2-ancestry mixing, positive discriminant

Percentage Accurate: 44.1% → 93.4%

Time: 47.4s

Alternatives: 9

Speedup: 2.1×

• could not determine a ground truth

  1. g = -1.5497881187685165e+220
  2. h = 1.3737837798162444e+92
  3. a = -8.166769568671368e+34

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 44.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 93.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{{g}^{2} - {h}^{2}}\\ \mathbf{if}\;g \leq -2 \cdot 10^{-99}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{t\_0 - g} + \sqrt[3]{\frac{{h}^{2}}{a} \cdot \frac{-0.5}{g - t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (sqrt (- (pow g 2.0) (pow h 2.0)))))
   (if (<= g -2e-99)
     (+
      (* (cbrt (/ 0.5 a)) (cbrt (- t_0 g)))
      (cbrt (* (/ (pow h 2.0) a) (/ -0.5 (- g t_0)))))
     (+ (/ (cbrt (- g)) (cbrt a)) (cbrt (* (- g g) (/ -0.5 a)))))))

C

double code(double g, double h, double a) {
	double t_0 = sqrt((pow(g, 2.0) - pow(h, 2.0)));
	double tmp;
	if (g <= -2e-99) {
		tmp = (cbrt((0.5 / a)) * cbrt((t_0 - g))) + cbrt(((pow(h, 2.0) / a) * (-0.5 / (g - t_0))));
	} else {
		tmp = (cbrt(-g) / cbrt(a)) + cbrt(((g - g) * (-0.5 / a)));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.sqrt((Math.pow(g, 2.0) - Math.pow(h, 2.0)));
	double tmp;
	if (g <= -2e-99) {
		tmp = (Math.cbrt((0.5 / a)) * Math.cbrt((t_0 - g))) + Math.cbrt(((Math.pow(h, 2.0) / a) * (-0.5 / (g - t_0))));
	} else {
		tmp = (Math.cbrt(-g) / Math.cbrt(a)) + Math.cbrt(((g - g) * (-0.5 / a)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = sqrt(Float64((g ^ 2.0) - (h ^ 2.0)))
	tmp = 0.0
	if (g <= -2e-99)
		tmp = Float64(Float64(cbrt(Float64(0.5 / a)) * cbrt(Float64(t_0 - g))) + cbrt(Float64(Float64((h ^ 2.0) / a) * Float64(-0.5 / Float64(g - t_0)))));
	else
		tmp = Float64(Float64(cbrt(Float64(-g)) / cbrt(a)) + cbrt(Float64(Float64(g - g) * Float64(-0.5 / a))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Sqrt[N[(N[Power[g, 2.0], $MachinePrecision] - N[Power[h, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[g, -2e-99], N[(N[(N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(t$95$0 - g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(N[Power[h, 2.0], $MachinePrecision] / a), $MachinePrecision] * N[(-0.5 / N[(g - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[(-g), 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(g - g), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if g < -2e-99

   1. Initial program 62.4%

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

   3. Simplified 62.4%

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

      1. cbrt-prod 65.0%

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

      2. pow2 65.0%

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

      3. pow2 65.0%

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

   6. Applied egg-rr 65.0%

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

      1. flip-+ 65.5%

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

      2. frac-2neg 65.5%

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

      3. metadata-eval 65.5%

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

      4. frac-times 62.2%

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

      5. add-sqr-sqrt 62.4%

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

      6. associate--r- 74.2%

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

      7. pow2 74.2%

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

      8. pow2 74.2%

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

      9. pow2 74.2%

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

   8. Applied egg-rr 74.2%

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

      1. times-frac 81.1%

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

      2. distribute-frac-neg2 81.1%

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

      3. distribute-neg-frac 81.1%

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

      4. metadata-eval 81.1%

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

      5. times-frac 74.2%

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

      6. *-commutative 74.2%

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

      7. times-frac 75.5%

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

      8. +-inverses 75.5%

         \[\leadsto \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\sqrt{{g}^{2} - {h}^{2}} - g} + \sqrt[3]{\frac{\color{blue}{0} + {h}^{2}}{a} \cdot \frac{-0.5}{g - \sqrt{{g}^{2} - {h}^{2}}}} \]

