2-ancestry mixing, positive discriminant

Percentage Accurate: 43.3% → 72.3%

Time: 10.6s

Alternatives: 6

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: 43.3% 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: 72.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{2 \cdot a}\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ t_2 := \sqrt[3]{\left(g + t\_1\right) \cdot \frac{-1}{2 \cdot a}}\\ \mathbf{if}\;\sqrt[3]{t\_0 \cdot \left(t\_1 - g\right)} + t\_2 \leq -\infty:\\ \;\;\;\;t\_2 + g \cdot \mathsf{fma}\left(0.16666666666666666, \frac{\sqrt[3]{\frac{1}{a \cdot {g}^{8}}} \cdot \left(\left(h \cdot h\right) \cdot \sqrt[3]{0.5}\right)}{{\left(\sqrt[3]{-2}\right)}^{2}}, \sqrt[3]{\frac{1}{a \cdot \left(g \cdot g\right)}} \cdot \left(\sqrt[3]{0.5} \cdot \sqrt[3]{-2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{t\_0 \cdot \frac{\left(h \cdot h\right) \cdot -0.5}{g}} + \sqrt[3]{t\_0 \cdot \left(g \cdot -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* 2.0 a)))
        (t_1 (sqrt (- (* g g) (* h h))))
        (t_2 (cbrt (* (+ g t_1) (/ -1.0 (* 2.0 a))))))
   (if (<= (+ (cbrt (* t_0 (- t_1 g))) t_2) (- INFINITY))
     (+
      t_2
      (*
       g
       (fma
        0.16666666666666666
        (/
         (* (cbrt (/ 1.0 (* a (pow g 8.0)))) (* (* h h) (cbrt 0.5)))
         (pow (cbrt -2.0) 2.0))
        (* (cbrt (/ 1.0 (* a (* g g)))) (* (cbrt 0.5) (cbrt -2.0))))))
     (+ (cbrt (* t_0 (/ (* (* h h) -0.5) g))) (cbrt (* t_0 (* g -2.0)))))))

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)));
	double t_2 = cbrt(((g + t_1) * (-1.0 / (2.0 * a))));
	double tmp;
	if ((cbrt((t_0 * (t_1 - g))) + t_2) <= -((double) INFINITY)) {
		tmp = t_2 + (g * fma(0.16666666666666666, ((cbrt((1.0 / (a * pow(g, 8.0)))) * ((h * h) * cbrt(0.5))) / pow(cbrt(-2.0), 2.0)), (cbrt((1.0 / (a * (g * g)))) * (cbrt(0.5) * cbrt(-2.0)))));
	} else {
		tmp = cbrt((t_0 * (((h * h) * -0.5) / g))) + cbrt((t_0 * (g * -2.0)));
	}
	return tmp;
}

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)))
	t_2 = cbrt(Float64(Float64(g + t_1) * Float64(-1.0 / Float64(2.0 * a))))
	tmp = 0.0
	if (Float64(cbrt(Float64(t_0 * Float64(t_1 - g))) + t_2) <= Float64(-Inf))
		tmp = Float64(t_2 + Float64(g * fma(0.16666666666666666, Float64(Float64(cbrt(Float64(1.0 / Float64(a * (g ^ 8.0)))) * Float64(Float64(h * h) * cbrt(0.5))) / (cbrt(-2.0) ^ 2.0)), Float64(cbrt(Float64(1.0 / Float64(a * Float64(g * g)))) * Float64(cbrt(0.5) * cbrt(-2.0))))));
	else
		tmp = Float64(cbrt(Float64(t_0 * Float64(Float64(Float64(h * h) * -0.5) / g))) + cbrt(Float64(t_0 * Float64(g * -2.0))));
	end
	return tmp
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]}, Block[{t$95$2 = N[Power[N[(N[(g + t$95$1), $MachinePrecision] * N[(-1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(t$95$0 * N[(t$95$1 - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + t$95$2), $MachinePrecision], (-Infinity)], N[(t$95$2 + N[(g * N[(0.16666666666666666 * N[(N[(N[Power[N[(1.0 / N[(a * N[Power[g, 8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(N[(h * h), $MachinePrecision] * N[Power[0.5, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Power[-2.0, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(1.0 / N[(a * N[(g * g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(N[Power[0.5, 1/3], $MachinePrecision] * N[Power[-2.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(t$95$0 * N[(N[(N[(h * h), $MachinePrecision] * -0.5), $MachinePrecision] / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(t$95$0 * N[(g * -2.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < -inf.0

