2-ancestry mixing, positive discriminant

Percentage Accurate: 43.4% → 95.8%

Time: 15.1s

Alternatives: 6

Speedup: 2.6×

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: 43.4% 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: 95.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 41.2%

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

3. Taylor expanded in g around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 25.3

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

5. Applied rewrites 25.3%

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

6. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-cbrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-cbrt.f64 71.4

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

8. Applied rewrites 71.4%

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

   1. lift-/.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-cbrt.f64 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. cbrt-prod N/A

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

   8. neg-mul-1 N/A

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

   9. lift-/.f64 N/A

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

   10. distribute-neg-frac N/A

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

   11. lift-neg.f64 N/A

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

   12. cbrt-div N/A

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

   13. lower-/.f64 N/A

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

   14. lower-cbrt.f64 N/A

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

   15. lower-cbrt.f64 94.8

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

10. Applied rewrites 94.8%

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

Alternative 2: 80.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{a \cdot 2} \leq 5 \cdot 10^{+50}:\\ \;\;\;\;-\sqrt[3]{\frac{g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-g} \cdot {a}^{-0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= (/ 1.0 (* a 2.0)) 5e+50)
   (- (cbrt (/ g a)))
   (* (cbrt (- g)) (pow a -0.3333333333333333))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((1.0 / (a * 2.0)) <= 5e+50) {
		tmp = -cbrt((g / a));
	} else {
		tmp = cbrt(-g) * pow(a, -0.3333333333333333);
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if ((1.0 / (a * 2.0)) <= 5e+50) {
		tmp = -Math.cbrt((g / a));
	} else {
		tmp = Math.cbrt(-g) * Math.pow(a, -0.3333333333333333);
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (Float64(1.0 / Float64(a * 2.0)) <= 5e+50)
		tmp = Float64(-cbrt(Float64(g / a)));
	else
		tmp = Float64(cbrt(Float64(-g)) * (a ^ -0.3333333333333333));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[N[(1.0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], 5e+50], (-N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision]), N[(N[Power[(-g), 1/3], $MachinePrecision] * N[Power[a, -0.3333333333333333], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) < 5e50

   1. Initial program 42.2%

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

   3. Taylor expanded in g around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 25.7

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

   5. Applied rewrites 25.7%

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

   6. Taylor expanded in g around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 73.3

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

   8. Applied rewrites 73.3%

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

   9. Taylor expanded in g around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. lower-cbrt.f64 N/A

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

       4. lower-/.f64 76.3

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

   11. Applied rewrites 76.3%

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

   if 5e50 < (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 36.4%

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

   3. Taylor expanded in g around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 23.2

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

   5. Applied rewrites 23.2%

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

   6. Taylor expanded in g around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cbrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-cbrt.f64 47.2

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

   8. Applied rewrites 47.2%

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

      1. lift-/.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-cbrt.f64 N/A

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

      6. lift-cbrt.f64 N/A

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

      7. cbrt-prod N/A

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

      8. neg-mul-1 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-neg-frac N/A

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

      11. lift-neg.f64 N/A

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

      12. div-inv N/A

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

      13. cbrt-prod N/A

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

      14. pow1/3 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cbrt.f64 N/A

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

      17. inv-pow N/A

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

      18. pow-pow N/A

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

      19. lower-pow.f64 N/A

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

      20. metadata-eval 87.2

          \[\leadsto \sqrt[3]{-g} \cdot {a}^{\color{blue}{-0.3333333333333333}} \]

   10. Applied rewrites 87.2%

       \[\leadsto \color{blue}{\sqrt[3]{-g} \cdot {a}^{-0.3333333333333333}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{a \cdot 2} \leq 5 \cdot 10^{+50}:\\ \;\;\;\;-\sqrt[3]{\frac{g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-g} \cdot {a}^{-0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.7% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 41.2%

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

3. Taylor expanded in g around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 25.3

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

5. Applied rewrites 25.3%

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

6. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 69.2

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

8. Applied rewrites 69.2%

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

9. Taylor expanded in g around -inf

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

    1. mul-1-neg N/A

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

    2. lower-neg.f64 N/A

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

    3. lower-cbrt.f64 N/A

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

    4. lower-/.f64 71.4

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

11. Applied rewrites 71.4%

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

Alternative 4: 1.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{g}{a}\right)}^{0.3333333333333333} \end{array} \]

FPCore

(FPCore (g h a) :precision binary64 (pow (/ g a) 0.3333333333333333))

C

double code(double g, double h, double a) {
	return pow((g / a), 0.3333333333333333);
}

Fortran

real(8) function code(g, h, a)
    real(8), intent (in) :: g
    real(8), intent (in) :: h
    real(8), intent (in) :: a
    code = (g / a) ** 0.3333333333333333d0
end function

Java

public static double code(double g, double h, double a) {
	return Math.pow((g / a), 0.3333333333333333);
}

Python

def code(g, h, a):
	return math.pow((g / a), 0.3333333333333333)

Julia

function code(g, h, a)
	return Float64(g / a) ^ 0.3333333333333333
end

MATLAB

function tmp = code(g, h, a)
	tmp = (g / a) ^ 0.3333333333333333;
end

Wolfram

code[g_, h_, a_] := N[Power[N[(g / a), $MachinePrecision], 0.3333333333333333], $MachinePrecision]