      9. +-lft-identity 75.5%

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

   10. Simplified 75.5%

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

   if -2e-99 < g

   1. Initial program 46.0%

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

   3. Simplified 46.0%

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

   5. Taylor expanded in g around -inf 11.6%

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

      1. *-commutative 11.6%

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

   7. Simplified 11.6%

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

   8. Taylor expanded in g around -inf 74.5%

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

      1. neg-mul-1 74.5%

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

   10. Simplified 74.5%

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

       1. associate-*l/ 74.5%

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

       2. cbrt-div 94.4%

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

       3. *-commutative 94.4%

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

       4. associate-*r* 94.4%

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

       5. metadata-eval 94.4%

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

       6. neg-mul-1 94.4%

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

   12. Applied egg-rr 94.4%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -2 \cdot 10^{-99}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\sqrt{{g}^{2} - {h}^{2}} - g} + \sqrt[3]{\frac{{h}^{2}}{a} \cdot \frac{-0.5}{g - \sqrt{{g}^{2} - {h}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 70.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.3 \cdot 10^{+269}:\\ \;\;\;\;\sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} - \sqrt[3]{\frac{g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{\frac{g}{-a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= a 3.3e+269)
   (- (cbrt (* (- g g) (/ -0.5 a))) (cbrt (/ g a)))
   (+ (/ (cbrt (- g)) (cbrt a)) (cbrt (/ g (- a))))))

C

double code(double g, double h, double a) {
	double tmp;
	if (a <= 3.3e+269) {
		tmp = cbrt(((g - g) * (-0.5 / a))) - cbrt((g / a));
	} else {
		tmp = (cbrt(-g) / cbrt(a)) + cbrt((g / -a));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if (a <= 3.3e+269) {
		tmp = Math.cbrt(((g - g) * (-0.5 / a))) - Math.cbrt((g / a));
	} else {
		tmp = (Math.cbrt(-g) / Math.cbrt(a)) + Math.cbrt((g / -a));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (a <= 3.3e+269)
		tmp = Float64(cbrt(Float64(Float64(g - g) * Float64(-0.5 / a))) - cbrt(Float64(g / a)));
	else
		tmp = Float64(Float64(cbrt(Float64(-g)) / cbrt(a)) + cbrt(Float64(g / Float64(-a))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[a, 3.3e+269], N[(N[Power[N[(N[(g - g), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[(-g), 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[(g / (-a)), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.2999999999999998e269

   1. Initial program 48.1%

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

   3. Simplified 48.1%

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

   5. Taylor expanded in g around -inf 14.4%

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

      1. *-commutative 14.4%

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

   7. Simplified 14.4%

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

   8. Taylor expanded in g around -inf 74.5%

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

      1. neg-mul-1 74.5%

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

   10. Simplified 74.5%

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

   11. Taylor expanded in g around -inf 74.6%

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

       1. mul-1-neg 16.0%

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

   13. Simplified 74.6%

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

   if 3.2999999999999998e269 < a

   1. Initial program 23.4%

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

   3. Simplified 23.4%

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

   5. Taylor expanded in g around -inf 7.5%

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

      1. *-commutative 7.5%

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

   7. Simplified 7.5%

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

   8. Taylor expanded in g around inf 7.5%

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

      1. mul-1-neg 7.5%

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

      2. distribute-frac-neg 7.5%

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

   10. Simplified 7.5%

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

       1. associate-*l/ 23.4%

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

       2. cbrt-div 85.5%

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

       3. *-commutative 85.5%

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

       4. associate-*r* 85.5%

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

       5. metadata-eval 85.5%

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

       6. neg-mul-1 85.5%

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

   12. Applied egg-rr 69.6%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.3 \cdot 10^{+269}:\\ \;\;\;\;\sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} - \sqrt[3]{\frac{g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{\frac{g}{-a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{g}{a}}\\ \mathbf{if}\;a \leq 3.3 \cdot 10^{+269}:\\ \;\;\;\;\sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{\sqrt[3]{g}}{\sqrt[3]{-a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (/ g a))))
   (if (<= a 3.3e+269)
     (- (cbrt (* (- g g) (/ -0.5 a))) t_0)
     (+ t_0 (/ (cbrt g) (cbrt (- a)))))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt((g / a));
	double tmp;
	if (a <= 3.3e+269) {
		tmp = cbrt(((g - g) * (-0.5 / a))) - t_0;
	} else {
		tmp = t_0 + (cbrt(g) / cbrt(-a));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.cbrt((g / a));
	double tmp;
	if (a <= 3.3e+269) {
		tmp = Math.cbrt(((g - g) * (-0.5 / a))) - t_0;
	} else {
		tmp = t_0 + (Math.cbrt(g) / Math.cbrt(-a));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(g / a))
	tmp = 0.0
	if (a <= 3.3e+269)
		tmp = Float64(cbrt(Float64(Float64(g - g) * Float64(-0.5 / a))) - t_0);
	else
		tmp = Float64(t_0 + Float64(cbrt(g) / cbrt(Float64(-a))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[a, 3.3e+269], N[(N[Power[N[(N[(g - g), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] - t$95$0), $MachinePrecision], N[(t$95$0 + N[(N[Power[g, 1/3], $MachinePrecision] / N[Power[(-a), 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 3.2999999999999998e269