   1. Initial program 4.4%

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

   3. Taylor expanded in g around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 4.4

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

   5. Simplified 4.4%

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

   6. Taylor expanded in g around inf

      \[\leadsto \color{blue}{g \cdot \left(\frac{1}{6} \cdot \left(\sqrt[3]{\frac{1}{a \cdot {g}^{8}}} \cdot \frac{{h}^{2} \cdot \sqrt[3]{\frac{1}{2}}}{{\left(\sqrt[3]{-2}\right)}^{2}}\right) + \sqrt[3]{\frac{1}{a \cdot {g}^{2}}} \cdot \left(\sqrt[3]{-2} \cdot \sqrt[3]{\frac{1}{2}}\right)\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(\mathsf{neg}\left(g\right)\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{g \cdot \left(\frac{1}{6} \cdot \left(\sqrt[3]{\frac{1}{a \cdot {g}^{8}}} \cdot \frac{{h}^{2} \cdot \sqrt[3]{\frac{1}{2}}}{{\left(\sqrt[3]{-2}\right)}^{2}}\right) + \sqrt[3]{\frac{1}{a \cdot {g}^{2}}} \cdot \left(\sqrt[3]{-2} \cdot \sqrt[3]{\frac{1}{2}}\right)\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(\mathsf{neg}\left(g\right)\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto g \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6}, \sqrt[3]{\frac{1}{a \cdot {g}^{8}}} \cdot \frac{{h}^{2} \cdot \sqrt[3]{\frac{1}{2}}}{{\left(\sqrt[3]{-2}\right)}^{2}}, \sqrt[3]{\frac{1}{a \cdot {g}^{2}}} \cdot \left(\sqrt[3]{-2} \cdot \sqrt[3]{\frac{1}{2}}\right)\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(\mathsf{neg}\left(g\right)\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   8. Simplified 97.8%

      \[\leadsto \color{blue}{g \cdot \mathsf{fma}\left(0.16666666666666666, \frac{\sqrt[3]{\frac{1}{a \cdot {g}^{8}}} \cdot \left(\left(h \cdot h\right) \cdot \sqrt[3]{0.5}\right)}{{\left(\sqrt[3]{-2}\right)}^{2}}, \sqrt[3]{\frac{1}{a \cdot \left(g \cdot g\right)}} \cdot \left(\sqrt[3]{0.5} \cdot \sqrt[3]{-2}\right)\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   if -inf.0 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))))

   1. Initial program 42.2%

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

   3. Taylor expanded in g around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 25.5

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

   5. Simplified 25.5%

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

   6. Taylor expanded in g around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 72.3

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

   8. Simplified 72.3%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} \leq -\infty:\\ \;\;\;\;\sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} + g \cdot \mathsf{fma}\left(0.16666666666666666, \frac{\sqrt[3]{\frac{1}{a \cdot {g}^{8}}} \cdot \left(\left(h \cdot h\right) \cdot \sqrt[3]{0.5}\right)}{{\left(\sqrt[3]{-2}\right)}^{2}}, \sqrt[3]{\frac{1}{a \cdot \left(g \cdot g\right)}} \cdot \left(\sqrt[3]{0.5} \cdot \sqrt[3]{-2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \frac{\left(h \cdot h\right) \cdot -0.5}{g}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 71.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{2 \cdot a}\\ \sqrt[3]{t\_0 \cdot \frac{\left(h \cdot h\right) \cdot -0.5}{g}} + \sqrt[3]{t\_0 \cdot \left(g \cdot -2\right)} \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* 2.0 a))))
   (+ (cbrt (* t_0 (/ (* (* h h) -0.5) g))) (cbrt (* t_0 (* g -2.0))))))

C

double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	return cbrt((t_0 * (((h * h) * -0.5) / g))) + cbrt((t_0 * (g * -2.0)));
}

Java

public static double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	return Math.cbrt((t_0 * (((h * h) * -0.5) / g))) + Math.cbrt((t_0 * (g * -2.0)));
}

Julia

function code(g, h, a)
	t_0 = Float64(1.0 / Float64(2.0 * a))
	return Float64(cbrt(Float64(t_0 * Float64(Float64(Float64(h * h) * -0.5) / g))) + cbrt(Float64(t_0 * Float64(g * -2.0))))
end

