2-ancestry mixing, positive discriminant

Percentage Accurate: 44.8% → 95.7%

Time: 15.1s

Alternatives: 7

Speedup: 2.0×

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: 44.8% 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: 95.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ 0 - \left(\sqrt[3]{-1} \cdot \frac{\sqrt[3]{g}}{\sqrt[3]{a}}\right) \cdot \left({2}^{0.3333333333333333} \cdot \sqrt[3]{-0.5}\right) \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (-
  0.0
  (*
   (* (cbrt -1.0) (/ (cbrt g) (cbrt a)))
   (* (pow 2.0 0.3333333333333333) (cbrt -0.5)))))

C

double code(double g, double h, double a) {
	return 0.0 - ((cbrt(-1.0) * (cbrt(g) / cbrt(a))) * (pow(2.0, 0.3333333333333333) * cbrt(-0.5)));
}

Java

public static double code(double g, double h, double a) {
	return 0.0 - ((Math.cbrt(-1.0) * (Math.cbrt(g) / Math.cbrt(a))) * (Math.pow(2.0, 0.3333333333333333) * Math.cbrt(-0.5)));
}

Julia

function code(g, h, a)
	return Float64(0.0 - Float64(Float64(cbrt(-1.0) * Float64(cbrt(g) / cbrt(a))) * Float64((2.0 ^ 0.3333333333333333) * cbrt(-0.5))))
end

Wolfram

code[g_, h_, a_] := N[(0.0 - N[(N[(N[Power[-1.0, 1/3], $MachinePrecision] * N[(N[Power[g, 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[2.0, 0.3333333333333333], $MachinePrecision] * N[Power[-0.5, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 45.1%

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

3. Simplified 45.1%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}} \]
4. Add Preprocessing

5. Taylor expanded in h around 0 67.9%

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

6. Taylor expanded in g around -inf 75.1%

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt[3]{\frac{g}{a}} \cdot \left(\sqrt[3]{-1} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right)\right)\right)} \]
7. Step-by-step derivation

   1. mul-1-neg 75.1%

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

   2. associate-*r* 75.1%

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

   3. distribute-rgt-neg-in 75.1%

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

   4. *-commutative 75.1%

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

   5. *-commutative 75.1%

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

8. Simplified 75.1%

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

10. Applied egg-rr 95.4%

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

12. Applied egg-rr 96.1%

    \[\leadsto \left(\sqrt[3]{-1} \cdot \frac{\sqrt[3]{g}}{\sqrt[3]{a}}\right) \cdot \left(-\color{blue}{{2}^{0.3333333333333333}} \cdot \sqrt[3]{-0.5}\right) \]

13. Final simplification 96.1%

    \[\leadsto 0 - \left(\sqrt[3]{-1} \cdot \frac{\sqrt[3]{g}}{\sqrt[3]{a}}\right) \cdot \left({2}^{0.3333333333333333} \cdot \sqrt[3]{-0.5}\right) \]
14. Add Preprocessing

Alternative 2: 95.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt[3]{g}}{\sqrt[3]{a}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \end{array} \]

FPCore

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

C

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

Java

public static double code(double g, double h, double a) {
	return (Math.cbrt(g) / Math.cbrt(a)) * (Math.cbrt(-0.5) * Math.cbrt(2.0));
}

Julia

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

Wolfram

code[g_, h_, a_] := N[(N[(N[Power[g, 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 45.1%

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

3. Simplified 45.1%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}} \]
4. Add Preprocessing

5. Taylor expanded in h around 0 67.9%

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

6. Taylor expanded in g around -inf 75.1%

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt[3]{\frac{g}{a}} \cdot \left(\sqrt[3]{-1} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right)\right)\right)} \]
7. Step-by-step derivation

   1. mul-1-neg 75.1%

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

   2. associate-*r* 75.1%

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

   3. distribute-rgt-neg-in 75.1%

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

   4. *-commutative 75.1%

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

   5. *-commutative 75.1%

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

8. Simplified 75.1%

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

10. Applied egg-rr 95.4%

    \[\leadsto \left(\sqrt[3]{-1} \cdot \color{blue}{\frac{\sqrt[3]{g}}{\sqrt[3]{a}}}\right) \cdot \left(-\sqrt[3]{2} \cdot \sqrt[3]{-0.5}\right) \]
11. Step-by-step derivation

    1. add-cube-cbrt 95.4%

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

    2. pow3 95.4%

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

    3. *-un-lft-identity 95.4%

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

    4. metadata-eval 95.4%

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

    5. metadata-eval 95.4%

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

    6. cbrt-unprod 95.4%

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

    7. pow1/3 0.0%

       \[\leadsto \left({\left(\sqrt[3]{\left(\color{blue}{{-1}^{0.3333333333333333}} \cdot \sqrt[3]{-1}\right) \cdot \sqrt[3]{-1}}\right)}^{3} \cdot \frac{\sqrt[3]{g}}{\sqrt[3]{a}}\right) \cdot \left(-\sqrt[3]{2} \cdot \sqrt[3]{-0.5}\right) \]

