2-ancestry mixing, positive discriminant

Percentage Accurate: 43.5% → 97.7%

Time: 12.1s

Alternatives: 5

Speedup: 2.6×

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: 43.5% 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: 97.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} h_m = \left|h\right| \\ \begin{array}{l} t_0 := {\left(0 \cdot g\right)}^{2}\\ \mathbf{if}\;h\_m \leq 1.48 \cdot 10^{-170}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}} \cdot \sqrt[3]{-0.5}, {2}^{0.3333333333333333}, \sqrt[3]{\frac{h\_m}{g} \cdot \frac{h\_m}{a}} \cdot \left(\sqrt[3]{0.5} \cdot \sqrt[3]{-0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{h\_m \cdot \mathsf{fma}\left(-0.5, h\_m \cdot \frac{\mathsf{fma}\left(0.25, \frac{t\_0}{g \cdot g}, 1\right)}{g}, 0.5 \cdot \frac{0 \cdot g}{g}\right)} + \sqrt[3]{\left(-g\right) - \mathsf{fma}\left(h\_m, \frac{\mathsf{fma}\left(-0.5, h\_m \cdot \mathsf{fma}\left(\frac{0.25}{g}, \frac{t\_0}{g}, 1\right), 0.5 \cdot \left(0 \cdot g\right)\right)}{g}, g\right)}}{\sqrt[3]{a \cdot 2}}\\ \end{array} \end{array} \]

FPCore

h_m = (fabs.f64 h)
(FPCore (g h_m a)
 :precision binary64
 (let* ((t_0 (pow (* 0.0 g) 2.0)))
   (if (<= h_m 1.48e-170)
     (fma
      (* (/ (cbrt g) (cbrt a)) (cbrt -0.5))
      (pow 2.0 0.3333333333333333)
      (* (cbrt (* (/ h_m g) (/ h_m a))) (* (cbrt 0.5) (cbrt -0.5))))
     (/
      (+
       (cbrt
        (*
         h_m
         (fma
          -0.5
          (* h_m (/ (fma 0.25 (/ t_0 (* g g)) 1.0) g))
          (* 0.5 (/ (* 0.0 g) g)))))
       (cbrt
        (-
         (- g)
         (fma
          h_m
          (/
           (fma -0.5 (* h_m (fma (/ 0.25 g) (/ t_0 g) 1.0)) (* 0.5 (* 0.0 g)))
           g)
          g))))
      (cbrt (* a 2.0))))))

C

h_m = fabs(h);
double code(double g, double h_m, double a) {
	double t_0 = pow((0.0 * g), 2.0);
	double tmp;
	if (h_m <= 1.48e-170) {
		tmp = fma(((cbrt(g) / cbrt(a)) * cbrt(-0.5)), pow(2.0, 0.3333333333333333), (cbrt(((h_m / g) * (h_m / a))) * (cbrt(0.5) * cbrt(-0.5))));
	} else {
		tmp = (cbrt((h_m * fma(-0.5, (h_m * (fma(0.25, (t_0 / (g * g)), 1.0) / g)), (0.5 * ((0.0 * g) / g))))) + cbrt((-g - fma(h_m, (fma(-0.5, (h_m * fma((0.25 / g), (t_0 / g), 1.0)), (0.5 * (0.0 * g))) / g), g)))) / cbrt((a * 2.0));
	}
	return tmp;
}

Julia

h_m = abs(h)
function code(g, h_m, a)
	t_0 = Float64(0.0 * g) ^ 2.0
	tmp = 0.0
	if (h_m <= 1.48e-170)
		tmp = fma(Float64(Float64(cbrt(g) / cbrt(a)) * cbrt(-0.5)), (2.0 ^ 0.3333333333333333), Float64(cbrt(Float64(Float64(h_m / g) * Float64(h_m / a))) * Float64(cbrt(0.5) * cbrt(-0.5))));
	else
		tmp = Float64(Float64(cbrt(Float64(h_m * fma(-0.5, Float64(h_m * Float64(fma(0.25, Float64(t_0 / Float64(g * g)), 1.0) / g)), Float64(0.5 * Float64(Float64(0.0 * g) / g))))) + cbrt(Float64(Float64(-g) - fma(h_m, Float64(fma(-0.5, Float64(h_m * fma(Float64(0.25 / g), Float64(t_0 / g), 1.0)), Float64(0.5 * Float64(0.0 * g))) / g), g)))) / cbrt(Float64(a * 2.0)));
	end
	return tmp
end

