2-ancestry mixing, positive discriminant

Percentage Accurate: 45.4% → 95.6%

Time: 15.1s

Alternatives: 6

Speedup: 2.1×

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: 45.4% 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: 95.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.5%

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

3. Simplified 49.5%

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

5. Applied egg-rr 12.9%

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

   1. +-commutative 12.9%

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

   2. +-commutative 12.9%

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

   3. associate-+l+ 12.9%

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

7. Simplified 12.9%

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

8. Taylor expanded in g around -inf 20.2%

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

   1. mul-1-neg 20.2%

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

10. Simplified 20.2%

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

    1. add-cbrt-cube 20.1%

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

    2. sqrt-unprod 20.1%

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

    3. sqrt-prod 20.1%

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

    4. add-cube-cbrt 20.2%

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

    5. unpow2 20.2%

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

    6. sqrt-prod 1.0%

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

    7. add-sqr-sqrt 1.5%

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

12. Applied egg-rr 95.0%

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

13. Final simplification 95.0%

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

Alternative 2: 95.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.5%

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

3. Simplified 49.5%

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

5. Applied egg-rr 36.3%

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

6. Taylor expanded in g around -inf 95.0%

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

   1. mul-1-neg 20.2%

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

8. Simplified 95.0%

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

9. Final simplification 95.0%

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

Alternative 3: 89.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.5%

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

3. Simplified 49.5%

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

   1. fma-neg 49.5%

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

   2. distribute-rgt-neg-out 49.5%

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

   3. add-sqr-sqrt 43.1%

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

   4. add-sqr-sqrt 27.9%

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

   5. difference-of-squares 27.9%

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

5. Applied egg-rr 28.0%

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

   1. difference-of-squares 28.0%

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

   2. rem-square-sqrt 21.6%

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

   3. rem-square-sqrt 29.2%

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

   4. unsub-neg 29.2%

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

   5. mul-1-neg 29.2%

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

   6. +-commutative 29.2%

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

   7. associate-+l+ 28.6%

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

   8. mul-1-neg 28.6%

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

   9. sub-neg 28.6%

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

   10. +-inverses 28.6%

       \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(h + \color{blue}{0}\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

   11. +-rgt-identity 28.6%

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

7. Simplified 28.6%

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

8. Taylor expanded in g around inf 73.8%

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

   1. *-commutative 73.8%

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

   2. cbrt-prod 90.3%

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

10. Applied egg-rr 90.3%

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

11. Final simplification 90.3%

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

Alternative 4: 89.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.5%

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

3. Simplified 49.5%

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

   1. fma-neg 49.5%

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

   2. distribute-rgt-neg-out 49.5%

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

   3. add-sqr-sqrt 43.1%

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

   4. add-sqr-sqrt 27.9%

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

   5. difference-of-squares 27.9%

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

5. Applied egg-rr 28.0%

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

   1. difference-of-squares 28.0%

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

   2. rem-square-sqrt 21.6%

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

   3. rem-square-sqrt 29.2%

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

   4. unsub-neg 29.2%

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

   5. mul-1-neg 29.2%

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

   6. +-commutative 29.2%

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

   7. associate-+l+ 28.6%

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

   8. mul-1-neg 28.6%

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

   9. sub-neg 28.6%

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

   10. +-inverses 28.6%

       \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(h + \color{blue}{0}\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

   11. +-rgt-identity 28.6%

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

7. Simplified 28.6%

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

8. Taylor expanded in g around inf 73.8%

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

   1. associate-*r/ 73.8%

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

   2. cbrt-div 90.3%

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

10. Applied egg-rr 90.3%

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

11. Final simplification 90.3%

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

Alternative 5: 75.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \cdot h \leq 2 \cdot 10^{+176}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(\left(\frac{0.5}{\frac{g}{h \cdot h}} - g\right) - g\right)} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(0.5 \cdot \frac{h}{\frac{g}{h}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= (* h h) 2e+176)
   (+
    (cbrt (* (/ 0.5 a) (- (- (/ 0.5 (/ g (* h h))) g) g)))
    (cbrt (* (/ -0.5 a) (* 0.5 (/ h (/ g h))))))
   (* (cbrt (/ g a)) (cbrt -1.0))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((h * h) <= 2e+176) {
		tmp = cbrt(((0.5 / a) * (((0.5 / (g / (h * h))) - g) - g))) + cbrt(((-0.5 / a) * (0.5 * (h / (g / h)))));
	} else {
		tmp = cbrt((g / a)) * cbrt(-1.0);
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if ((h * h) <= 2e+176) {
		tmp = Math.cbrt(((0.5 / a) * (((0.5 / (g / (h * h))) - g) - g))) + Math.cbrt(((-0.5 / a) * (0.5 * (h / (g / h)))));
	} else {
		tmp = Math.cbrt((g / a)) * Math.cbrt(-1.0);
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (Float64(h * h) <= 2e+176)
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(Float64(Float64(0.5 / Float64(g / Float64(h * h))) - g) - g))) + cbrt(Float64(Float64(-0.5 / a) * Float64(0.5 * Float64(h / Float64(g / h))))));
	else
		tmp = Float64(cbrt(Float64(g / a)) * cbrt(-1.0));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[N[(h * h), $MachinePrecision], 2e+176], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(N[(N[(0.5 / N[(g / N[(h * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - g), $MachinePrecision] - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(-0.5 / a), $MachinePrecision] * N[(0.5 * N[(h / N[(g / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[-1.0, 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 h h) < 2e176