Derivation

1. Initial program 41.2%

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

3. Taylor expanded in g around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 25.3

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

5. Applied rewrites 25.3%

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

6. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-cbrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-cbrt.f64 71.4

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

8. Applied rewrites 71.4%

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

   1. lift-/.f64 N/A

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

   2. cbrt-unprod N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. associate-*l/ N/A

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

   6. metadata-eval N/A

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

   7. cbrt-div N/A

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

   8. pow1/3 N/A

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

   9. sqr-pow N/A

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

   10. pow-prod-down N/A

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

   11. metadata-eval N/A

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

   12. pow-base-1 N/A

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

   13. metadata-eval N/A

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

   14. cbrt-div N/A

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

   15. clear-num N/A

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

   16. lift-/.f64 N/A

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

   17. pow1/3 N/A

       \[\leadsto \color{blue}{{\left(\frac{g}{a}\right)}^{\frac{1}{3}}} \]

   18. lower-pow.f64 1.4

       \[\leadsto \color{blue}{{\left(\frac{g}{a}\right)}^{0.3333333333333333}} \]

10. Applied rewrites 1.4%

    \[\leadsto \color{blue}{{\left(\frac{g}{a}\right)}^{0.3333333333333333}} \]
11. Add Preprocessing

Alternative 5: 1.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{a}{g}\right)}^{-0.3333333333333333} \end{array} \]

FPCore

(FPCore (g h a) :precision binary64 (pow (/ a g) -0.3333333333333333))

C

double code(double g, double h, double a) {
	return pow((a / g), -0.3333333333333333);
}

Fortran

real(8) function code(g, h, a)
    real(8), intent (in) :: g
    real(8), intent (in) :: h
    real(8), intent (in) :: a
    code = (a / g) ** (-0.3333333333333333d0)
end function

Java

public static double code(double g, double h, double a) {
	return Math.pow((a / g), -0.3333333333333333);
}

Python

def code(g, h, a):
	return math.pow((a / g), -0.3333333333333333)

Julia

function code(g, h, a)
	return Float64(a / g) ^ -0.3333333333333333
end

MATLAB

function tmp = code(g, h, a)
	tmp = (a / g) ^ -0.3333333333333333;
end

Wolfram

code[g_, h_, a_] := N[Power[N[(a / g), $MachinePrecision], -0.3333333333333333], $MachinePrecision]

Derivation

1. Initial program 41.2%

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

3. Taylor expanded in g around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 25.3

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

5. Applied rewrites 25.3%

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

6. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-cbrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-cbrt.f64 71.4

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

8. Applied rewrites 71.4%

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

   1. lift-/.f64 N/A

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

   2. cbrt-unprod N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. associate-*l/ N/A

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

   6. metadata-eval N/A

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

   7. cbrt-div N/A

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

   8. pow1/3 N/A

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

   9. sqr-pow N/A

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

   10. pow-prod-down N/A

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

   11. metadata-eval N/A

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

   12. pow-base-1 N/A

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

   13. metadata-eval N/A

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

   14. cbrt-div N/A

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

   15. clear-num N/A

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

   16. lift-/.f64 N/A

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

   17. pow1/3 N/A

       \[\leadsto \color{blue}{{\left(\frac{g}{a}\right)}^{\frac{1}{3}}} \]

   18. lift-/.f64 N/A

       \[\leadsto {\color{blue}{\left(\frac{g}{a}\right)}}^{\frac{1}{3}} \]

   19. clear-num N/A

       \[\leadsto {\color{blue}{\left(\frac{1}{\frac{a}{g}}\right)}}^{\frac{1}{3}} \]

   20. inv-pow N/A

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

   21. pow-pow N/A

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

   22. lower-pow.f64 N/A

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

   23. lower-/.f64 N/A

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

   24. metadata-eval 1.3

       \[\leadsto {\left(\frac{a}{g}\right)}^{\color{blue}{-0.3333333333333333}} \]

10. Applied rewrites 1.3%

    \[\leadsto \color{blue}{{\left(\frac{a}{g}\right)}^{-0.3333333333333333}} \]
11. Add Preprocessing

Alternative 6: 1.4% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (g h a) :precision binary64 (cbrt (/ g a)))

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 41.2%

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

3. Taylor expanded in g around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 25.3

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

5. Applied rewrites 25.3%

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

6. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-cbrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-cbrt.f64 71.4

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

8. Applied rewrites 71.4%

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

   1. lift-/.f64 N/A

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

   2. cbrt-unprod N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. associate-*l/ N/A

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

   6. metadata-eval N/A

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

   7. cbrt-div N/A

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

   8. pow1/3 N/A

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

   9. sqr-pow N/A

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

   10. pow-prod-down N/A

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

   11. metadata-eval N/A

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

   12. pow-base-1 N/A

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

   13. metadata-eval N/A

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

   14. cbrt-div N/A

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

   15. clear-num N/A

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

   16. lift-/.f64 N/A

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

   17. lift-cbrt.f64 1.3

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

10. Applied rewrites 1.3%

    \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a}}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(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))))))))