Wolfram

h_m = N[Abs[h], $MachinePrecision]
code[g_, h$95$m_, a_] := Block[{t$95$0 = N[Power[N[(0.0 * g), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[h$95$m, 1.48e-170], N[(N[(N[(N[Power[g, 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[-0.5, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[2.0, 0.3333333333333333], $MachinePrecision] + N[(N[Power[N[(N[(h$95$m / g), $MachinePrecision] * N[(h$95$m / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(N[Power[0.5, 1/3], $MachinePrecision] * N[Power[-0.5, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(h$95$m * N[(-0.5 * N[(h$95$m * N[(N[(0.25 * N[(t$95$0 / N[(g * g), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / g), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(0.0 * g), $MachinePrecision] / g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[((-g) - N[(h$95$m * N[(N[(-0.5 * N[(h$95$m * N[(N[(0.25 / g), $MachinePrecision] * N[(t$95$0 / g), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(0.0 * g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / g), $MachinePrecision] + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[N[(a * 2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < 1.48000000000000005e-170

   1. Initial program 48.4%

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

   3. Taylor expanded in h around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cbrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. unpow2 N/A

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

      11. *-commutative N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cbrt.f64 N/A

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

      19. lower-cbrt.f64 75.4

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

   5. Applied rewrites 75.4%

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

   7. Applied rewrites 93.5%

      \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}} \cdot \sqrt[3]{-0.5}, \sqrt[3]{2}, \sqrt[3]{\frac{h}{g} \cdot \frac{h}{a}} \cdot \left(\sqrt[3]{0.5} \cdot \sqrt[3]{-0.5}\right)\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 94.2%

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

   if 1.48000000000000005e-170 < h

   1. Initial program 42.6%

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

   4. Applied rewrites 9.7%

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

   5. Taylor expanded in h around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 20.9%

      \[\leadsto \frac{\sqrt[3]{\color{blue}{h \cdot \mathsf{fma}\left(-0.5, h \cdot \frac{\mathsf{fma}\left(0.25, \frac{{\left(0 \cdot g\right)}^{2}}{g \cdot g}, 1\right)}{g}, 0.5 \cdot \frac{0 \cdot g}{g}\right)}} + \sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}}{\sqrt[3]{a \cdot 2}} \]

   8. Taylor expanded in h around 0

      \[\leadsto \frac{\sqrt[3]{h \cdot \mathsf{fma}\left(\frac{-1}{2}, h \cdot \frac{\mathsf{fma}\left(\frac{1}{4}, \frac{{\left(0 \cdot g\right)}^{2}}{g \cdot g}, 1\right)}{g}, \frac{1}{2} \cdot \frac{0 \cdot g}{g}\right)} + \sqrt[3]{\left(-g\right) - \color{blue}{\left(g + h \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot \left(1 + \frac{1}{4} \cdot \frac{{\left(g + -1 \cdot g\right)}^{2}}{{g}^{2}}\right)}{g} + \frac{1}{2} \cdot \frac{g + -1 \cdot g}{g}\right)\right)}}}{\sqrt[3]{a \cdot 2}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sqrt[3]{h \cdot \mathsf{fma}\left(\frac{-1}{2}, h \cdot \frac{\mathsf{fma}\left(\frac{1}{4}, \frac{{\left(0 \cdot g\right)}^{2}}{g \cdot g}, 1\right)}{g}, \frac{1}{2} \cdot \frac{0 \cdot g}{g}\right)} + \sqrt[3]{\left(-g\right) - \color{blue}{\left(h \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot \left(1 + \frac{1}{4} \cdot \frac{{\left(g + -1 \cdot g\right)}^{2}}{{g}^{2}}\right)}{g} + \frac{1}{2} \cdot \frac{g + -1 \cdot g}{g}\right) + g\right)}}}{\sqrt[3]{a \cdot 2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt[3]{h \cdot \mathsf{fma}\left(\frac{-1}{2}, h \cdot \frac{\mathsf{fma}\left(\frac{1}{4}, \frac{{\left(0 \cdot g\right)}^{2}}{g \cdot g}, 1\right)}{g}, \frac{1}{2} \cdot \frac{0 \cdot g}{g}\right)} + \sqrt[3]{\left(-g\right) - \color{blue}{\mathsf{fma}\left(h, \frac{-1}{2} \cdot \frac{h \cdot \left(1 + \frac{1}{4} \cdot \frac{{\left(g + -1 \cdot g\right)}^{2}}{{g}^{2}}\right)}{g} + \frac{1}{2} \cdot \frac{g + -1 \cdot g}{g}, g\right)}}}{\sqrt[3]{a \cdot 2}} \]