   1. Initial program 48.1%

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

   3. Simplified 48.1%

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

   5. Taylor expanded in g around -inf 14.4%

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

      1. *-commutative 14.4%

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

   7. Simplified 14.4%

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

   8. Taylor expanded in g around -inf 74.5%

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

      1. neg-mul-1 74.5%

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

   10. Simplified 74.5%

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

   11. Taylor expanded in g around -inf 74.6%

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

       1. mul-1-neg 16.0%

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

   13. Simplified 74.6%

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

   if 3.2999999999999998e269 < a

   1. Initial program 23.4%

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

   3. Simplified 23.4%

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

   5. Taylor expanded in g around -inf 7.5%

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

      1. *-commutative 7.5%

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

   7. Simplified 7.5%

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

   8. Taylor expanded in g around inf 7.5%

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

      1. mul-1-neg 7.5%

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

      2. distribute-frac-neg 7.5%

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

   10. Simplified 7.5%

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

       1. associate-*l/ 7.5%

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

       2. *-commutative 7.5%

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

       3. associate-*r* 7.5%

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

       4. metadata-eval 7.5%

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

       5. neg-mul-1 7.5%

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

       6. add-sqr-sqrt 0.5%

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

       7. sqrt-unprod 4.5%

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

       8. sqr-neg 4.5%

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

       9. sqrt-prod 4.6%

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

       10. add-sqr-sqrt 4.5%

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

   12. Applied egg-rr 4.5%

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

       1. frac-2neg 4.5%

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

       2. cbrt-div 67.9%

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

       3. remove-double-neg 67.9%

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

   14. Applied egg-rr 67.9%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.3 \cdot 10^{+269}:\\ \;\;\;\;\sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} - \sqrt[3]{\frac{g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{g}{a}} + \frac{\sqrt[3]{g}}{\sqrt[3]{-a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 47.1%

