2-ancestry mixing, positive discriminant

Percentage Accurate: 43.7% → 96.4%

Time: 38.5s

Alternatives: 8

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: 43.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 96.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ \mathbf{if}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(t\_0 - g\right)} + \sqrt[3]{\left(g + t\_0\right) \cdot \frac{-1}{2 \cdot a}} \leq 5 \cdot 10^{+37}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{g \cdot -2} + \sqrt[3]{\frac{0.5 \cdot {h}^{2}}{g} \cdot \frac{-0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} - \sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (sqrt (- (* g g) (* h h)))))
   (if (<=
        (+
         (cbrt (* (/ 1.0 (* 2.0 a)) (- t_0 g)))
         (cbrt (* (+ g t_0) (/ -1.0 (* 2.0 a)))))
        5e+37)
     (+
      (* (cbrt (/ 0.5 a)) (cbrt (* g -2.0)))
      (cbrt (* (/ (* 0.5 (pow h 2.0)) g) (/ -0.5 a))))
     (- (cbrt (* (/ 0.5 a) (- g g))) (* (cbrt g) (cbrt (/ 1.0 a)))))))

C

double code(double g, double h, double a) {
	double t_0 = sqrt(((g * g) - (h * h)));
	double tmp;
	if ((cbrt(((1.0 / (2.0 * a)) * (t_0 - g))) + cbrt(((g + t_0) * (-1.0 / (2.0 * a))))) <= 5e+37) {
		tmp = (cbrt((0.5 / a)) * cbrt((g * -2.0))) + cbrt((((0.5 * pow(h, 2.0)) / g) * (-0.5 / a)));
	} else {
		tmp = cbrt(((0.5 / a) * (g - g))) - (cbrt(g) * cbrt((1.0 / a)));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.sqrt(((g * g) - (h * h)));
	double tmp;
	if ((Math.cbrt(((1.0 / (2.0 * a)) * (t_0 - g))) + Math.cbrt(((g + t_0) * (-1.0 / (2.0 * a))))) <= 5e+37) {
		tmp = (Math.cbrt((0.5 / a)) * Math.cbrt((g * -2.0))) + Math.cbrt((((0.5 * Math.pow(h, 2.0)) / g) * (-0.5 / a)));
	} else {
		tmp = Math.cbrt(((0.5 / a) * (g - g))) - (Math.cbrt(g) * Math.cbrt((1.0 / a)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	tmp = 0.0
	if (Float64(cbrt(Float64(Float64(1.0 / Float64(2.0 * a)) * Float64(t_0 - g))) + cbrt(Float64(Float64(g + t_0) * Float64(-1.0 / Float64(2.0 * a))))) <= 5e+37)
		tmp = Float64(Float64(cbrt(Float64(0.5 / a)) * cbrt(Float64(g * -2.0))) + cbrt(Float64(Float64(Float64(0.5 * (h ^ 2.0)) / g) * Float64(-0.5 / a))));
	else
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(g - g))) - Float64(cbrt(g) * cbrt(Float64(1.0 / a))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(g + t$95$0), $MachinePrecision] * N[(-1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 5e+37], N[(N[(N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(g * -2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(N[(0.5 * N[Power[h, 2.0], $MachinePrecision]), $MachinePrecision] / g), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(g - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] - N[(N[Power[g, 1/3], $MachinePrecision] * N[Power[N[(1.0 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < 4.99999999999999989e37

   1. Initial program 77.7%

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

   3. Simplified 77.7%

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

      1. cbrt-prod 81.7%

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

      2. pow2 81.7%

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

      3. pow2 81.7%

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

   6. Applied egg-rr 81.7%

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

   7. Taylor expanded in g around -inf 43.1%

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

      1. associate-*r/ 43.1%

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

   9. Simplified 43.1%

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

   10. Taylor expanded in g around -inf 96.3%

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

       1. *-commutative 96.3%

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

   12. Simplified 96.3%

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

   if 4.99999999999999989e37 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))))