   10. Applied rewrites 98.8%

       \[\leadsto \frac{\sqrt[3]{h \cdot \mathsf{fma}\left(-0.5, h \cdot \frac{\mathsf{fma}\left(0.25, \frac{{\left(0 \cdot g\right)}^{2}}{g \cdot g}, 1\right)}{g}, 0.5 \cdot \frac{0 \cdot g}{g}\right)} + \sqrt[3]{\left(-g\right) - \color{blue}{\mathsf{fma}\left(h, \frac{\mathsf{fma}\left(-0.5, h \cdot \mathsf{fma}\left(\frac{0.25}{g}, \frac{{\left(0 \cdot g\right)}^{2}}{g}, 1\right), 0.5 \cdot \left(0 \cdot g\right)\right)}{g}, g\right)}}}{\sqrt[3]{a \cdot 2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 97.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} h_m = \left|h\right| \\ \begin{array}{l} t_0 := {\left(0 \cdot g\right)}^{2}\\ \mathbf{if}\;h\_m \leq 5 \cdot 10^{-169}:\\ \;\;\;\;\frac{\sqrt[3]{g \cdot -1}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{h\_m \cdot \mathsf{fma}\left(-0.5, h\_m \cdot \frac{\mathsf{fma}\left(0.25, \frac{t\_0}{g \cdot g}, 1\right)}{g}, 0.5 \cdot \frac{0 \cdot g}{g}\right)} + \sqrt[3]{\left(-g\right) - \mathsf{fma}\left(h\_m, \frac{\mathsf{fma}\left(-0.5, h\_m \cdot \mathsf{fma}\left(\frac{0.25}{g}, \frac{t\_0}{g}, 1\right), 0.5 \cdot \left(0 \cdot g\right)\right)}{g}, g\right)}}{\sqrt[3]{a \cdot 2}}\\ \end{array} \end{array} \]

FPCore

h_m = (fabs.f64 h)
(FPCore (g h_m a)
 :precision binary64
 (let* ((t_0 (pow (* 0.0 g) 2.0)))
   (if (<= h_m 5e-169)
     (/ (cbrt (* g -1.0)) (cbrt a))
     (/
      (+
       (cbrt
        (*
         h_m
         (fma
          -0.5
          (* h_m (/ (fma 0.25 (/ t_0 (* g g)) 1.0) g))
          (* 0.5 (/ (* 0.0 g) g)))))
       (cbrt
        (-
         (- g)
         (fma
          h_m
          (/
           (fma -0.5 (* h_m (fma (/ 0.25 g) (/ t_0 g) 1.0)) (* 0.5 (* 0.0 g)))
           g)
          g))))
      (cbrt (* a 2.0))))))

C

h_m = fabs(h);
double code(double g, double h_m, double a) {
	double t_0 = pow((0.0 * g), 2.0);
	double tmp;
	if (h_m <= 5e-169) {
		tmp = cbrt((g * -1.0)) / cbrt(a);
	} else {
		tmp = (cbrt((h_m * fma(-0.5, (h_m * (fma(0.25, (t_0 / (g * g)), 1.0) / g)), (0.5 * ((0.0 * g) / g))))) + cbrt((-g - fma(h_m, (fma(-0.5, (h_m * fma((0.25 / g), (t_0 / g), 1.0)), (0.5 * (0.0 * g))) / g), g)))) / cbrt((a * 2.0));
	}
	return tmp;
}

