2-ancestry mixing, positive discriminant

Percentage Accurate: 44.7% → 95.8%

Time: 12.7s

Alternatives: 5

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: 44.7% 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

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 41.5%

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

3. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 24.8

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

5. Applied rewrites 24.8%

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

6. Taylor expanded in g around -inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cbrt.f64 N/A

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

   10. lower-cbrt.f64 73.4

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

8. Applied rewrites 73.4%

   \[\leadsto \color{blue}{\left(-\sqrt[3]{\frac{g}{a}}\right) \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{0.5}\right)} \]
9. Step-by-step derivation

10. Applied rewrites 96.7%

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

12. Applied rewrites 96.7%

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

13. Final simplification 96.7%

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

Alternative 2: 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(Float64(-cbrt(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 41.5%

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

3. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 24.8

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

5. Applied rewrites 24.8%

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

6. Taylor expanded in g around -inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cbrt.f64 N/A

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

   10. lower-cbrt.f64 73.4

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

8. Applied rewrites 73.4%

   \[\leadsto \color{blue}{\left(-\sqrt[3]{\frac{g}{a}}\right) \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{0.5}\right)} \]
9. Step-by-step derivation

10. Applied rewrites 96.7%

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

Alternative 3: 74.3% accurate, 2.5× 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(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 41.5%

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

3. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 24.8

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

5. Applied rewrites 24.8%

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

6. Taylor expanded in g around -inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cbrt.f64 N/A

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

   10. lower-cbrt.f64 73.4

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

8. Applied rewrites 73.4%

   \[\leadsto \color{blue}{\left(-\sqrt[3]{\frac{g}{a}}\right) \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{0.5}\right)} \]
9. Step-by-step derivation

10. Applied rewrites 96.7%

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

12. Applied rewrites 75.0%

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

13. Final simplification 75.0%

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

Alternative 4: 73.5% 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 41.5%

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

3. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 24.8

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

5. Applied rewrites 24.8%

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

6. Taylor expanded in g around -inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cbrt.f64 N/A

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

   10. lower-cbrt.f64 73.4

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

8. Applied rewrites 73.4%

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

10. Applied rewrites 74.0%

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

Alternative 5: 2.9% accurate, 302.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[g_, h_, a_] := 0.0

Derivation

1. Initial program 41.5%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

   8. cbrt-div N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 44.1%

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

5. Taylor expanded in g around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. *-commutative N/A

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

   4. +-commutative N/A

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

   5. unpow2 N/A

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

   6. rem-square-sqrt N/A

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

   7. metadata-eval N/A

      \[\leadsto -\sqrt[3]{\frac{\color{blue}{0} \cdot g}{a}} \cdot \sqrt[3]{\frac{1}{2}} \]

   8. mul0-lft N/A

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

   9. mul0-lft N/A

      \[\leadsto -\sqrt[3]{\frac{\color{blue}{0 \cdot h}}{a}} \cdot \sqrt[3]{\frac{1}{2}} \]

   10. metadata-eval N/A

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

   11. distribute-rgt1-in N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cbrt.f64 N/A

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

   14. distribute-rgt1-in N/A

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

   15. metadata-eval N/A

       \[\leadsto -\sqrt[3]{\frac{\color{blue}{0} \cdot h}{a}} \cdot \sqrt[3]{\frac{1}{2}} \]

   16. mul0-lft N/A

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

   17. lower-/.f64 N/A

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

   18. lower-cbrt.f64 2.9

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

7. Applied rewrites 2.9%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 2.9%

    \[\leadsto 0 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024277 
(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))))))))