) * (g - h)));
	double t_1 = cbrt((a * 2.0));
	double tmp;
	if ((1.0 / (a * 2.0)) <= -4e+273) {
		tmp = (cbrt((-g - t_0)) / t_1) + (cbrt((t_0 - g)) / t_1);
	} else {
		tmp = cbrt((((h / a) * (h / g)) * -0.25)) + cbrt((pow(a, -1.0) * -g));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.sqrt(((h + g) * (g - h)));
	double t_1 = Math.cbrt((a * 2.0));
	double tmp;
	if ((1.0 / (a * 2.0)) <= -4e+273) {
		tmp = (Math.cbrt((-g - t_0)) / t_1) + (Math.cbrt((t_0 - g)) / t_1);
	} else {
		tmp = Math.cbrt((((h / a) * (h / g)) * -0.25)) + Math.cbrt((Math.pow(a, -1.0) * -g));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = sqrt(Float64(Float64(h + g) * Float64(g - h)))
	t_1 = cbrt(Float64(a * 2.0))
	tmp = 0.0
	if (Float64(1.0 / Float64(a * 2.0)) <= -4e+273)
		tmp = Float64(Float64(cbrt(Float64(Float64(-g) - t_0)) / t_1) + Float64(cbrt(Float64(t_0 - g)) / t_1));
	else
		tmp = Float64(cbrt(Float64(Float64(Float64(h / a) * Float64(h / g)) * -0.25)) + cbrt(Float64((a ^ -1.0) * Float64(-g))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Sqrt[N[(N[(h + g), $MachinePrecision] * N[(g - h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(a * 2.0), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(1.0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -4e+273], N[(N[(N[Power[N[((-g) - t$95$0), $MachinePrecision], 1/3], $MachinePrecision] / t$95$1), $MachinePrecision] + N[(N[Power[N[(t$95$0 - g), $MachinePrecision], 1/3], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(N[(h / a), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[Power[a, -1.0], $MachinePrecision] * (-g)), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) < -3.99999999999999978e273

   1. Initial program 18.9%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. cbrt-div N/A

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

      6. *-lft-identity N/A

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

      7. pow1/3 N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 42.5%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. cbrt-div N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cbrt.f64 N/A

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

      11. lower-/.f64 N/A

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

   6. Applied rewrites 65.8%

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

   if -3.99999999999999978e273 < (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 50.7%

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

   3. Taylor expanded in 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 31.9

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

   5. Applied rewrites 31.9%

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

   6. Taylor expanded in g around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cbrt.f64 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. lower-cbrt.f64 80.8

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

   8. Applied rewrites 80.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 80.8

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

   10. Applied rewrites 80.8%

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

   12. Applied rewrites 80.8%

       \[\leadsto \sqrt[3]{{a}^{-1} \cdot \color{blue}{\left(-g\right)}} + \sqrt[3]{-0.25 \cdot \left(\frac{h}{g} \cdot \frac{h}{a}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{a \cdot 2} \leq -4 \cdot 10^{+273}:\\ \;\;\;\;\frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(h + g\right) \cdot \left(g - h\right)}}}{\sqrt[3]{a \cdot 2}} + \frac{\sqrt[3]{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}}{\sqrt[3]{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\frac{h}{a} \cdot \frac{h}{g}\right) \cdot -0.25} + \sqrt[3]{{a}^{-1} \cdot \left(-g\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.2%

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

3. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 30.7

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

5. Applied rewrites 30.7%

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

6. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-cbrt.f64 N/A

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

   3. unpow2 N/A

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

   4. times-frac N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cbrt.f64 N/A

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

   10. lower-cbrt.f64 78.0

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

8. Applied rewrites 78.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 78.0

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

10. Applied rewrites 78.0%

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

12. Applied rewrites 78.0%

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

13. Final simplification 78.0%

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

Alternative 4: 31.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\left(-2 \cdot g\right) \cdot \frac{1}{a \cdot 2}} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{if}\;g \leq -3.5 \cdot 10^{-160}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;g \leq 5.6 \cdot 10^{+151}:\\ \;\;\;\;\sqrt[3]{-0.5} \cdot \sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} + g}{a}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (+ (cbrt (* (* -2.0 g) (/ 1.0 (* a 2.0)))) (cbrt (/ (- g) a)))))
   (if (<= g -3.5e-160)
     t_0
     (if (<= g 5.6e+151)
       (* (cbrt -0.5) (cbrt (/ (+ (sqrt (- (* g g) (* h h))) g) a)))
       t_0))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt(((-2.0 * g) * (1.0 / (a * 2.0)))) + cbrt((-g / a));
	double tmp;
	if (g <= -3.5e-160) {
		tmp = t_0;
	} else if (g <= 5.6e+151) {
		tmp = cbrt(-0.5) * cbrt(((sqrt(((g * g) - (h * h))) + g) / a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.cbrt(((-2.0 * g) * (1.0 / (a * 2.0)))) + Math.cbrt((-g / a));
	double tmp;
	if (g <= -3.5e-160) {
		tmp = t_0;
	} else if (g <= 5.6e+151) {
		tmp = Math.cbrt(-0.5) * Math.cbrt(((Math.sqrt(((g * g) - (h * h))) + g) / a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64(cbrt(Float64(Float64(-2.0 * g) * Float64(1.0 / Float64(a * 2.0)))) + cbrt(Float64(Float64(-g) / a)))
	tmp = 0.0
	if (g <= -3.5e-160)
		tmp = t_0;
	elseif (g <= 5.6e+151)
		tmp = Float64(cbrt(-0.5) * cbrt(Float64(Float64(sqrt(Float64(Float64(g * g) - Float64(h * h))) + g) / a)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(N[Power[N[(N[(-2.0 * g), $MachinePrecision] * N[(1.0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[g, -3.5e-160], t$95$0, If[LessEqual[g, 5.6e+151], N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[N[(N[(N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + g), $MachinePrecision] / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if g < -3.5000000000000003e-160 or 5.59999999999999975e151 < g

