2-ancestry mixing, positive discriminant

Percentage Accurate: 44.7% → 47.2%

Time: 8.7s

Alternatives: 8

Speedup: 1.3×

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: 44.7% 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: 47.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := h \cdot \left(-h\right)\\ t_1 := \sqrt{\mathsf{fma}\left(g, g, t_0\right)}\\ \mathbf{if}\;h \cdot h \leq 0:\\ \;\;\;\;\sqrt[3]{\left(g - \sqrt{\left(h + g\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + \mathsf{hypot}\left(g, \sqrt{t_0}\right)\right)}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(t_1 - g\right)} + \sqrt[3]{\frac{g + t_1}{\frac{a}{-0.5}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (* h (- h))) (t_1 (sqrt (fma g g t_0))))
   (if (<= (* h h) 0.0)
     (+
      (cbrt (* (- g (sqrt (* (+ h g) (- g h)))) (/ -0.5 a)))
      (/ (cbrt (* -0.5 (+ g (hypot g (sqrt t_0))))) (cbrt a)))
     (+ (cbrt (* (/ 0.5 a) (- t_1 g))) (cbrt (/ (+ g t_1) (/ a -0.5)))))))

C

double code(double g, double h, double a) {
	double t_0 = h * -h;
	double t_1 = sqrt(fma(g, g, t_0));
	double tmp;
	if ((h * h) <= 0.0) {
		tmp = cbrt(((g - sqrt(((h + g) * (g - h)))) * (-0.5 / a))) + (cbrt((-0.5 * (g + hypot(g, sqrt(t_0))))) / cbrt(a));
	} else {
		tmp = cbrt(((0.5 / a) * (t_1 - g))) + cbrt(((g + t_1) / (a / -0.5)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64(h * Float64(-h))
	t_1 = sqrt(fma(g, g, t_0))
	tmp = 0.0
	if (Float64(h * h) <= 0.0)
		tmp = Float64(cbrt(Float64(Float64(g - sqrt(Float64(Float64(h + g) * Float64(g - h)))) * Float64(-0.5 / a))) + Float64(cbrt(Float64(-0.5 * Float64(g + hypot(g, sqrt(t_0))))) / cbrt(a)));
	else
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(t_1 - g))) + cbrt(Float64(Float64(g + t_1) / Float64(a / -0.5))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(h * (-h)), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(g * g + t$95$0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(h * h), $MachinePrecision], 0.0], N[(N[Power[N[(N[(g - N[Sqrt[N[(N[(h + g), $MachinePrecision] * N[(g - h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(-0.5 * N[(g + N[Sqrt[g ^ 2 + N[Sqrt[t$95$0], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(t$95$1 - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(g + t$95$1), $MachinePrecision] / N[(a / -0.5), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 h h) < 0.0

   1. Initial program 48.1%

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

   3. Simplified 48.1%

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

      1. associate-*r/ 48.1%

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

      2. cbrt-div 52.7%

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

      3. difference-of-squares 52.7%

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

      4. sub-neg 52.7%

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

      5. add-sqr-sqrt 52.7%

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

      6. hypot-def 57.7%

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

      7. distribute-rgt-neg-in 57.7%

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

   5. Applied egg-rr 57.7%

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

   if 0.0 < (*.f64 h h)

   1. Initial program 33.2%

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

      1. associate-/r* 33.2%

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

      2. metadata-eval 33.2%

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

      3. +-commutative 33.2%

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

      4. unsub-neg 33.2%

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

      5. fma-neg 33.2%

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

      6. sub-neg 33.2%

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

      7. distribute-neg-out 33.2%

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

      8. neg-mul-1 33.2%

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

      9. associate-*r* 33.2%

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

   3. Simplified 33.2%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \cdot h \leq 0:\\ \;\;\;\;\sqrt[3]{\left(g - \sqrt{\left(h + g\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + \mathsf{hypot}\left(g, \sqrt{h \cdot \left(-h\right)}\right)\right)}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(g, g, h \cdot \left(-h\right)\right)} - g\right)} + \sqrt[3]{\frac{g + \sqrt{\mathsf{fma}\left(g, g, h \cdot \left(-h\right)\right)}}{\frac{a}{-0.5}}}\\ \end{array} \]

