2-ancestry mixing, positive discriminant

Percentage Accurate: 44.2% → 77.7%

Time: 22.2s

Alternatives: 5

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 44.2% 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: 77.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ t_1 := t_0 - g\\ t_2 := \sqrt[3]{\frac{1}{2 \cdot a} \cdot t_1} + \sqrt[3]{\left(g + t_0\right) \cdot \frac{-1}{2 \cdot a}}\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{-100} \lor \neg \left(t_2 \leq 0\right):\\ \;\;\;\;\sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\left(0.5 \cdot \frac{h}{\frac{g}{h}}\right) \cdot \frac{-0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{t_1 \cdot \frac{0.5}{a}} - \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot \frac{\sqrt[3]{g}}{-\sqrt[3]{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (sqrt (- (* g g) (* h h))))
        (t_1 (- t_0 g))
        (t_2
         (+
          (cbrt (* (/ 1.0 (* 2.0 a)) t_1))
          (cbrt (* (+ g t_0) (/ -1.0 (* 2.0 a)))))))
   (if (or (<= t_2 -5e-100) (not (<= t_2 0.0)))
     (+
      (cbrt (* (+ g g) (/ (- 0.5) a)))
      (cbrt (* (* 0.5 (/ h (/ g h))) (/ -0.5 a))))
     (-
      (cbrt (* t_1 (/ 0.5 a)))
      (* (* (cbrt -0.5) (cbrt 2.0)) (/ (cbrt g) (- (cbrt a))))))))

C

double code(double g, double h, double a) {
	double t_0 = sqrt(((g * g) - (h * h)));
	double t_1 = t_0 - g;
	double t_2 = cbrt(((1.0 / (2.0 * a)) * t_1)) + cbrt(((g + t_0) * (-1.0 / (2.0 * a))));
	double tmp;
	if ((t_2 <= -5e-100) || !(t_2 <= 0.0)) {
		tmp = cbrt(((g + g) * (-0.5 / a))) + cbrt(((0.5 * (h / (g / h))) * (-0.5 / a)));
	} else {
		tmp = cbrt((t_1 * (0.5 / a))) - ((cbrt(-0.5) * cbrt(2.0)) * (cbrt(g) / -cbrt(a)));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.sqrt(((g * g) - (h * h)));
	double t_1 = t_0 - g;
	double t_2 = Math.cbrt(((1.0 / (2.0 * a)) * t_1)) + Math.cbrt(((g + t_0) * (-1.0 / (2.0 * a))));
	double tmp;
	if ((t_2 <= -5e-100) || !(t_2 <= 0.0)) {
		tmp = Math.cbrt(((g + g) * (-0.5 / a))) + Math.cbrt(((0.5 * (h / (g / h))) * (-0.5 / a)));
	} else {
		tmp = Math.cbrt((t_1 * (0.5 / a))) - ((Math.cbrt(-0.5) * Math.cbrt(2.0)) * (Math.cbrt(g) / -Math.cbrt(a)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	t_1 = Float64(t_0 - g)
	t_2 = Float64(cbrt(Float64(Float64(1.0 / Float64(2.0 * a)) * t_1)) + cbrt(Float64(Float64(g + t_0) * Float64(-1.0 / Float64(2.0 * a)))))
	tmp = 0.0
	if ((t_2 <= -5e-100) || !(t_2 <= 0.0))
		tmp = Float64(cbrt(Float64(Float64(g + g) * Float64(Float64(-0.5) / a))) + cbrt(Float64(Float64(0.5 * Float64(h / Float64(g / h))) * Float64(-0.5 / a))));
	else
		tmp = Float64(cbrt(Float64(t_1 * Float64(0.5 / a))) - Float64(Float64(cbrt(-0.5) * cbrt(2.0)) * Float64(cbrt(g) / Float64(-cbrt(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]}, Block[{t$95$1 = N[(t$95$0 - g), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[(N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(g + t$95$0), $MachinePrecision] * N[(-1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$2, -5e-100], N[Not[LessEqual[t$95$2, 0.0]], $MachinePrecision]], N[(N[Power[N[(N[(g + g), $MachinePrecision] * N[((-0.5) / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(0.5 * N[(h / N[(g / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(t$95$1 * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] - N[(N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[Power[g, 1/3], $MachinePrecision] / (-N[Power[a, 1/3], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < -5.0000000000000001e-100 or 0.0 < (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))))