Wolfram

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

Derivation

1. Initial program 41.5%

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

3. Taylor expanded in g around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 25.0

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

5. Simplified 25.0%

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

6. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 71.0

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

8. Simplified 71.0%

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

9. Final simplification 71.0%

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \frac{\left(h \cdot h\right) \cdot -0.5}{g}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]
10. Add Preprocessing

Alternative 3: 71.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{1}{2 \cdot a} \cdot \frac{\left(h \cdot h\right) \cdot -0.5}{g}} - \sqrt[3]{\frac{g}{a}} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (- (cbrt (* (/ 1.0 (* 2.0 a)) (/ (* (* h h) -0.5) g))) (cbrt (/ g a))))

C

double code(double g, double h, double a) {
	return cbrt(((1.0 / (2.0 * a)) * (((h * h) * -0.5) / g))) - cbrt((g / a));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt(((1.0 / (2.0 * a)) * (((h * h) * -0.5) / g))) - Math.cbrt((g / a));
}

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(1.0 / Float64(2.0 * a)) * Float64(Float64(Float64(h * h) * -0.5) / g))) - cbrt(Float64(g / a)))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(h * h), $MachinePrecision] * -0.5), $MachinePrecision] / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.5%

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

3. Taylor expanded in g around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 25.0

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

5. Simplified 25.0%

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

6. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 71.0

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

8. Simplified 71.0%

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

9. Taylor expanded in g around -inf

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

    1. mul-1-neg N/A

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

    2. lower-neg.f64 N/A

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

    3. lower-cbrt.f64 N/A

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

    4. lower-/.f64 71.0

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

11. Simplified 71.0%

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

12. Final simplification 71.0%

    \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \frac{\left(h \cdot h\right) \cdot -0.5}{g}} - \sqrt[3]{\frac{g}{a}} \]
13. Add Preprocessing

Alternative 4: 68.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} + \sqrt[3]{\frac{\left(h \cdot h\right) \cdot -0.25}{a \cdot g}} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (+
  (cbrt (* (/ 1.0 (* 2.0 a)) (* g -2.0)))
  (cbrt (/ (* (* h h) -0.25) (* a g)))))

C

double code(double g, double h, double a) {
	return cbrt(((1.0 / (2.0 * a)) * (g * -2.0))) + cbrt((((h * h) * -0.25) / (a * g)));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt(((1.0 / (2.0 * a)) * (g * -2.0))) + Math.cbrt((((h * h) * -0.25) / (a * g)));
}

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(1.0 / Float64(2.0 * a)) * Float64(g * -2.0))) + cbrt(Float64(Float64(Float64(h * h) * -0.25) / Float64(a * g))))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * N[(g * -2.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(N[(h * h), $MachinePrecision] * -0.25), $MachinePrecision] / N[(a * g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.5%

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

3. Taylor expanded in g around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 25.0

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

5. Simplified 25.0%

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

6. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 71.0

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

8. Simplified 71.0%

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

9. Taylor expanded in a around 0

   \[\leadsto \sqrt[3]{\color{blue}{\frac{-1}{4} \cdot \frac{{h}^{2}}{a \cdot g}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]
10. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{-1}{4} \cdot {h}^{2}}{a \cdot g}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]

    2. lower-/.f64 N/A

       \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{-1}{4} \cdot {h}^{2}}{a \cdot g}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]

    3. lower-*.f64 N/A

       \[\leadsto \sqrt[3]{\frac{\color{blue}{\frac{-1}{4} \cdot {h}^{2}}}{a \cdot g}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]

    4. unpow2 N/A

       \[\leadsto \sqrt[3]{\frac{\frac{-1}{4} \cdot \color{blue}{\left(h \cdot h\right)}}{a \cdot g}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]

    5. lower-*.f64 N/A

       \[\leadsto \sqrt[3]{\frac{\frac{-1}{4} \cdot \color{blue}{\left(h \cdot h\right)}}{a \cdot g}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]

    6. lower-*.f64 66.9

       \[\leadsto \sqrt[3]{\frac{-0.25 \cdot \left(h \cdot h\right)}{\color{blue}{a \cdot g}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]

11. Simplified 66.9%

    \[\leadsto \sqrt[3]{\color{blue}{\frac{-0.25 \cdot \left(h \cdot h\right)}{a \cdot g}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} \]

12. Final simplification 66.9%

    \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(g \cdot -2\right)} + \sqrt[3]{\frac{\left(h \cdot h\right) \cdot -0.25}{a \cdot g}} \]
13. Add Preprocessing

