2-ancestry mixing, positive discriminant

Percentage Accurate: 43.2% → 95.9%

Time: 8.2s

Alternatives: 3

Speedup: 3.6×

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: 43.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 95.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 \cdot \left(g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right)\right)\\ \mathsf{fma}\left(\frac{1}{\sqrt[3]{2 \cdot a}}, \sqrt[3]{\left(-g\right) + t\_0}, \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - t\_0\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (* -1.0 (* g (+ 1.0 (* 0.5 (/ (+ h (* -1.0 h)) g)))))))
   (fma
    (/ 1.0 (cbrt (* 2.0 a)))
    (cbrt (+ (- g) t_0))
    (cbrt (* (/ 1.0 (* 2.0 a)) (- (- g) t_0))))))

C

double code(double g, double h, double a) {
	double t_0 = -1.0 * (g * (1.0 + (0.5 * ((h + (-1.0 * h)) / g))));
	return fma((1.0 / cbrt((2.0 * a))), cbrt((-g + t_0)), cbrt(((1.0 / (2.0 * a)) * (-g - t_0))));
}

Julia

function code(g, h, a)
	t_0 = Float64(-1.0 * Float64(g * Float64(1.0 + Float64(0.5 * Float64(Float64(h + Float64(-1.0 * h)) / g)))))
	return fma(Float64(1.0 / cbrt(Float64(2.0 * a))), cbrt(Float64(Float64(-g) + t_0)), cbrt(Float64(Float64(1.0 / Float64(2.0 * a)) * Float64(Float64(-g) - t_0))))
end

Wolfram

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

Derivation

1. Initial program 43.2%

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

3. Applied rewrites 46.5%

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

4. Taylor expanded in g around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower-*.f64 32.9

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

6. Applied rewrites 32.9%

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

7. Taylor expanded in g around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower-*.f64 95.7

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

9. Applied rewrites 95.7%

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

    1. lift-cbrt.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-/.f64 N/A

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

    4. cbrt-div N/A

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

    5. metadata-eval N/A

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

    6. lower-/.f64 N/A

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

    7. lower-cbrt.f64 N/A

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

    8. lift-*.f64 95.7

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

11. Applied rewrites 95.7%

    \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\sqrt[3]{2 \cdot a}}}, \sqrt[3]{\left(-g\right) + -1 \cdot \left(g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right)\right)}, \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - -1 \cdot \left(g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right)\right)\right)}\right) \]
12. Add Preprocessing

Alternative 2: 95.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 43.2%

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

2. Taylor expanded in g around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. cbrt-unprod N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. lower-cbrt.f64 95.9

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

4. Applied rewrites 95.9%

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

5. Taylor expanded in g around 0

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

   1. lift-cbrt.f64 95.9

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

7. Applied rewrites 95.9%

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

Alternative 3: 73.0% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 43.2%

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

2. Taylor expanded in g around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. cbrt-unprod N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. lower-cbrt.f64 95.9

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

4. Applied rewrites 95.9%

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

5. Taylor expanded in g around 0

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

   1. cbrt-undiv N/A

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

   2. lower-cbrt.f64 N/A

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

   3. lift-/.f64 73.0

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

7. Applied rewrites 73.0%

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

Reproduce

herbie shell --seed 2025139 
(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))))))))