   1. Initial program 36.0%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   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 8.2

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

   5. Applied rewrites 8.2%

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

   6. Taylor expanded in g around -inf

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

      1. lower-*.f64 15.0

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

   8. Applied rewrites 15.0%

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

   if -3.5000000000000003e-160 < g < 5.59999999999999975e151

   1. Initial program 75.3%

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

   3. Taylor expanded in g around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 75.3

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

   5. Applied rewrites 75.3%

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

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. difference-of-squares N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. lift--.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. sub-neg N/A

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

      17. lift--.f64 N/A

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

      18. lift-/.f64 N/A

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

   7. Applied rewrites 35.2%

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

   8. Taylor expanded in a around 0

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

      1. distribute-lft-out-- N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-log.f64 35.2

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

   10. Applied rewrites 35.2%

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

   11. Taylor expanded in a around 0

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

       1. lower-*.f64 N/A

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

       2. lower-cbrt.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-+.f64 N/A

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

       6. lower-sqrt.f64 N/A

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

       7. lower--.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. lower-cbrt.f64 75.5

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

   13. Applied rewrites 75.5%

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

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -3.5 \cdot 10^{-160}:\\ \;\;\;\;\sqrt[3]{\left(-2 \cdot g\right) \cdot \frac{1}{a \cdot 2}} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{elif}\;g \leq 5.6 \cdot 10^{+151}:\\ \;\;\;\;\sqrt[3]{-0.5} \cdot \sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} + g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(-2 \cdot g\right) \cdot \frac{1}{a \cdot 2}} + \sqrt[3]{\frac{-g}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.2%

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

3. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 30.7

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

5. Applied rewrites 30.7%

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

6. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-cbrt.f64 N/A

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

   3. unpow2 N/A

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

   4. times-frac N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cbrt.f64 N/A

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

   10. lower-cbrt.f64 78.0

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

8. Applied rewrites 78.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 78.0

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

10. Applied rewrites 78.0%

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

11. Final simplification 78.0%

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

Alternative 6: 23.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{-0.5} \cdot \sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} + g}{a}} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.2%

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

3. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 30.7

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

5. Applied rewrites 30.7%

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

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. difference-of-squares N/A

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

   10. +-commutative N/A

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

   11. lift-+.f64 N/A

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

   12. lift--.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lift-neg.f64 N/A

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

   16. sub-neg N/A

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

   17. lift--.f64 N/A

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

   18. lift-/.f64 N/A

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

7. Applied rewrites 14.5%

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

8. Taylor expanded in a around 0

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

   1. distribute-lft-out-- N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-log.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-log.f64 N/A

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

   12. lower-log.f64 14.5

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

10. Applied rewrites 14.5%

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

11. Taylor expanded in a around 0

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

    1. lower-*.f64 N/A

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

    2. lower-cbrt.f64 N/A

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

    3. lower-/.f64 N/A

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

    4. +-commutative N/A

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

    5. lower-+.f64 N/A

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

    6. lower-sqrt.f64 N/A

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

    7. lower--.f64 N/A

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

    8. unpow2 N/A

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

    9. lower-*.f64 N/A

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

    10. unpow2 N/A

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

    11. lower-*.f64 N/A

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

    12. lower-cbrt.f64 26.9

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

13. Applied rewrites 26.9%

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

14. Final simplification 26.9%

    \[\leadsto \sqrt[3]{-0.5} \cdot \sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} + g}{a}} \]
15. Add Preprocessing

Alternative 7: 3.0% accurate, 302.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[g_, h_, a_] := 0.0

Derivation

1. Initial program 49.2%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

   8. cbrt-div N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 52.7%

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

5. Taylor expanded in g around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-rgt-neg-in N/A

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

   3. lower-*.f64 N/A

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

   4. distribute-rgt-in N/A

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

   5. *-lft-identity N/A

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

   6. unpow2 N/A

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

   7. rem-square-sqrt N/A

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

   8. distribute-rgt1-in N/A

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

   9. metadata-eval N/A

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

   10. mul0-lft N/A

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

   11. lower-cbrt.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-cbrt.f64 3.0

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

7. Applied rewrites 3.0%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 3.0%

    \[\leadsto 0 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024242 
(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))))))))