Alternative 2: 47.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ \mathbf{if}\;h \cdot h \leq 0:\\ \;\;\;\;\sqrt[3]{\left(g - \sqrt{\left(h + g\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + \mathsf{hypot}\left(g, \sqrt{h \cdot \left(-h\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(t_0 - g\right)} + \sqrt[3]{\left(g + t_0\right) \cdot \frac{-0.5}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (sqrt (- (* g g) (* h h)))))
   (if (<= (* h h) 0.0)
     (+
      (cbrt (* (- g (sqrt (* (+ h g) (- g h)))) (/ -0.5 a)))
      (* (cbrt (/ -0.5 a)) (cbrt (+ g (hypot g (sqrt (* h (- h))))))))
     (+ (cbrt (* (/ 0.5 a) (- t_0 g))) (cbrt (* (+ g t_0) (/ (- 0.5) a)))))))

C

double code(double g, double h, double a) {
	double t_0 = sqrt(((g * g) - (h * h)));
	double tmp;
	if ((h * h) <= 0.0) {
		tmp = cbrt(((g - sqrt(((h + g) * (g - h)))) * (-0.5 / a))) + (cbrt((-0.5 / a)) * cbrt((g + hypot(g, sqrt((h * -h))))));
	} else {
		tmp = cbrt(((0.5 / a) * (t_0 - g))) + cbrt(((g + t_0) * (-0.5 / 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 ((h * h) <= 0.0) {
		tmp = Math.cbrt(((g - Math.sqrt(((h + g) * (g - h)))) * (-0.5 / a))) + (Math.cbrt((-0.5 / a)) * Math.cbrt((g + Math.hypot(g, Math.sqrt((h * -h))))));
	} else {
		tmp = Math.cbrt(((0.5 / a) * (t_0 - g))) + Math.cbrt(((g + t_0) * (-0.5 / a)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	tmp = 0.0
	if (Float64(h * h) <= 0.0)
		tmp = Float64(cbrt(Float64(Float64(g - sqrt(Float64(Float64(h + g) * Float64(g - h)))) * Float64(-0.5 / a))) + Float64(cbrt(Float64(-0.5 / a)) * cbrt(Float64(g + hypot(g, sqrt(Float64(h * Float64(-h))))))));
	else
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(t_0 - g))) + cbrt(Float64(Float64(g + t_0) * Float64(Float64(-0.5) / 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[(h * h), $MachinePrecision], 0.0], N[(N[Power[N[(N[(g - N[Sqrt[N[(N[(h + g), $MachinePrecision] * N[(g - h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(-0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(g + N[Sqrt[g ^ 2 + N[Sqrt[N[(h * (-h)), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(t$95$0 - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(g + t$95$0), $MachinePrecision] * N[((-0.5) / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 h h) < 0.0

   1. Initial program 48.1%

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

   3. Simplified 48.1%

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

      1. cbrt-prod 52.7%

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

      2. difference-of-squares 52.7%

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

      3. fma-neg 52.7%

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

      4. fma-neg 52.7%

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

      5. difference-of-squares 52.7%

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

      6. difference-of-squares 52.7%

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

      7. sub-neg 52.7%

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

      8. add-sqr-sqrt 52.7%

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

      9. hypot-def 57.7%

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

      10. distribute-rgt-neg-in 57.7%

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

   5. Applied egg-rr 57.7%

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

      1. *-commutative 57.7%

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

   7. Simplified 57.7%

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

   if 0.0 < (*.f64 h h)

