2-ancestry mixing, positive discriminant

Percentage Accurate: 44.3% → 75.4%

Time: 13.7s

Alternatives: 4

Speedup: 1.5×

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: 75.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ t_1 := \frac{1}{2 \cdot a}\\ \mathbf{if}\;\sqrt[3]{t\_1 \cdot \left(\left(-g\right) + t\_0\right)} + \sqrt[3]{t\_1 \cdot \left(\left(-g\right) - t\_0\right)} \leq -\infty:\\ \;\;\;\;g \cdot \mathsf{fma}\left(\sqrt[3]{\frac{1}{a \cdot {g}^{2}}}, \sqrt[3]{-0.5} \cdot \sqrt[3]{2}, \sqrt[3]{\frac{{h}^{2}}{a \cdot {g}^{4}}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{t\_1 \cdot \left(\left(-g\right) + g\right)} + \sqrt[3]{-1 \cdot \frac{g}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (sqrt (- (* g g) (* h h)))) (t_1 (/ 1.0 (* 2.0 a))))
   (if (<=
        (+ (cbrt (* t_1 (+ (- g) t_0))) (cbrt (* t_1 (- (- g) t_0))))
        (- INFINITY))
     (*
      g
      (fma
       (cbrt (/ 1.0 (* a (pow g 2.0))))
       (* (cbrt -0.5) (cbrt 2.0))
       (*
        (cbrt (/ (pow h 2.0) (* a (pow g 4.0))))
        (* (cbrt -0.5) (cbrt 0.5)))))
     (+ (cbrt (* t_1 (+ (- g) g))) (cbrt (* -1.0 (/ g a)))))))

C

double code(double g, double h, double a) {
	double t_0 = sqrt(((g * g) - (h * h)));
	double t_1 = 1.0 / (2.0 * a);
	double tmp;
	if ((cbrt((t_1 * (-g + t_0))) + cbrt((t_1 * (-g - t_0)))) <= -((double) INFINITY)) {
		tmp = g * fma(cbrt((1.0 / (a * pow(g, 2.0)))), (cbrt(-0.5) * cbrt(2.0)), (cbrt((pow(h, 2.0) / (a * pow(g, 4.0)))) * (cbrt(-0.5) * cbrt(0.5))));
	} else {
		tmp = cbrt((t_1 * (-g + g))) + cbrt((-1.0 * (g / a)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	t_1 = Float64(1.0 / Float64(2.0 * a))
	tmp = 0.0
	if (Float64(cbrt(Float64(t_1 * Float64(Float64(-g) + t_0))) + cbrt(Float64(t_1 * Float64(Float64(-g) - t_0)))) <= Float64(-Inf))
		tmp = Float64(g * fma(cbrt(Float64(1.0 / Float64(a * (g ^ 2.0)))), Float64(cbrt(-0.5) * cbrt(2.0)), Float64(cbrt(Float64((h ^ 2.0) / Float64(a * (g ^ 4.0)))) * Float64(cbrt(-0.5) * cbrt(0.5)))));
	else
		tmp = Float64(cbrt(Float64(t_1 * Float64(Float64(-g) + g))) + cbrt(Float64(-1.0 * Float64(g / a))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(t$95$1 * N[((-g) + t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(t$95$1 * N[((-g) - t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(g * N[(N[Power[N[(1.0 / N[(a * N[Power[g, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(N[Power[h, 2.0], $MachinePrecision] / N[(a * N[Power[g, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[0.5, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(t$95$1 * N[((-g) + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(-1.0 * N[(g / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < -inf.0

   1. Initial program 4.2%

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

   2. Taylor expanded in g around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 97.1%

      \[\leadsto \color{blue}{g \cdot \mathsf{fma}\left(\sqrt[3]{\frac{1}{a \cdot {g}^{2}}}, \sqrt[3]{-0.5} \cdot \sqrt[3]{2}, \sqrt[3]{\frac{{h}^{2}}{a \cdot {g}^{4}}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right)\right)} \]

   if -inf.0 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))))

   1. Initial program 45.2%

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

   2. Taylor expanded in g around inf

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

   4. Applied rewrites 24.4%

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

   5. Taylor expanded in g around inf

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

   7. Applied rewrites 74.8%

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

   8. Taylor expanded in g around inf

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

      1. lower-*.f64 N/A

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

      2. lift-/.f64 74.9

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

   10. Applied rewrites 74.9%

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

Alternative 2: 73.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + g\right)} + \sqrt[3]{-1 \cdot \frac{g}{a}} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.3%

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

2. Taylor expanded in g around inf

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

4. Applied rewrites 23.9%

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

5. Taylor expanded in g around inf

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

7. Applied rewrites 73.2%

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

8. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lift-/.f64 73.3

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

10. Applied rewrites 73.3%

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

Alternative 3: 15.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(-2 \cdot g\right)} + \sqrt[3]{-1 \cdot \frac{g}{a}} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.3%

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

2. Taylor expanded in g around inf

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

4. Applied rewrites 23.9%

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

5. Taylor expanded in g around inf

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

7. Applied rewrites 73.2%

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

8. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lift-/.f64 73.3

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

10. Applied rewrites 73.3%

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

11. Taylor expanded in g around -inf

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

    1. lower-*.f64 15.3

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

13. Applied rewrites 15.3%

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

Alternative 4: 0.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{0.5 \cdot \frac{h \cdot \sqrt{-1}}{a}} + \sqrt[3]{-1 \cdot \frac{g}{a}} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (+ (cbrt (* 0.5 (/ (* h (sqrt -1.0)) a))) (cbrt (* -1.0 (/ g a)))))

C

double code(double g, double h, double a) {
	return cbrt((0.5 * ((h * sqrt(-1.0)) / a))) + cbrt((-1.0 * (g / a)));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt((0.5 * ((h * Math.sqrt(-1.0)) / a))) + Math.cbrt((-1.0 * (g / a)));
}

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(0.5 * Float64(Float64(h * sqrt(-1.0)) / a))) + cbrt(Float64(-1.0 * Float64(g / a))))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(0.5 * N[(N[(h * N[Sqrt[-1.0], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(-1.0 * N[(g / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 44.3%

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

2. Taylor expanded in g around inf

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

4. Applied rewrites 23.9%

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

5. Taylor expanded in g around inf

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

7. Applied rewrites 73.2%

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

8. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lift-/.f64 73.3

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

10. Applied rewrites 73.3%

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

11. Taylor expanded in g around 0

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

    1. lower-*.f64 N/A

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

    2. lower-/.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower-sqrt.f64 0.0

       \[\leadsto \sqrt[3]{0.5 \cdot \frac{h \cdot \sqrt{-1}}{a}} + \sqrt[3]{-1 \cdot \frac{g}{a}} \]

13. Applied rewrites 0.0%

    \[\leadsto \sqrt[3]{\color{blue}{0.5 \cdot \frac{h \cdot \sqrt{-1}}{a}}} + \sqrt[3]{-1 \cdot \frac{g}{a}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025124 
(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))))))))