    8. pow1/3 0.0%

       \[\leadsto \left({\left(\sqrt[3]{\left({-1}^{0.3333333333333333} \cdot \color{blue}{{-1}^{0.3333333333333333}}\right) \cdot \sqrt[3]{-1}}\right)}^{3} \cdot \frac{\sqrt[3]{g}}{\sqrt[3]{a}}\right) \cdot \left(-\sqrt[3]{2} \cdot \sqrt[3]{-0.5}\right) \]

    9. pow1/3 0.0%

       \[\leadsto \left({\left(\sqrt[3]{\left({-1}^{0.3333333333333333} \cdot {-1}^{0.3333333333333333}\right) \cdot \color{blue}{{-1}^{0.3333333333333333}}}\right)}^{3} \cdot \frac{\sqrt[3]{g}}{\sqrt[3]{a}}\right) \cdot \left(-\sqrt[3]{2} \cdot \sqrt[3]{-0.5}\right) \]

    10. add-cbrt-cube 0.0%

        \[\leadsto \left({\color{blue}{\left({-1}^{0.3333333333333333}\right)}}^{3} \cdot \frac{\sqrt[3]{g}}{\sqrt[3]{a}}\right) \cdot \left(-\sqrt[3]{2} \cdot \sqrt[3]{-0.5}\right) \]

    11. pow1/3 95.4%

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

12. Applied egg-rr 95.4%

    \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{-1}\right)}^{3}} \cdot \frac{\sqrt[3]{g}}{\sqrt[3]{a}}\right) \cdot \left(-\sqrt[3]{2} \cdot \sqrt[3]{-0.5}\right) \]
13. Step-by-step derivation

    1. rem-cube-cbrt 95.4%

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

14. Simplified 95.4%

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

15. Final simplification 95.4%

    \[\leadsto \frac{\sqrt[3]{g}}{\sqrt[3]{a}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \]
16. Add Preprocessing

Alternative 3: 74.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{g}{-a}} + \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 45.1%

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

3. Simplified 45.1%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}} \]
4. Add Preprocessing

5. Taylor expanded in g around -inf 28.4%

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

   1. mul-1-neg 28.4%

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

   2. distribute-neg-frac2 28.4%

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

7. Simplified 28.4%

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

8. Taylor expanded in g around -inf 75.7%

   \[\leadsto \sqrt[3]{\frac{g}{-a}} + \sqrt[3]{\left(g + \color{blue}{-1 \cdot g}\right) \cdot \frac{-0.5}{a}} \]
9. Step-by-step derivation

   1. neg-mul-1 75.7%

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

10. Simplified 75.7%

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

11. Final simplification 75.7%

    \[\leadsto \sqrt[3]{\frac{g}{-a}} + \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} \]
12. Add Preprocessing

Alternative 4: 36.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 45.1%

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

3. Simplified 45.1%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}} \]
4. Add Preprocessing

5. Taylor expanded in g around -inf 28.4%

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

   1. mul-1-neg 28.4%

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

   2. distribute-neg-frac2 28.4%

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

7. Simplified 28.4%

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

8. Taylor expanded in g around inf 15.5%

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

   1. expm1-log1p-u 10.5%

      \[\leadsto \sqrt[3]{\frac{g}{-a}} + \sqrt[3]{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(g + g\right) \cdot \frac{-0.5}{a}\right)\right)}} \]

   2. expm1-undefine 31.1%

      \[\leadsto \sqrt[3]{\frac{g}{-a}} + \sqrt[3]{\color{blue}{e^{\mathsf{log1p}\left(\left(g + g\right) \cdot \frac{-0.5}{a}\right)} - 1}} \]

   3. add-cube-cbrt 31.1%

      \[\leadsto \sqrt[3]{\frac{g}{-a}} + \sqrt[3]{e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \cdot \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}}\right) \cdot \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}}}\right)} - 1} \]

   4. pow3 31.1%

      \[\leadsto \sqrt[3]{\frac{g}{-a}} + \sqrt[3]{e^{\mathsf{log1p}\left(\color{blue}{{\left(\sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}}\right)}^{3}}\right)} - 1} \]

   5. pow3 31.1%

      \[\leadsto \sqrt[3]{\frac{g}{-a}} + \sqrt[3]{e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \cdot \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}}\right) \cdot \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}}}\right)} - 1} \]

   6. add-cube-cbrt 31.1%

      \[\leadsto \sqrt[3]{\frac{g}{-a}} + \sqrt[3]{e^{\mathsf{log1p}\left(\color{blue}{\left(g + g\right) \cdot \frac{-0.5}{a}}\right)} - 1} \]