Julia

h_m = abs(h)
function code(g, h_m, a)
	t_0 = Float64(0.0 * g) ^ 2.0
	tmp = 0.0
	if (h_m <= 5e-169)
		tmp = Float64(cbrt(Float64(g * -1.0)) / cbrt(a));
	else
		tmp = Float64(Float64(cbrt(Float64(h_m * fma(-0.5, Float64(h_m * Float64(fma(0.25, Float64(t_0 / Float64(g * g)), 1.0) / g)), Float64(0.5 * Float64(Float64(0.0 * g) / g))))) + cbrt(Float64(Float64(-g) - fma(h_m, Float64(fma(-0.5, Float64(h_m * fma(Float64(0.25 / g), Float64(t_0 / g), 1.0)), Float64(0.5 * Float64(0.0 * g))) / g), g)))) / cbrt(Float64(a * 2.0)));
	end
	return tmp
end

Wolfram

h_m = N[Abs[h], $MachinePrecision]
code[g_, h$95$m_, a_] := Block[{t$95$0 = N[Power[N[(0.0 * g), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[h$95$m, 5e-169], N[(N[Power[N[(g * -1.0), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(h$95$m * N[(-0.5 * N[(h$95$m * N[(N[(0.25 * N[(t$95$0 / N[(g * g), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / g), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(0.0 * g), $MachinePrecision] / g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[((-g) - N[(h$95$m * N[(N[(-0.5 * N[(h$95$m * N[(N[(0.25 / g), $MachinePrecision] * N[(t$95$0 / g), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(0.0 * g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / g), $MachinePrecision] + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[N[(a * 2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < 5.0000000000000002e-169

   1. Initial program 48.4%

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

   4. Applied rewrites 10.8%

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

   5. Taylor expanded in g around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cbrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-cbrt.f64 74.7

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

   7. Applied rewrites 74.7%

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

   9. Applied rewrites 96.0%

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

   if 5.0000000000000002e-169 < h

   1. Initial program 42.6%

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

   4. Applied rewrites 9.7%

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

   5. Taylor expanded in h around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 20.9%

      \[\leadsto \frac{\sqrt[3]{\color{blue}{h \cdot \mathsf{fma}\left(-0.5, h \cdot \frac{\mathsf{fma}\left(0.25, \frac{{\left(0 \cdot g\right)}^{2}}{g \cdot g}, 1\right)}{g}, 0.5 \cdot \frac{0 \cdot g}{g}\right)}} + \sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}}{\sqrt[3]{a \cdot 2}} \]

   8. Taylor expanded in h around 0

      \[\leadsto \frac{\sqrt[3]{h \cdot \mathsf{fma}\left(\frac{-1}{2}, h \cdot \frac{\mathsf{fma}\left(\frac{1}{4}, \frac{{\left(0 \cdot g\right)}^{2}}{g \cdot g}, 1\right)}{g}, \frac{1}{2} \cdot \frac{0 \cdot g}{g}\right)} + \sqrt[3]{\left(-g\right) - \color{blue}{\left(g + h \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot \left(1 + \frac{1}{4} \cdot \frac{{\left(g + -1 \cdot g\right)}^{2}}{{g}^{2}}\right)}{g} + \frac{1}{2} \cdot \frac{g + -1 \cdot g}{g}\right)\right)}}}{\sqrt[3]{a \cdot 2}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sqrt[3]{h \cdot \mathsf{fma}\left(\frac{-1}{2}, h \cdot \frac{\mathsf{fma}\left(\frac{1}{4}, \frac{{\left(0 \cdot g\right)}^{2}}{g \cdot g}, 1\right)}{g}, \frac{1}{2} \cdot \frac{0 \cdot g}{g}\right)} + \sqrt[3]{\left(-g\right) - \color{blue}{\left(h \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot \left(1 + \frac{1}{4} \cdot \frac{{\left(g + -1 \cdot g\right)}^{2}}{{g}^{2}}\right)}{g} + \frac{1}{2} \cdot \frac{g + -1 \cdot g}{g}\right) + g\right)}}}{\sqrt[3]{a \cdot 2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt[3]{h \cdot \mathsf{fma}\left(\frac{-1}{2}, h \cdot \frac{\mathsf{fma}\left(\frac{1}{4}, \frac{{\left(0 \cdot g\right)}^{2}}{g \cdot g}, 1\right)}{g}, \frac{1}{2} \cdot \frac{0 \cdot g}{g}\right)} + \sqrt[3]{\left(-g\right) - \color{blue}{\mathsf{fma}\left(h, \frac{-1}{2} \cdot \frac{h \cdot \left(1 + \frac{1}{4} \cdot \frac{{\left(g + -1 \cdot g\right)}^{2}}{{g}^{2}}\right)}{g} + \frac{1}{2} \cdot \frac{g + -1 \cdot g}{g}, g\right)}}}{\sqrt[3]{a \cdot 2}} \]