   1. Initial program 33.2%

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

      1. associate-/r* 33.2%

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

      2. metadata-eval 33.2%

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

      3. +-commutative 33.2%

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

      4. unsub-neg 33.2%

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

      5. associate-/r* 33.2%

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

      6. metadata-eval 33.2%

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

   3. Simplified 33.2%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \cdot h \leq 0:\\ \;\;\;\;\sqrt[3]{\left(g - \sqrt{\left(h + g\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + \mathsf{hypot}\left(g, \sqrt{h \cdot \left(-h\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}\\ \end{array} \]

Alternative 3: 47.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ \mathbf{if}\;h \cdot h \leq 0:\\ \;\;\;\;\sqrt[3]{\left(g - \sqrt{\left(h + g\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + \mathsf{hypot}\left(g, \sqrt{h \cdot \left(-h\right)}\right)\right)}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(t_0 - g\right)} + \sqrt[3]{\left(g + t_0\right) \cdot \frac{-0.5}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (sqrt (- (* g g) (* h h)))))
   (if (<= (* h h) 0.0)
     (+
      (cbrt (* (- g (sqrt (* (+ h g) (- g h)))) (/ -0.5 a)))
      (/ (cbrt (* -0.5 (+ g (hypot g (sqrt (* h (- h))))))) (cbrt a)))
     (+ (cbrt (* (/ 0.5 a) (- t_0 g))) (cbrt (* (+ g t_0) (/ (- 0.5) a)))))))

C

double code(double g, double h, double a) {
	double t_0 = sqrt(((g * g) - (h * h)));
	double tmp;
	if ((h * h) <= 0.0) {
		tmp = cbrt(((g - sqrt(((h + g) * (g - h)))) * (-0.5 / a))) + (cbrt((-0.5 * (g + hypot(g, sqrt((h * -h)))))) / cbrt(a));
	} else {
		tmp = cbrt(((0.5 / a) * (t_0 - g))) + cbrt(((g + t_0) * (-0.5 / 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 ((h * h) <= 0.0) {
		tmp = Math.cbrt(((g - Math.sqrt(((h + g) * (g - h)))) * (-0.5 / a))) + (Math.cbrt((-0.5 * (g + Math.hypot(g, Math.sqrt((h * -h)))))) / Math.cbrt(a));
	} else {
		tmp = Math.cbrt(((0.5 / a) * (t_0 - g))) + Math.cbrt(((g + t_0) * (-0.5 / a)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	tmp = 0.0
	if (Float64(h * h) <= 0.0)
		tmp = Float64(cbrt(Float64(Float64(g - sqrt(Float64(Float64(h + g) * Float64(g - h)))) * Float64(-0.5 / a))) + Float64(cbrt(Float64(-0.5 * Float64(g + hypot(g, sqrt(Float64(h * Float64(-h))))))) / cbrt(a)));
	else
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(t_0 - g))) + cbrt(Float64(Float64(g + t_0) * Float64(Float64(-0.5) / 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[(h * h), $MachinePrecision], 0.0], N[(N[Power[N[(N[(g - N[Sqrt[N[(N[(h + g), $MachinePrecision] * N[(g - h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(-0.5 * N[(g + N[Sqrt[g ^ 2 + N[Sqrt[N[(h * (-h)), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(t$95$0 - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(g + t$95$0), $MachinePrecision] * N[((-0.5) / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 h h) < 0.0

   1. Initial program 48.1%

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

   3. Simplified 48.1%

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

      1. associate-*r/ 48.1%

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

      2. cbrt-div 52.7%

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

      3. difference-of-squares 52.7%

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

      4. sub-neg 52.7%

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

      5. add-sqr-sqrt 52.7%

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

      6. hypot-def 57.7%

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

      7. distribute-rgt-neg-in 57.7%

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

   5. Applied egg-rr 57.7%

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

   if 0.0 < (*.f64 h h)

   1. Initial program 33.2%

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

      1. associate-/r* 33.2%

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

      2. metadata-eval 33.2%

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

      3. +-commutative 33.2%

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

      4. unsub-neg 33.2%

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

      5. associate-/r* 33.2%

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

      6. metadata-eval 33.2%

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

   3. Simplified 33.2%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \cdot h \leq 0:\\ \;\;\;\;\sqrt[3]{\left(g - \sqrt{\left(h + g\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + \mathsf{hypot}\left(g, \sqrt{h \cdot \left(-h\right)}\right)\right)}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}\\ \end{array} \]