   1. Initial program 54.7%

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

   3. Simplified 54.7%

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

   4. Taylor expanded in g around -inf 30.8%

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

      1. +-commutative 30.8%

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

      2. mul-1-neg 30.8%

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

      3. unsub-neg 30.8%

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

      4. associate-*r/ 30.8%

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

      5. associate-/l* 30.8%

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

      6. unpow2 30.8%

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

   6. Simplified 30.8%

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

   7. Taylor expanded in g around -inf 81.2%

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

      1. unpow2 81.2%

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

      2. associate-/l* 81.6%

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

   9. Simplified 81.6%

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

   if 2e176 < (*.f64 h h)

   1. Initial program 9.9%

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

   3. Simplified 9.9%

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

      1. fma-neg 9.9%

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

      2. distribute-rgt-neg-out 9.9%

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

      3. add-sqr-sqrt 9.9%

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

      4. add-sqr-sqrt 6.6%

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

      5. difference-of-squares 6.6%

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

   5. Applied egg-rr 7.2%

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

      1. difference-of-squares 7.2%

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

      2. rem-square-sqrt 7.2%

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

      3. rem-square-sqrt 7.4%

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

      4. unsub-neg 7.4%

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

      5. mul-1-neg 7.4%

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

      6. +-commutative 7.4%

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

      7. associate-+l+ 6.3%

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

      8. mul-1-neg 6.3%

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

      9. sub-neg 6.3%

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

      10. +-inverses 6.3%

          \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(h + \color{blue}{0}\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

      11. +-rgt-identity 6.3%

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

   7. Simplified 6.3%

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

   8. Taylor expanded in g around inf 48.3%

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

   9. Taylor expanded in g around 0 48.3%

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

   10. Taylor expanded in h around 0 27.9%

       \[\leadsto \color{blue}{{\left(\frac{1 \cdot g}{a}\right)}^{0.3333333333333333} \cdot \sqrt[3]{-1}} \]
   11. Step-by-step derivation

       1. unpow1/3 55.2%

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

       2. *-lft-identity 55.2%

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

   12. Simplified 55.2%

       \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \cdot h \leq 2 \cdot 10^{+176}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(\left(\frac{0.5}{\frac{g}{h \cdot h}} - g\right) - g\right)} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(0.5 \cdot \frac{h}{\frac{g}{h}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}\\ \end{array} \]

Alternative 6: 74.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.5%

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

3. Simplified 49.5%

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

   1. fma-neg 49.5%

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

   2. distribute-rgt-neg-out 49.5%

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

   3. add-sqr-sqrt 43.1%

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

   4. add-sqr-sqrt 27.9%

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

   5. difference-of-squares 27.9%

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

5. Applied egg-rr 28.0%

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

   1. difference-of-squares 28.0%

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

   2. rem-square-sqrt 21.6%

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

   3. rem-square-sqrt 29.2%

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

   4. unsub-neg 29.2%

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

   5. mul-1-neg 29.2%

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

   6. +-commutative 29.2%

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

   7. associate-+l+ 28.6%

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

   8. mul-1-neg 28.6%

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

   9. sub-neg 28.6%

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

   10. +-inverses 28.6%

       \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(h + \color{blue}{0}\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

   11. +-rgt-identity 28.6%

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

7. Simplified 28.6%

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

8. Taylor expanded in g around inf 73.8%

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

9. Taylor expanded in g around 0 73.8%

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

10. Taylor expanded in h around 0 40.8%

    \[\leadsto \color{blue}{{\left(\frac{1 \cdot g}{a}\right)}^{0.3333333333333333} \cdot \sqrt[3]{-1}} \]
11. Step-by-step derivation

    1. unpow1/3 76.8%

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

    2. *-lft-identity 76.8%

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

12. Simplified 76.8%

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

13. Final simplification 76.8%

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

Reproduce

herbie shell --seed 2023292 
(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))))))))