   1. Initial program 13.1%

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

   3. Simplified 13.1%

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

   5. Taylor expanded in g around inf 8.2%

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

   6. Taylor expanded in g around inf 64.3%

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

      1. *-commutative 64.3%

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

      2. associate-*l* 64.1%

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

   8. Simplified 64.1%

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

   10. Applied egg-rr 97.7%

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

   12. Applied egg-rr 44.1%

       \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \color{blue}{\frac{{g}^{0.16666666666666666}}{-1} \cdot \frac{{g}^{0.16666666666666666}}{\sqrt[3]{a}}} \]
   13. Step-by-step derivation

       1. times-frac 44.1%

          \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \color{blue}{\frac{{g}^{0.16666666666666666} \cdot {g}^{0.16666666666666666}}{-1 \cdot \sqrt[3]{a}}} \]

       2. pow-sqr 44.1%

          \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \frac{\color{blue}{{g}^{\left(2 \cdot 0.16666666666666666\right)}}}{-1 \cdot \sqrt[3]{a}} \]

       3. metadata-eval 44.1%

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

       4. unpow1/3 97.7%

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

       5. neg-mul-1 97.7%

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

       6. distribute-frac-neg2 97.7%

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

       7. distribute-neg-frac 97.7%

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

   14. Simplified 97.7%

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

       1. frac-2neg 97.7%

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

       2. div-inv 97.6%

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

       3. remove-double-neg 97.6%

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

   16. Applied egg-rr 97.6%

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

   17. Taylor expanded in a around 0 97.7%

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

       1. mul-1-neg 97.7%

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

   19. Simplified 97.7%

       \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \sqrt[3]{g} \cdot \color{blue}{\left(-\sqrt[3]{\frac{1}{a}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} \leq 5 \cdot 10^{+37}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{g \cdot -2} + \sqrt[3]{\frac{0.5 \cdot {h}^{2}}{g} \cdot \frac{-0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} - \sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ \mathbf{if}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(t\_1 - g\right)} + \sqrt[3]{\left(g + t\_1\right) \cdot \frac{-1}{2 \cdot a}} \leq 5 \cdot 10^{+33}:\\ \;\;\;\;\sqrt[3]{\frac{h}{g} \cdot \frac{h}{a}} \cdot \left(\sqrt[3]{0.5} \cdot \sqrt[3]{-0.5}\right) + t\_0 \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (* (cbrt g) (cbrt (/ 1.0 a)))) (t_1 (sqrt (- (* g g) (* h h)))))
   (if (<=
        (+
         (cbrt (* (/ 1.0 (* 2.0 a)) (- t_1 g)))
         (cbrt (* (+ g t_1) (/ -1.0 (* 2.0 a)))))
        5e+33)
     (+
      (* (cbrt (* (/ h g) (/ h a))) (* (cbrt 0.5) (cbrt -0.5)))
      (* t_0 (* (cbrt -0.5) (cbrt 2.0))))
     (- (cbrt (* (/ 0.5 a) (- g g))) t_0))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt(g) * cbrt((1.0 / a));
	double t_1 = sqrt(((g * g) - (h * h)));
	double tmp;
	if ((cbrt(((1.0 / (2.0 * a)) * (t_1 - g))) + cbrt(((g + t_1) * (-1.0 / (2.0 * a))))) <= 5e+33) {
		tmp = (cbrt(((h / g) * (h / a))) * (cbrt(0.5) * cbrt(-0.5))) + (t_0 * (cbrt(-0.5) * cbrt(2.0)));
	} else {
		tmp = cbrt(((0.5 / a) * (g - g))) - t_0;
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.cbrt(g) * Math.cbrt((1.0 / a));
	double t_1 = Math.sqrt(((g * g) - (h * h)));
	double tmp;
	if ((Math.cbrt(((1.0 / (2.0 * a)) * (t_1 - g))) + Math.cbrt(((g + t_1) * (-1.0 / (2.0 * a))))) <= 5e+33) {
		tmp = (Math.cbrt(((h / g) * (h / a))) * (Math.cbrt(0.5) * Math.cbrt(-0.5))) + (t_0 * (Math.cbrt(-0.5) * Math.cbrt(2.0)));
	} else {
		tmp = Math.cbrt(((0.5 / a) * (g - g))) - t_0;
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64(cbrt(g) * cbrt(Float64(1.0 / a)))
	t_1 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	tmp = 0.0
	if (Float64(cbrt(Float64(Float64(1.0 / Float64(2.0 * a)) * Float64(t_1 - g))) + cbrt(Float64(Float64(g + t_1) * Float64(-1.0 / Float64(2.0 * a))))) <= 5e+33)
		tmp = Float64(Float64(cbrt(Float64(Float64(h / g) * Float64(h / a))) * Float64(cbrt(0.5) * cbrt(-0.5))) + Float64(t_0 * Float64(cbrt(-0.5) * cbrt(2.0))));
	else
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(g - g))) - t_0);
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(N[Power[g, 1/3], $MachinePrecision] * N[Power[N[(1.0 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(g + t$95$1), $MachinePrecision] * N[(-1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 5e+33], N[(N[(N[Power[N[(N[(h / g), $MachinePrecision] * N[(h / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(N[Power[0.5, 1/3], $MachinePrecision] * N[Power[-0.5, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(g - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < 4.99999999999999973e33