   7. count-2 31.1%

      \[\leadsto \sqrt[3]{\frac{g}{-a}} + \sqrt[3]{e^{\mathsf{log1p}\left(\color{blue}{\left(2 \cdot g\right)} \cdot \frac{-0.5}{a}\right)} - 1} \]

10. Applied egg-rr 31.1%

    \[\leadsto \sqrt[3]{\frac{g}{-a}} + \sqrt[3]{\color{blue}{e^{\mathsf{log1p}\left(\left(2 \cdot g\right) \cdot \frac{-0.5}{a}\right)} - 1}} \]
11. Step-by-step derivation

    1. sub-neg 31.1%

       \[\leadsto \sqrt[3]{\frac{g}{-a}} + \sqrt[3]{\color{blue}{e^{\mathsf{log1p}\left(\left(2 \cdot g\right) \cdot \frac{-0.5}{a}\right)} + \left(-1\right)}} \]

    2. metadata-eval 31.1%

       \[\leadsto \sqrt[3]{\frac{g}{-a}} + \sqrt[3]{e^{\mathsf{log1p}\left(\left(2 \cdot g\right) \cdot \frac{-0.5}{a}\right)} + \color{blue}{-1}} \]

    3. +-commutative 31.1%

       \[\leadsto \sqrt[3]{\frac{g}{-a}} + \sqrt[3]{\color{blue}{-1 + e^{\mathsf{log1p}\left(\left(2 \cdot g\right) \cdot \frac{-0.5}{a}\right)}}} \]

    4. log1p-undefine 31.1%

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

    5. rem-exp-log 36.1%

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

    6. associate-*r/ 36.1%

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

    7. *-commutative 36.1%

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

    8. associate-*r* 36.1%

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

    9. metadata-eval 36.1%

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

    10. neg-mul-1 36.1%

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

    11. distribute-neg-frac 36.1%

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

    12. unsub-neg 36.1%

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

12. Simplified 36.1%

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

Alternative 5: 15.4% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{g}{a}} \cdot -2 \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 45.1%

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

3. Simplified 45.1%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}} \]
4. Add Preprocessing

5. Taylor expanded in g around -inf 28.4%

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

   1. mul-1-neg 28.4%

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

   2. distribute-neg-frac2 28.4%

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

7. Simplified 28.4%

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

8. Taylor expanded in g around inf 15.5%

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

9. Taylor expanded in g around -inf 15.5%

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

    1. *-commutative 15.5%

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

11. Simplified 15.5%

    \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a}} \cdot -2} \]
12. Add Preprocessing

Alternative 6: 5.8% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 45.1%

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

3. Simplified 45.1%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}} \]
4. Add Preprocessing

5. Taylor expanded in h around 0 67.9%

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

6. Taylor expanded in g around -inf 75.1%

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt[3]{\frac{g}{a}} \cdot \left(\sqrt[3]{-1} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right)\right)\right)} \]
7. Step-by-step derivation

   1. mul-1-neg 75.1%

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

   2. associate-*r* 75.1%

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

   3. distribute-rgt-neg-in 75.1%

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

   4. *-commutative 75.1%

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

   5. *-commutative 75.1%

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

8. Simplified 75.1%

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

10. Applied egg-rr 95.4%

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

12. Applied egg-rr 5.9%

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

    1. associate-*l* 5.9%

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

    2. *-commutative 5.9%

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

    3. neg-mul-1 5.9%

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

14. Simplified 5.9%

    \[\leadsto \color{blue}{\sqrt[3]{a \cdot \left(-g\right)}} \]
15. Add Preprocessing

Alternative 7: 3.0% accurate, 433.0× 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 45.1%

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

3. Simplified 45.1%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}} \]
4. Add Preprocessing

5. Taylor expanded in g around -inf 28.4%

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

   1. mul-1-neg 28.4%

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

   2. distribute-neg-frac2 28.4%

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

7. Simplified 28.4%

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

8. Taylor expanded in g around inf 15.5%

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

   1. flip-+ 1.4%

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

   2. pow2 1.4%

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

   3. pow2 1.4%

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

   4. count-2 1.4%

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

   5. count-2 1.4%

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

10. Applied egg-rr 1.4%

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

    1. div-sub 1.4%

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

    2. distribute-frac-neg2 1.4%

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

    3. mul-1-neg 1.4%

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

    4. *-commutative 1.4%

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

    5. metadata-eval 1.4%

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

    6. associate-*l* 1.4%

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

    7. *-commutative 1.4%

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

    8. associate-*r/ 1.4%

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

    9. associate-*l/ 1.4%

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

    10. associate-*r/ 0.2%

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

    11. +-inverses 2.9%

        \[\leadsto \color{blue}{0} \]

12. Simplified 2.9%

    \[\leadsto \color{blue}{0} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024117 
(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))))))))