   1. Initial program 44.9%

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

   3. Simplified 44.9%

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

   4. Taylor expanded in g around -inf 27.7%

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

      1. mul-1-neg 27.7%

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

   6. Simplified 27.7%

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

   7. Taylor expanded in g around -inf 73.5%

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

      1. unpow2 73.5%

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

      2. associate-/l* 76.6%

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

   9. Simplified 76.6%

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

   if -5.0000000000000001e-100 < (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < 0.0

   1. Initial program 18.9%

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

   3. Simplified 26.8%

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

   4. Taylor expanded in h around 0 18.0%

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

      1. unpow1/3 20.4%

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

      2. *-lft-identity 20.4%

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

   6. Simplified 20.4%

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

      1. cbrt-div 92.2%

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

      2. frac-2neg 92.2%

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

   8. Applied egg-rr 92.2%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} \leq -5 \cdot 10^{-100} \lor \neg \left(\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} \leq 0\right):\\ \;\;\;\;\sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\left(0.5 \cdot \frac{h}{\frac{g}{h}}\right) \cdot \frac{-0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} - g\right) \cdot \frac{0.5}{a}} - \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot \frac{\sqrt[3]{g}}{-\sqrt[3]{a}}\\ \end{array} \]

Alternative 2: 77.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ t_1 := t_0 - g\\ t_2 := \sqrt[3]{\frac{1}{2 \cdot a} \cdot t_1} + \sqrt[3]{\left(g + t_0\right) \cdot \frac{-1}{2 \cdot a}}\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{-100} \lor \neg \left(t_2 \leq 0\right):\\ \;\;\;\;\sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\left(0.5 \cdot \frac{h}{\frac{g}{h}}\right) \cdot \frac{-0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{t_1 \cdot \frac{0.5}{a}} + \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (sqrt (- (* g g) (* h h))))
        (t_1 (- t_0 g))
        (t_2
         (+
          (cbrt (* (/ 1.0 (* 2.0 a)) t_1))
          (cbrt (* (+ g t_0) (/ -1.0 (* 2.0 a)))))))
   (if (or (<= t_2 -5e-100) (not (<= t_2 0.0)))
     (+
      (cbrt (* (+ g g) (/ (- 0.5) a)))
      (cbrt (* (* 0.5 (/ h (/ g h))) (/ -0.5 a))))
     (+
      (cbrt (* t_1 (/ 0.5 a)))
      (* (* (cbrt -0.5) (cbrt 2.0)) (* (cbrt g) (cbrt (/ 1.0 a))))))))