Alternative 4: 46.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if g < -2e-176

   1. Initial program 43.2%

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

      1. associate-/r* 43.2%

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

      2. metadata-eval 43.2%

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

      3. +-commutative 43.2%

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

      4. unsub-neg 43.2%

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

      5. associate-/r* 43.2%

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

      6. metadata-eval 43.2%

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

   3. Simplified 43.2%

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

   4. Taylor expanded in g around -inf 46.4%

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

      1. unpow2 46.4%

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

   6. Simplified 46.4%

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

   if -2e-176 < g

   1. Initial program 36.9%

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

   3. Simplified 36.9%

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

   4. Taylor expanded in g around inf 38.2%

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

4. Final simplification 42.0%

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

Alternative 5: 45.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;g \leq -8.5 \cdot 10^{-182}:\\ \;\;\;\;\sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{0.5}{a} \cdot \left(\left(-g\right) - g\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{-0.5}{a} \cdot \left(g - g\right)} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + \sqrt{\left(h + g\right) \cdot \left(g - h\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= g -8.5e-182)
   (+
    (cbrt (* (+ g (sqrt (- (* g g) (* h h)))) (/ (- 0.5) a)))
    (cbrt (* (/ 0.5 a) (- (- g) g))))
   (+
    (cbrt (* (/ -0.5 a) (- g g)))
    (cbrt (* (/ -0.5 a) (+ g (sqrt (* (+ h g) (- g h)))))))))

C

double code(double g, double h, double a) {
	double tmp;
	if (g <= -8.5e-182) {
		tmp = cbrt(((g + sqrt(((g * g) - (h * h)))) * (-0.5 / a))) + cbrt(((0.5 / a) * (-g - g)));
	} else {
		tmp = cbrt(((-0.5 / a) * (g - g))) + cbrt(((-0.5 / a) * (g + sqrt(((h + g) * (g - h))))));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if (g <= -8.5e-182) {
		tmp = Math.cbrt(((g + Math.sqrt(((g * g) - (h * h)))) * (-0.5 / a))) + Math.cbrt(((0.5 / a) * (-g - g)));
	} else {
		tmp = Math.cbrt(((-0.5 / a) * (g - g))) + Math.cbrt(((-0.5 / a) * (g + Math.sqrt(((h + g) * (g - h))))));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (g <= -8.5e-182)
		tmp = Float64(cbrt(Float64(Float64(g + sqrt(Float64(Float64(g * g) - Float64(h * h)))) * Float64(Float64(-0.5) / a))) + cbrt(Float64(Float64(0.5 / a) * Float64(Float64(-g) - g))));
	else
		tmp = Float64(cbrt(Float64(Float64(-0.5 / a) * Float64(g - g))) + cbrt(Float64(Float64(-0.5 / a) * Float64(g + sqrt(Float64(Float64(h + g) * Float64(g - h)))))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[g, -8.5e-182], N[(N[Power[N[(N[(g + N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[((-0.5) / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[((-g) - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(-0.5 / a), $MachinePrecision] * N[(g - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(-0.5 / a), $MachinePrecision] * N[(g + N[Sqrt[N[(N[(h + g), $MachinePrecision] * N[(g - h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if g < -8.5000000000000001e-182