   1. Initial program 78.1%

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

   3. Simplified 78.1%

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

   5. Taylor expanded in g around inf 51.8%

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

   6. Taylor expanded in g around inf 81.4%

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

      1. *-commutative 81.4%

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

      2. *-commutative 81.4%

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

   8. Simplified 81.4%

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

      1. pow1/3 46.7%

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

      2. div-inv 46.7%

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

      3. unpow-prod-down 35.3%

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

      4. pow1/3 56.1%

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

   10. Applied egg-rr 56.1%

       \[\leadsto \sqrt[3]{\frac{{h}^{2}}{g \cdot a}} \cdot \left(\sqrt[3]{0.5} \cdot \sqrt[3]{-0.5}\right) + \color{blue}{\left(\sqrt[3]{g} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right)} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \]
   11. Step-by-step derivation

       1. unpow1/3 92.3%

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

   12. Simplified 92.3%

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

       1. unpow2 92.3%

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

       2. times-frac 96.6%

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

   14. Applied egg-rr 96.6%

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

   if 4.99999999999999973e33 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))))

   1. Initial program 14.1%

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

   3. Simplified 14.1%

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

   5. Taylor expanded in g around inf 9.0%

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

   6. Taylor expanded in g around inf 64.6%

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

      1. *-commutative 64.6%

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

      2. associate-*l* 64.4%

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

   8. Simplified 64.4%

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

   10. Applied egg-rr 97.3%

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

   12. Applied egg-rr 44.5%

       \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \color{blue}{\frac{{g}^{0.16666666666666666}}{-1} \cdot \frac{{g}^{0.16666666666666666}}{\sqrt[3]{a}}} \]
   13. Step-by-step derivation

       1. times-frac 44.5%

          \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \color{blue}{\frac{{g}^{0.16666666666666666} \cdot {g}^{0.16666666666666666}}{-1 \cdot \sqrt[3]{a}}} \]

       2. pow-sqr 44.5%

          \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \frac{\color{blue}{{g}^{\left(2 \cdot 0.16666666666666666\right)}}}{-1 \cdot \sqrt[3]{a}} \]

       3. metadata-eval 44.5%

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

       4. unpow1/3 97.3%

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

       5. neg-mul-1 97.3%

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

       6. distribute-frac-neg2 97.3%

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

       7. distribute-neg-frac 97.3%

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

   14. Simplified 97.3%

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

       1. frac-2neg 97.3%

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

       2. div-inv 97.2%

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

       3. remove-double-neg 97.2%

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

   16. Applied egg-rr 97.2%

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

   17. Taylor expanded in a around 0 97.3%

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

       1. mul-1-neg 97.3%

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

   19. Simplified 97.3%

       \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \sqrt[3]{g} \cdot \color{blue}{\left(-\sqrt[3]{\frac{1}{a}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} \leq 5 \cdot 10^{+33}:\\ \;\;\;\;\sqrt[3]{\frac{h}{g} \cdot \frac{h}{a}} \cdot \left(\sqrt[3]{0.5} \cdot \sqrt[3]{-0.5}\right) + \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right) \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} - \sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.8% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (g h a)
 :precision binary64
 (- (cbrt (* (/ 0.5 a) (- g g))) (* (cbrt g) (cbrt (/ 1.0 a)))))