C

double code(double g, double h, double a) {
	double t_0 = sqrt(((g * g) - (h * h)));
	double t_1 = t_0 - g;
	double t_2 = cbrt(((1.0 / (2.0 * a)) * t_1)) + cbrt(((g + t_0) * (-1.0 / (2.0 * a))));
	double tmp;
	if ((t_2 <= -5e-100) || !(t_2 <= 0.0)) {
		tmp = cbrt(((g + g) * (-0.5 / a))) + cbrt(((0.5 * (h / (g / h))) * (-0.5 / a)));
	} else {
		tmp = cbrt((t_1 * (0.5 / a))) + ((cbrt(-0.5) * cbrt(2.0)) * (cbrt(g) * cbrt((1.0 / a))));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.sqrt(((g * g) - (h * h)));
	double t_1 = t_0 - g;
	double t_2 = Math.cbrt(((1.0 / (2.0 * a)) * t_1)) + Math.cbrt(((g + t_0) * (-1.0 / (2.0 * a))));
	double tmp;
	if ((t_2 <= -5e-100) || !(t_2 <= 0.0)) {
		tmp = Math.cbrt(((g + g) * (-0.5 / a))) + Math.cbrt(((0.5 * (h / (g / h))) * (-0.5 / a)));
	} else {
		tmp = Math.cbrt((t_1 * (0.5 / a))) + ((Math.cbrt(-0.5) * Math.cbrt(2.0)) * (Math.cbrt(g) * Math.cbrt((1.0 / a))));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	t_1 = Float64(t_0 - g)
	t_2 = Float64(cbrt(Float64(Float64(1.0 / Float64(2.0 * a)) * t_1)) + cbrt(Float64(Float64(g + t_0) * Float64(-1.0 / Float64(2.0 * a)))))
	tmp = 0.0
	if ((t_2 <= -5e-100) || !(t_2 <= 0.0))
		tmp = Float64(cbrt(Float64(Float64(g + g) * Float64(Float64(-0.5) / a))) + cbrt(Float64(Float64(0.5 * Float64(h / Float64(g / h))) * Float64(-0.5 / a))));
	else
		tmp = Float64(cbrt(Float64(t_1 * Float64(0.5 / a))) + Float64(Float64(cbrt(-0.5) * cbrt(2.0)) * Float64(cbrt(g) * cbrt(Float64(1.0 / a)))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - g), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[(N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(g + t$95$0), $MachinePrecision] * N[(-1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$2, -5e-100], N[Not[LessEqual[t$95$2, 0.0]], $MachinePrecision]], N[(N[Power[N[(N[(g + g), $MachinePrecision] * N[((-0.5) / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(0.5 * N[(h / N[(g / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(t$95$1 * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[Power[g, 1/3], $MachinePrecision] * N[Power[N[(1.0 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < -5.0000000000000001e-100 or 0.0 < (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))))

   1. Initial program 44.9%

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

   3. Simplified 44.9%

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

   4. Taylor expanded in g around -inf 27.7%

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

      1. mul-1-neg 27.7%

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

   6. Simplified 27.7%

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

   7. Taylor expanded in g around -inf 73.5%

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

      1. unpow2 73.5%

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

      2. associate-/l* 76.6%

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

   9. Simplified 76.6%

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

   if -5.0000000000000001e-100 < (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < 0.0

   1. Initial program 18.9%

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

   3. Simplified 26.8%

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

   4. Taylor expanded in h around 0 18.0%

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

      1. unpow1/3 20.4%

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

      2. *-lft-identity 20.4%

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

   6. Simplified 20.4%

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

      1. div-inv 20.4%

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

      2. cbrt-prod 92.2%

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

   8. Applied egg-rr 92.2%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} \leq -5 \cdot 10^{-100} \lor \neg \left(\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} \leq 0\right):\\ \;\;\;\;\sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\left(0.5 \cdot \frac{h}{\frac{g}{h}}\right) \cdot \frac{-0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} - g\right) \cdot \frac{0.5}{a}} + \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right)\\ \end{array} \]

Alternative 3: 74.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Derivation

1. Initial program 43.7%

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

3. Simplified 44.1%

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

4. Taylor expanded in g around -inf 26.8%

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

   1. mul-1-neg 26.8%

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

6. Simplified 26.8%

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

7. Taylor expanded in g around -inf 71.0%

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

   1. unpow2 71.0%

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

   2. associate-/l* 74.0%

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

9. Simplified 74.0%

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

10. Final simplification 74.0%

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

Alternative 4: 15.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(g + g) * Float64(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 + g), $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 43.7%

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

3. Simplified 44.1%

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

4. Taylor expanded in g around -inf 26.8%

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

   1. mul-1-neg 26.8%

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

6. Simplified 26.8%

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

7. Taylor expanded in g around inf 15.1%

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

8. Final simplification 15.1%

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

Alternative 5: 73.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(g + g) * Float64(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 + g), $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 43.7%

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

3. Simplified 44.1%

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

4. Taylor expanded in g around -inf 26.8%

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

   1. mul-1-neg 26.8%

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

6. Simplified 26.8%

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

7. Taylor expanded in g around -inf 72.8%

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

   1. mul-1-neg 26.8%

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

9. Simplified 72.8%

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

10. Final simplification 72.8%

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

Reproduce

herbie shell --seed 2023274 
(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))))))))