   1. Initial program 42.9%

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

      1. associate-/r* 42.9%

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

      2. metadata-eval 42.9%

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

      3. +-commutative 42.9%

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

      4. unsub-neg 42.9%

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

      5. associate-/r* 42.9%

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

      6. metadata-eval 42.9%

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

   3. Simplified 42.9%

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

   4. Taylor expanded in g around -inf 43.1%

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

      1. neg-mul-1 43.1%

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

   6. Simplified 43.1%

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

   if -8.5000000000000001e-182 < g

   1. Initial program 37.1%

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

   3. Simplified 37.1%

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

   4. Taylor expanded in g around inf 38.4%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -8.5 \cdot 10^{-182}:\\ \;\;\;\;\sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{0.5}{a} \cdot \left(\left(-g\right) - g\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{-0.5}{a} \cdot \left(g - g\right)} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + \sqrt{\left(h + g\right) \cdot \left(g - h\right)}\right)}\\ \end{array} \]

Alternative 6: 45.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;g \leq -2 \cdot 10^{-155}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{-0.5}{a} \cdot \left(g - g\right)} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + \sqrt{\left(h + g\right) \cdot \left(g - h\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= g -2e-155)
   (+
    (cbrt (* (/ 0.5 a) (- (sqrt (- (* g g) (* h h))) g)))
    (cbrt (* (/ 0.5 a) (- g g))))
   (+
    (cbrt (* (/ -0.5 a) (- g g)))
    (cbrt (* (/ -0.5 a) (+ g (sqrt (* (+ h g) (- g h)))))))))

C

double code(double g, double h, double a) {
	double tmp;
	if (g <= -2e-155) {
		tmp = cbrt(((0.5 / a) * (sqrt(((g * g) - (h * h))) - g))) + cbrt(((0.5 / a) * (g - g)));
	} else {
		tmp = cbrt(((-0.5 / a) * (g - g))) + cbrt(((-0.5 / a) * (g + sqrt(((h + g) * (g - h))))));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if (g <= -2e-155) {
		tmp = Math.cbrt(((0.5 / a) * (Math.sqrt(((g * g) - (h * h))) - g))) + Math.cbrt(((0.5 / a) * (g - g)));
	} else {
		tmp = Math.cbrt(((-0.5 / a) * (g - g))) + Math.cbrt(((-0.5 / a) * (g + Math.sqrt(((h + g) * (g - h))))));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (g <= -2e-155)
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(sqrt(Float64(Float64(g * g) - Float64(h * h))) - g))) + cbrt(Float64(Float64(0.5 / a) * Float64(g - g))));
	else
		tmp = Float64(cbrt(Float64(Float64(-0.5 / a) * Float64(g - g))) + cbrt(Float64(Float64(-0.5 / a) * Float64(g + sqrt(Float64(Float64(h + g) * Float64(g - h)))))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[g, -2e-155], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(g - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(-0.5 / a), $MachinePrecision] * N[(g - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(-0.5 / a), $MachinePrecision] * N[(g + N[Sqrt[N[(N[(h + g), $MachinePrecision] * N[(g - h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if g < -2.00000000000000003e-155

   1. Initial program 43.4%

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

      1. associate-/r* 43.4%

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

      2. metadata-eval 43.4%

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

      3. +-commutative 43.4%

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

      4. unsub-neg 43.4%

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

      5. associate-/r* 43.4%

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

      6. metadata-eval 43.4%

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

   3. Simplified 43.4%

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

   4. Taylor expanded in g around -inf 45.1%

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

      1. neg-mul-1 43.5%

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

   6. Simplified 45.1%

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

   if -2.00000000000000003e-155 < g

   1. Initial program 36.8%

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

   3. Simplified 36.8%

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

   4. Taylor expanded in g around inf 38.1%

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

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -2 \cdot 10^{-155}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{-0.5}{a} \cdot \left(g - g\right)} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + \sqrt{\left(h + g\right) \cdot \left(g - h\right)}\right)}\\ \end{array} \]

Alternative 7: 25.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Derivation

1. Initial program 39.8%

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

3. Simplified 39.8%

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

4. Taylor expanded in g around inf 21.6%

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

5. Final simplification 21.6%

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

Alternative 8: 28.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 39.8%

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

3. Simplified 39.8%

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

4. Taylor expanded in g around inf 24.7%

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

5. Final simplification 24.7%

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

Reproduce

herbie shell --seed 2023258 
(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))))))))