C

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

Java

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

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(0.5 / a) * Float64(g - g))) - Float64(cbrt(g) * cbrt(Float64(1.0 / 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[g, 1/3], $MachinePrecision] * N[Power[N[(1.0 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 42.6%

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

3. Simplified 42.6%

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

5. Taylor expanded in g around inf 26.0%

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

6. Taylor expanded in g around inf 72.2%

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

   1. *-commutative 72.2%

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

   2. associate-*l* 72.0%

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

8. Simplified 72.0%

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

10. Applied egg-rr 95.4%

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

12. Applied egg-rr 46.9%

    \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \color{blue}{\frac{{g}^{0.16666666666666666}}{-1} \cdot \frac{{g}^{0.16666666666666666}}{\sqrt[3]{a}}} \]
13. Step-by-step derivation

    1. times-frac 46.9%

       \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \color{blue}{\frac{{g}^{0.16666666666666666} \cdot {g}^{0.16666666666666666}}{-1 \cdot \sqrt[3]{a}}} \]

    2. pow-sqr 46.9%

       \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \frac{\color{blue}{{g}^{\left(2 \cdot 0.16666666666666666\right)}}}{-1 \cdot \sqrt[3]{a}} \]

    3. metadata-eval 46.9%

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

    4. unpow1/3 95.4%

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

    5. neg-mul-1 95.4%

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

    6. distribute-frac-neg2 95.4%

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

    7. distribute-neg-frac 95.4%

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

14. Simplified 95.4%

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

    1. frac-2neg 95.4%

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

    2. div-inv 95.4%

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

    3. remove-double-neg 95.4%

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

16. Applied egg-rr 95.4%

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

17. Taylor expanded in a around 0 95.4%

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

    1. mul-1-neg 95.4%

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

19. Simplified 95.4%

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

20. Final simplification 95.4%

    \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} - \sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}} \]
21. Add Preprocessing

Alternative 4: 95.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(0.5 / a) * Float64(g - g))) - Float64(cbrt(g) / 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[g, 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 42.6%

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

3. Simplified 42.6%

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

5. Taylor expanded in g around inf 26.0%

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

6. Taylor expanded in g around inf 72.2%

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

   1. *-commutative 72.2%

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

   2. associate-*l* 72.0%

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

8. Simplified 72.0%

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

10. Applied egg-rr 95.4%

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

12. Applied egg-rr 46.9%

    \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \color{blue}{\frac{{g}^{0.16666666666666666}}{-1} \cdot \frac{{g}^{0.16666666666666666}}{\sqrt[3]{a}}} \]
13. Step-by-step derivation

    1. times-frac 46.9%

       \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \color{blue}{\frac{{g}^{0.16666666666666666} \cdot {g}^{0.16666666666666666}}{-1 \cdot \sqrt[3]{a}}} \]

    2. pow-sqr 46.9%

       \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \frac{\color{blue}{{g}^{\left(2 \cdot 0.16666666666666666\right)}}}{-1 \cdot \sqrt[3]{a}} \]

    3. metadata-eval 46.9%

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

    4. unpow1/3 95.4%

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

    5. neg-mul-1 95.4%

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

    6. distribute-frac-neg2 95.4%

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

    7. distribute-neg-frac 95.4%

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

14. Simplified 95.4%

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

15. Final simplification 95.4%

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

Alternative 5: 6.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)}\\ \mathbf{if}\;a \leq -9 \cdot 10^{-32}:\\ \;\;\;\;\sqrt[3]{g} + t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \sqrt[3]{-g}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (* (/ 0.5 a) (- g g)))))
   (if (<= a -9e-32) (+ (cbrt g) t_0) (+ t_0 (cbrt (- g))))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt(((0.5 / a) * (g - g)));
	double tmp;
	if (a <= -9e-32) {
		tmp = cbrt(g) + t_0;
	} else {
		tmp = t_0 + cbrt(-g);
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.cbrt(((0.5 / a) * (g - g)));
	double tmp;
	if (a <= -9e-32) {
		tmp = Math.cbrt(g) + t_0;
	} else {
		tmp = t_0 + Math.cbrt(-g);
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(Float64(0.5 / a) * Float64(g - g)))
	tmp = 0.0
	if (a <= -9e-32)
		tmp = Float64(cbrt(g) + t_0);
	else
		tmp = Float64(t_0 + cbrt(Float64(-g)));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(g - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[a, -9e-32], N[(N[Power[g, 1/3], $MachinePrecision] + t$95$0), $MachinePrecision], N[(t$95$0 + N[Power[(-g), 1/3], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -9.00000000000000009e-32

