2-ancestry mixing, positive discriminant

Percentage Accurate: 44.6% → 93.6%

Time: 8.0s

Alternatives: 3

Speedup: 2.6×

Specification

\[\mathsf{TRUE}\left(\right)\]

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: 44.6% 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: 93.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\left(\frac{h}{a} \cdot \frac{h}{g}\right) \cdot -0.25} + \sqrt[3]{{\left(2 \cdot a\right)}^{-1}} \cdot \sqrt[3]{-\left(g + g\right)} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (+
  (cbrt (* (* (/ h a) (/ h g)) -0.25))
  (* (cbrt (pow (* 2.0 a) -1.0)) (cbrt (- (+ g g))))))

C

double code(double g, double h, double a) {
	return cbrt((((h / a) * (h / g)) * -0.25)) + (cbrt(pow((2.0 * a), -1.0)) * cbrt(-(g + g)));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt((((h / a) * (h / g)) * -0.25)) + (Math.cbrt(Math.pow((2.0 * a), -1.0)) * Math.cbrt(-(g + g)));
}

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(Float64(h / a) * Float64(h / g)) * -0.25)) + Float64(cbrt((Float64(2.0 * a) ^ -1.0)) * cbrt(Float64(-Float64(g + g)))))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(N[(N[(h / a), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[Power[N[(2.0 * a), $MachinePrecision], -1.0], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[(-N[(g + g), $MachinePrecision]), 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 47.6%

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

3. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-cbrt.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. cbrt-unprod N/A

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

   9. metadata-eval N/A

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

   10. lower-cbrt.f64 30.3

       \[\leadsto \sqrt[3]{\frac{\frac{h \cdot h}{a}}{g}} \cdot \sqrt[3]{-0.25} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

5. Applied rewrites 30.3%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{\frac{h \cdot h}{a}}{g}} \cdot \sqrt[3]{-0.25}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
6. Step-by-step derivation

   1. lift-cbrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. cbrt-prod N/A

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

   10. lower-*.f64 N/A

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

7. Applied rewrites 32.2%

   \[\leadsto \sqrt[3]{\frac{\frac{h \cdot h}{a}}{g}} \cdot \sqrt[3]{-0.25} + \color{blue}{\sqrt[3]{{\left(2 \cdot a\right)}^{-1}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}} \]

8. Taylor expanded in g around inf

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

10. Applied rewrites 90.1%

    \[\leadsto \sqrt[3]{\frac{\frac{h \cdot h}{a}}{g}} \cdot \sqrt[3]{-0.25} + \sqrt[3]{{\left(2 \cdot a\right)}^{-1}} \cdot \sqrt[3]{\left(-g\right) - \color{blue}{g}} \]

11. Taylor expanded in g around inf

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

    1. cbrt-unprod N/A

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

    2. metadata-eval N/A

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

    3. cbrt-unprod N/A

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

    4. lower-cbrt.f64 N/A

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

    5. lower-*.f64 N/A

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

    6. pow2 N/A

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

    7. times-frac N/A

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

    8. lower-*.f64 N/A

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

    9. lower-/.f64 N/A

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

    10. lower-/.f64 95.8

        \[\leadsto \sqrt[3]{\left(\frac{h}{a} \cdot \frac{h}{g}\right) \cdot -0.25} + \sqrt[3]{{\left(2 \cdot a\right)}^{-1}} \cdot \sqrt[3]{\left(-g\right) - g} \]

13. Applied rewrites 95.8%

    \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{h}{a} \cdot \frac{h}{g}\right) \cdot -0.25}} + \sqrt[3]{{\left(2 \cdot a\right)}^{-1}} \cdot \sqrt[3]{\left(-g\right) - g} \]

14. Final simplification 95.8%

    \[\leadsto \sqrt[3]{\left(\frac{h}{a} \cdot \frac{h}{g}\right) \cdot -0.25} + \sqrt[3]{{\left(2 \cdot a\right)}^{-1}} \cdot \sqrt[3]{-\left(g + g\right)} \]
15. 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 47.6%

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

3. Taylor expanded in g around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. cbrt-unprod N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. lower-/.f64 77.0

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

5. Applied rewrites 77.0%

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

   1. lift-/.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. cbrt-div N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 94.2

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

7. Applied rewrites 94.2%

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

8. Final simplification 94.2%

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

Alternative 3: 73.4% 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 47.6%

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

3. Taylor expanded in g around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. cbrt-unprod N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. lower-/.f64 77.0

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

5. Applied rewrites 77.0%

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

6. Taylor expanded in g around 0

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 77.0

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

8. Applied rewrites 77.0%

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

Reproduce

herbie shell --seed 2025050 
(FPCore (g h a)
  :name "2-ancestry mixing, positive discriminant"
  :precision binary64
  :pre (TRUE)
  (+ (cbrt (* (/ 1.0 (* 2.0 a)) (+ (- g) (sqrt (- (* g g) (* h h)))))) (cbrt (* (/ 1.0 (* 2.0 a)) (- (- g) (sqrt (- (* g g) (* h h))))))))