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

3. Simplified 47.1%

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

5. Taylor expanded in g around -inf 14.2%

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

   1. *-commutative 14.2%

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

7. Simplified 14.2%

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

8. Taylor expanded in g around -inf 72.5%

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

   1. neg-mul-1 72.5%

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

10. Simplified 72.5%

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

    1. associate-*l/ 72.6%

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

    2. cbrt-div 91.4%

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

    3. *-commutative 91.4%

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

    4. associate-*r* 91.4%

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

    5. metadata-eval 91.4%

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

    6. neg-mul-1 91.4%

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

12. Applied egg-rr 91.4%

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

13. Final simplification 91.4%

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

Alternative 5: 71.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 47.1%

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

3. Simplified 47.1%

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

5. Taylor expanded in g around -inf 14.2%

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

   1. *-commutative 14.2%

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

7. Simplified 14.2%

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

8. Taylor expanded in g around -inf 72.5%

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

   1. neg-mul-1 72.5%

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

10. Simplified 72.5%

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

11. Taylor expanded in g around -inf 72.6%

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

    1. mul-1-neg 15.6%

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

13. Simplified 72.6%

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

14. Final simplification 72.6%

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

Alternative 6: 15.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 47.1%

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

3. Simplified 47.1%

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

5. Taylor expanded in g around -inf 14.2%

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

   1. *-commutative 14.2%

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

7. Simplified 14.2%

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

8. Taylor expanded in g around inf 15.6%

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

   1. mul-1-neg 15.6%

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

   2. distribute-frac-neg 15.6%

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

10. Simplified 15.6%

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

    1. associate-*l/ 15.6%

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

    2. *-commutative 15.6%

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

    3. associate-*r* 15.6%

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

    4. metadata-eval 15.6%

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

    5. neg-mul-1 15.6%

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

    6. clear-num 15.7%

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

    7. frac-2neg 15.7%

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

    8. metadata-eval 15.7%

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

    9. add-sqr-sqrt 1.5%

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

    10. sqrt-unprod 8.6%

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

    11. sqr-neg 8.6%

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

    12. sqrt-prod 3.0%

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

    13. add-sqr-sqrt 3.0%

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

    14. distribute-frac-neg2 3.0%

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

    15. add-sqr-sqrt 0.4%

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

    16. sqrt-unprod 15.7%

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

    17. sqr-neg 15.7%

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

    18. sqrt-prod 14.3%

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

    19. add-sqr-sqrt 15.7%

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

12. Applied egg-rr 15.7%

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

13. Final simplification 15.7%

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

Alternative 7: 2.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 47.1%

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

3. Simplified 47.1%

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

5. Taylor expanded in g around -inf 14.2%

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

   1. *-commutative 14.2%

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

7. Simplified 14.2%

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

8. Taylor expanded in g around inf 15.6%

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

   1. mul-1-neg 15.6%

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

   2. distribute-frac-neg 15.6%

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

10. Simplified 15.6%

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

    1. associate-*l/ 15.6%

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

    2. *-commutative 15.6%

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

    3. associate-*r* 15.6%

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

    4. metadata-eval 15.6%

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

    5. neg-mul-1 15.6%

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

    6. add-sqr-sqrt 1.5%

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

    7. sqrt-unprod 8.4%

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

    8. sqr-neg 8.4%

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

    9. sqrt-prod 2.8%

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

    10. add-sqr-sqrt 2.6%

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

12. Applied egg-rr 2.6%

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

13. Final simplification 2.6%

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

Alternative 8: 15.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 47.1%

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

3. Simplified 47.1%

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

5. Taylor expanded in g around -inf 14.2%

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

   1. *-commutative 14.2%

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

7. Simplified 14.2%

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

8. Taylor expanded in g around inf 15.6%

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

   1. mul-1-neg 15.6%

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

   2. distribute-frac-neg 15.6%

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

10. Simplified 15.6%

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

11. Taylor expanded in g around -inf 15.6%

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

    1. mul-1-neg 15.6%

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

13. Simplified 15.6%

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

14. Final simplification 15.6%

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

Alternative 9: 1.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{g}{a}}\\ t\_0 + t\_0 \end{array} \end{array} \]

FPCore

(FPCore (g h a) :precision binary64 (let* ((t_0 (cbrt (/ g a)))) (+ t_0 t_0)))

C

double code(double g, double h, double a) {
	double t_0 = cbrt((g / a));
	return t_0 + t_0;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.cbrt((g / a));
	return t_0 + t_0;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(g / a))
	return Float64(t_0 + t_0)
end

Wolfram

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

Derivation

1. Initial program 47.1%

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

3. Simplified 47.1%

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

5. Taylor expanded in g around -inf 14.2%

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

   1. *-commutative 14.2%

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

7. Simplified 14.2%

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

8. Taylor expanded in g around inf 15.6%

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

   1. mul-1-neg 15.6%

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

   2. distribute-frac-neg 15.6%

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

10. Simplified 15.6%

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

    1. associate-*l/ 15.6%

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

    2. *-commutative 15.6%

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

    3. associate-*r* 15.6%

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

    4. metadata-eval 15.6%

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

    5. neg-mul-1 15.6%

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

    6. add-sqr-sqrt 1.5%

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

    7. sqrt-unprod 8.4%

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

    8. sqr-neg 8.4%

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

    9. sqrt-prod 2.8%

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

    10. add-sqr-sqrt 2.6%

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

12. Applied egg-rr 2.6%

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

    1. div-inv 2.9%

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

    2. neg-mul-1 2.9%

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

    3. metadata-eval 2.9%

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

    4. associate-*r* 2.9%

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

    5. *-commutative 2.9%

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

    6. div-inv 2.6%

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

    7. associate-*l/ 2.9%

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

    8. *-un-lft-identity 2.9%

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

    9. associate-*l/ 2.6%

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

    10. *-commutative 2.6%

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

    11. associate-*r* 2.6%

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

    12. metadata-eval 2.6%

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

    13. neg-mul-1 2.6%

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

    14. add-sqr-sqrt 0.2%

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

    15. sqrt-unprod 1.4%

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

    16. sqr-neg 1.4%

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

    17. sqrt-prod 1.3%

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

    18. add-sqr-sqrt 1.4%

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

14. Applied egg-rr 1.4%

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

    1. *-lft-identity 1.4%

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

16. Simplified 1.4%

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

17. Final simplification 1.4%

    \[\leadsto \sqrt[3]{\frac{g}{a}} + \sqrt[3]{\frac{g}{a}} \]
18. Add Preprocessing

Reproduce

herbie shell --seed 2024073 
(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))))))))