2-ancestry mixing, positive discriminant

Percentage Accurate: 44.3% → 97.7%

Time: 16.2s

Alternatives: 7

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: 44.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{a \cdot 2}\\ \frac{\mathsf{fma}\left(\sqrt[3]{h \cdot \frac{h \cdot -0.5}{g}}, \sqrt[3]{a}, t\_0 \cdot \sqrt[3]{-g}\right)}{\sqrt[3]{a} \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (* a 2.0))))
   (/
    (fma (cbrt (* h (/ (* h -0.5) g))) (cbrt a) (* t_0 (cbrt (- g))))
    (* (cbrt a) t_0))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt((a * 2.0));
	return fma(cbrt((h * ((h * -0.5) / g))), cbrt(a), (t_0 * cbrt(-g))) / (cbrt(a) * t_0);
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(a * 2.0))
	return Float64(fma(cbrt(Float64(h * Float64(Float64(h * -0.5) / g))), cbrt(a), Float64(t_0 * cbrt(Float64(-g)))) / Float64(cbrt(a) * t_0))
end

Wolfram

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

Derivation

1. Initial program 39.7%

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

3. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 22.0

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

5. Simplified 22.0%

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

6. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 69.2

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

8. Simplified 69.2%

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

10. Applied egg-rr 98.2%

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

11. Final simplification 98.2%

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

Alternative 2: 81.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{a \cdot 2} \leq 10^{+57}:\\ \;\;\;\;-\sqrt[3]{\frac{g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-g} \cdot {a}^{-0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= (/ 1.0 (* a 2.0)) 1e+57)
   (- (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)) <= 1e+57) {
		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)) <= 1e+57) {
		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)) <= 1e+57)
		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], 1e+57], (-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)) < 1.00000000000000005e57

   1. Initial program 43.9%

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

   3. Taylor expanded in g around inf

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(\mathsf{neg}\left(g\right)\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
   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 28.0

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

   5. Simplified 28.0%

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

   6. 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 80.5

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

   8. Simplified 80.5%

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

   9. Taylor expanded in a around -inf

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

       1. rem-cube-cbrt N/A

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

       2. *-commutative N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 N/A

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

       5. lower-cbrt.f64 N/A

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

       6. lower-/.f64 80.5

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

   11. Simplified 80.5%

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

   if 1.00000000000000005e57 < (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 24.4%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   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 10.7

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

   5. Simplified 10.7%

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

   6. Taylor expanded in g around inf

      \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}} \]
   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 37.6

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

   8. Simplified 37.6%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}} \]
   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-unprod 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-frac-neg 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. clear-num N/A

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

      13. associate-/r/ N/A

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

      14. cbrt-prod N/A

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

      15. pow1/3 N/A

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

      16. inv-pow N/A

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

      17. pow-pow N/A

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

      18. metadata-eval N/A

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

      19. lower-*.f64 N/A

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

      20. lower-pow.f64 N/A

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

      21. lower-cbrt.f64 88.3

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

   10. Applied egg-rr 88.3%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{a \cdot 2} \leq 10^{+57}:\\ \;\;\;\;-\sqrt[3]{\frac{g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-g} \cdot {a}^{-0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 39.7%

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

3. Taylor expanded in g around inf

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(\mathsf{neg}\left(g\right)\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
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 24.2

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

5. Simplified 24.2%

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

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 71.1

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

8. Simplified 71.1%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}} \]
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-unprod 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-frac-neg 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. lift-/.f64 N/A

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

   13. pow1/3 N/A

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

   14. pow-to-exp N/A

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

   15. lower-exp.f64 N/A

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

   16. rem-log-exp N/A

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

   17. pow-to-exp N/A

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

   18. log-pow N/A

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

   19. lower-*.f64 N/A

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

   20. lower-log.f64 33.7

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

   21. lift-/.f64 N/A

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

   22. lift-neg.f64 N/A

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

   23. distribute-frac-neg N/A

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

   24. distribute-frac-neg2 N/A

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

   25. lower-/.f64 N/A

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

   26. lower-neg.f64 33.7

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

10. Applied egg-rr 33.7%

    \[\leadsto \color{blue}{e^{0.3333333333333333 \cdot \log \left(\frac{g}{-a}\right)}} \]
11. Step-by-step derivation

    1. rem-log-exp N/A

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

    2. lift-neg.f64 N/A

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

    3. lift-/.f64 N/A

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

    4. lift-log.f64 N/A

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

    5. rem-log-exp N/A

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

    6. *-commutative N/A

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

    7. lift-log.f64 N/A

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

    8. exp-to-pow N/A

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

    9. pow1/3 N/A

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

    10. lift-/.f64 N/A

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

    11. clear-num N/A

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

    12. associate-/r/ N/A

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

    13. cbrt-prod N/A

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

    14. pow1/3 N/A

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

    15. lower-*.f64 N/A

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

12. Applied egg-rr 96.4%

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

Alternative 4: 95.7% 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 39.7%

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

3. Taylor expanded in g around inf

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(\mathsf{neg}\left(g\right)\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
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 24.2