   1. Initial program 43.1%

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

   3. Simplified 43.1%

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

   5. Taylor expanded in g around inf 24.9%

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

   6. Taylor expanded in g around inf 92.7%

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

      1. *-commutative 92.7%

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

      2. distribute-lft-in 92.7%

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

   8. Applied egg-rr 92.7%

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

   10. Simplified 1.5%

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

       1. add-cube-cbrt 1.5%

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

       2. distribute-rgt-neg-in 1.5%

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

       3. sqr-neg 1.5%

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

       4. add-cbrt-cube 1.5%

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

       5. neg-sub0 1.5%

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

       6. sub-neg 1.5%

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

       7. add-sqr-sqrt 0.7%

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

       8. sqrt-unprod 5.1%

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

       9. sqr-neg 5.1%

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

       10. sqrt-unprod 4.4%

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

       11. add-sqr-sqrt 8.3%

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

   12. Applied egg-rr 8.3%

       \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \color{blue}{\left(0 + \sqrt[3]{g}\right)} \]
   13. Step-by-step derivation

       1. +-lft-identity 8.3%

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

   14. Simplified 8.3%

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

   if -9.00000000000000009e-32 < a

   1. Initial program 42.4%

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

   3. Simplified 42.4%

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

   5. Taylor expanded in g around inf 26.3%

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

   6. Taylor expanded in g around inf 66.8%

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

      1. *-commutative 66.8%

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

      2. distribute-lft-in 66.7%

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

   8. Applied egg-rr 66.7%

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

   10. Simplified 6.3%

       \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \sqrt[3]{\color{blue}{-g}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 6.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{-32}:\\ \;\;\;\;\sqrt[3]{g} + \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \sqrt[3]{-g}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.6%

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

3. Simplified 42.6%

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

5. Taylor expanded in g around inf 26.0%

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

6. Taylor expanded in g around inf 72.2%

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

   1. *-commutative 72.2%

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

   2. associate-*l* 72.0%

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

8. Simplified 72.0%

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

10. Applied egg-rr 95.4%

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

12. Applied egg-rr 46.9%

    \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \color{blue}{\frac{{g}^{0.16666666666666666}}{-1} \cdot \frac{{g}^{0.16666666666666666}}{\sqrt[3]{a}}} \]

14. Applied egg-rr 73.2%

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

Alternative 7: 73.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(0.5 / a) * Float64(g - g))) - cbrt(Float64(g / 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[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 42.6%

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

3. Simplified 42.6%

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

5. Taylor expanded in g around inf 26.0%

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

6. Taylor expanded in g around inf 72.8%

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

7. Taylor expanded in g around 0 72.9%

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

9. Simplified 72.9%

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

10. Final simplification 72.9%

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

Alternative 8: 4.5% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.6%

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

3. Simplified 42.6%

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

5. Taylor expanded in g around inf 26.0%

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

6. Taylor expanded in g around inf 72.8%

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

   1. cbrt-prod 95.3%

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

   2. flip-+ 0.0%

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

   3. cbrt-div 0.0%

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

   4. unpow2 0.0%

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

   5. unpow2 0.0%

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

   6. +-inverses 0.0%

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

   7. metadata-eval 0.0%

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

   8. metadata-eval 0.0%

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

   9. +-inverses 0.0%

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

   10. +-inverses 0.0%

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

   11. +-inverses 0.0%

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

   12. add-cbrt-cube 0.0%

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

   13. +-inverses 0.0%

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

   14. +-inverses 0.0%

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

   15. metadata-eval 0.0%

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

   16. metadata-eval 0.0%

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

   17. +-inverses 0.0%

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

   18. +-inverses 0.0%

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

   19. +-inverses 0.0%

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

   20. add-cbrt-cube 0.0%

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

   21. +-inverses 0.0%

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

8. Applied egg-rr 0.0%

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

10. Simplified 4.5%

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

Reproduce

herbie shell --seed 2024107 
(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))))))))