Alternative 5: 68.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{\left(h \cdot h\right) \cdot -0.25}{a \cdot g}} - \sqrt[3]{\frac{g}{a}} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (- (cbrt (/ (* (* h h) -0.25) (* a g))) (cbrt (/ g a))))

C

double code(double g, double h, double a) {
	return cbrt((((h * h) * -0.25) / (a * g))) - cbrt((g / a));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt((((h * h) * -0.25) / (a * g))) - Math.cbrt((g / a));
}

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(Float64(h * h) * -0.25) / Float64(a * g))) - cbrt(Float64(g / a)))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(N[(N[(h * h), $MachinePrecision] * -0.25), $MachinePrecision] / N[(a * g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.5%

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

3. Taylor expanded in g around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 25.0

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

5. Simplified 25.0%

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

6. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 71.0

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

8. Simplified 71.0%

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

9. Taylor expanded in g around -inf

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

    1. mul-1-neg N/A

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

    2. lower-neg.f64 N/A

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

    3. lower-cbrt.f64 N/A

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

    4. lower-/.f64 71.0

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

11. Simplified 71.0%

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

12. Taylor expanded in a around 0

    \[\leadsto \sqrt[3]{\color{blue}{\frac{-1}{4} \cdot \frac{{h}^{2}}{a \cdot g}}} + \left(\mathsf{neg}\left(\sqrt[3]{\frac{g}{a}}\right)\right) \]
13. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{-1}{4} \cdot {h}^{2}}{a \cdot g}}} + \left(\mathsf{neg}\left(\sqrt[3]{\frac{g}{a}}\right)\right) \]

    2. lower-/.f64 N/A

       \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{-1}{4} \cdot {h}^{2}}{a \cdot g}}} + \left(\mathsf{neg}\left(\sqrt[3]{\frac{g}{a}}\right)\right) \]

    3. lower-*.f64 N/A

       \[\leadsto \sqrt[3]{\frac{\color{blue}{\frac{-1}{4} \cdot {h}^{2}}}{a \cdot g}} + \left(\mathsf{neg}\left(\sqrt[3]{\frac{g}{a}}\right)\right) \]

    4. unpow2 N/A

       \[\leadsto \sqrt[3]{\frac{\frac{-1}{4} \cdot \color{blue}{\left(h \cdot h\right)}}{a \cdot g}} + \left(\mathsf{neg}\left(\sqrt[3]{\frac{g}{a}}\right)\right) \]

    5. lower-*.f64 N/A

       \[\leadsto \sqrt[3]{\frac{\frac{-1}{4} \cdot \color{blue}{\left(h \cdot h\right)}}{a \cdot g}} + \left(\mathsf{neg}\left(\sqrt[3]{\frac{g}{a}}\right)\right) \]

    6. lower-*.f64 66.9

       \[\leadsto \sqrt[3]{\frac{-0.25 \cdot \left(h \cdot h\right)}{\color{blue}{a \cdot g}}} + \left(-\sqrt[3]{\frac{g}{a}}\right) \]

14. Simplified 66.9%

    \[\leadsto \sqrt[3]{\color{blue}{\frac{-0.25 \cdot \left(h \cdot h\right)}{a \cdot g}}} + \left(-\sqrt[3]{\frac{g}{a}}\right) \]

15. Final simplification 66.9%

    \[\leadsto \sqrt[3]{\frac{\left(h \cdot h\right) \cdot -0.25}{a \cdot g}} - \sqrt[3]{\frac{g}{a}} \]
16. Add Preprocessing

Alternative 6: 15.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{g}{a}}\\ \left(-t\_0\right) - t\_0 \end{array} \end{array} \]

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(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 41.5%

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

3. Taylor expanded in g around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 25.0

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

5. Simplified 25.0%

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

6. Taylor expanded in g around -inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 15.2

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

8. Simplified 15.2%

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

9. Taylor expanded in g around -inf

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

    1. mul-1-neg N/A

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

    2. lower-neg.f64 N/A

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

    3. lower-cbrt.f64 N/A

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

    4. lower-/.f64 15.2

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

11. Simplified 15.2%

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

12. Taylor expanded in g around -inf

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

    1. mul-1-neg N/A

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

    2. lower-neg.f64 N/A

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

    3. lower-cbrt.f64 N/A

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

    4. lower-/.f64 15.2

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

14. Simplified 15.2%

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

15. Final simplification 15.2%

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

Reproduce

herbie shell --seed 2024215 
(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))))))))