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

5. Simplified 24.2%

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

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 71.1

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

8. Simplified 71.1%

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

   1. cbrt-div N/A

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

   2. lift-cbrt.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lift-cbrt.f64 N/A

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

   5. cbrt-prod N/A

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

   6. *-commutative N/A

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

   7. neg-mul-1 N/A

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

   8. lift-neg.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-cbrt.f64 N/A

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

   11. lower-cbrt.f64 96.3

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

10. Applied egg-rr 96.3%

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

Alternative 5: 74.3% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 39.7%

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

3. Taylor expanded in g around inf

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(\mathsf{neg}\left(g\right)\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
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 24.2

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

5. Simplified 24.2%

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

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 71.1

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

8. Simplified 71.1%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}} \]
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-unprod 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-frac-neg 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. lift-/.f64 N/A

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

   13. pow1/3 N/A

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

   14. pow-to-exp N/A

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

   15. lower-exp.f64 N/A

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

   16. rem-log-exp N/A

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

   17. pow-to-exp N/A

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

   18. log-pow N/A

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

   19. lower-*.f64 N/A

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

   20. lower-log.f64 33.7

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

   21. lift-/.f64 N/A

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

   22. lift-neg.f64 N/A

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

   23. distribute-frac-neg N/A

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

   24. distribute-frac-neg2 N/A

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

   25. lower-/.f64 N/A

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

   26. lower-neg.f64 33.7

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

10. Applied egg-rr 33.7%

    \[\leadsto \color{blue}{e^{0.3333333333333333 \cdot \log \left(\frac{g}{-a}\right)}} \]
11. Step-by-step derivation

    1. rem-log-exp N/A

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

    2. lift-neg.f64 N/A

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

    3. lift-/.f64 N/A

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

    4. lift-log.f64 N/A

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

    5. rem-log-exp N/A

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

    6. *-commutative N/A

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

    7. lift-log.f64 N/A

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

    8. exp-to-pow N/A

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

    9. pow1/3 N/A

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

    10. lift-/.f64 N/A

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

    11. clear-num N/A

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

    12. cbrt-div N/A

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

    13. metadata-eval N/A

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

    14. lower-/.f64 N/A

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

    15. lower-cbrt.f64 N/A

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

    16. frac-2neg N/A

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

    17. lift-neg.f64 N/A

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

    18. remove-double-neg N/A

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

    19. lift-neg.f64 N/A

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

    20. lower-/.f64 71.4

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

12. Applied egg-rr 71.4%

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

13. Final simplification 71.4%

    \[\leadsto \frac{1}{\sqrt[3]{\frac{-a}{g}}} \]
14. Add Preprocessing

Alternative 6: 73.6% 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 39.7%

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

3. Taylor expanded in g around inf

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(\mathsf{neg}\left(g\right)\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
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 24.2

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

5. Simplified 24.2%

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

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 71.1

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

8. Simplified 71.1%

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

9. Taylor expanded in a around -inf

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

    1. rem-cube-cbrt N/A

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

    2. *-commutative N/A

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

    3. mul-1-neg N/A

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

    4. lower-neg.f64 N/A

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

    5. lower-cbrt.f64 N/A

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

    6. lower-/.f64 71.1

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

11. Simplified 71.1%

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

Alternative 7: 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 39.7%

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

3. Taylor expanded in g around inf

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(\mathsf{neg}\left(g\right)\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
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 24.2

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

5. Simplified 24.2%

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

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 71.1

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

8. Simplified 71.1%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}} \]
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. pow1/3 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-*l/ N/A

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

   6. *-commutative N/A

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

   7. neg-mul-1 N/A

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

   8. lift-neg.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. sqr-pow N/A

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

   11. pow-prod-down N/A

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

   12. lift-/.f64 N/A

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

   13. lift-neg.f64 N/A

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

   14. distribute-frac-neg N/A

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

   15. lift-/.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. lift-neg.f64 N/A

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

   18. distribute-frac-neg N/A

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

   19. lift-/.f64 N/A

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

   20. sqr-neg N/A

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

   21. pow-prod-down N/A

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

   22. sqr-pow N/A

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

   23. pow1/3 N/A

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

   24. lift-cbrt.f64 1.4

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

10. Applied egg-rr 1.4%

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

Reproduce

herbie shell --seed 2024212 
(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))))))))