   10. Applied rewrites 98.8%

       \[\leadsto \frac{\sqrt[3]{h \cdot \mathsf{fma}\left(-0.5, h \cdot \frac{\mathsf{fma}\left(0.25, \frac{{\left(0 \cdot g\right)}^{2}}{g \cdot g}, 1\right)}{g}, 0.5 \cdot \frac{0 \cdot g}{g}\right)} + \sqrt[3]{\left(-g\right) - \color{blue}{\mathsf{fma}\left(h, \frac{\mathsf{fma}\left(-0.5, h \cdot \mathsf{fma}\left(\frac{0.25}{g}, \frac{{\left(0 \cdot g\right)}^{2}}{g}, 1\right), 0.5 \cdot \left(0 \cdot g\right)\right)}{g}, g\right)}}}{\sqrt[3]{a \cdot 2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} h_m = \left|h\right| \\ \begin{array}{l} \mathbf{if}\;h\_m \leq 5 \cdot 10^{-169}:\\ \;\;\;\;\frac{\sqrt[3]{g \cdot -1}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{h\_m \cdot \mathsf{fma}\left(-0.5, h\_m \cdot \frac{\mathsf{fma}\left(0.25, \frac{{\left(0 \cdot g\right)}^{2}}{g \cdot g}, 1\right)}{g}, 0.5 \cdot \frac{0 \cdot g}{g}\right)} + \sqrt[3]{\left(-g\right) - \left(-g\right) \cdot -1}}{\sqrt[3]{a \cdot 2}}\\ \end{array} \end{array} \]

FPCore

h_m = (fabs.f64 h)
(FPCore (g h_m a)
 :precision binary64
 (if (<= h_m 5e-169)
   (/ (cbrt (* g -1.0)) (cbrt a))
   (/
    (+
     (cbrt
      (*
       h_m
       (fma
        -0.5
        (* h_m (/ (fma 0.25 (/ (pow (* 0.0 g) 2.0) (* g g)) 1.0) g))
        (* 0.5 (/ (* 0.0 g) g)))))
     (cbrt (- (- g) (* (- g) -1.0))))
    (cbrt (* a 2.0)))))

C

h_m = fabs(h);
double code(double g, double h_m, double a) {
	double tmp;
	if (h_m <= 5e-169) {
		tmp = cbrt((g * -1.0)) / cbrt(a);
	} else {
		tmp = (cbrt((h_m * fma(-0.5, (h_m * (fma(0.25, (pow((0.0 * g), 2.0) / (g * g)), 1.0) / g)), (0.5 * ((0.0 * g) / g))))) + cbrt((-g - (-g * -1.0)))) / cbrt((a * 2.0));
	}
	return tmp;
}

Julia

h_m = abs(h)
function code(g, h_m, a)
	tmp = 0.0
	if (h_m <= 5e-169)
		tmp = Float64(cbrt(Float64(g * -1.0)) / cbrt(a));
	else
		tmp = Float64(Float64(cbrt(Float64(h_m * fma(-0.5, Float64(h_m * Float64(fma(0.25, Float64((Float64(0.0 * g) ^ 2.0) / Float64(g * g)), 1.0) / g)), Float64(0.5 * Float64(Float64(0.0 * g) / g))))) + cbrt(Float64(Float64(-g) - Float64(Float64(-g) * -1.0)))) / cbrt(Float64(a * 2.0)));
	end
	return tmp
end

Wolfram

h_m = N[Abs[h], $MachinePrecision]
code[g_, h$95$m_, a_] := If[LessEqual[h$95$m, 5e-169], N[(N[Power[N[(g * -1.0), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(h$95$m * N[(-0.5 * N[(h$95$m * N[(N[(0.25 * N[(N[Power[N[(0.0 * g), $MachinePrecision], 2.0], $MachinePrecision] / N[(g * g), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / g), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(0.0 * g), $MachinePrecision] / g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[((-g) - N[((-g) * -1.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[N[(a * 2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 5.0000000000000002e-169

   1. Initial program 48.4%

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

   4. Applied rewrites 10.8%

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

   5. Taylor expanded in g around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cbrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-cbrt.f64 74.7

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

   7. Applied rewrites 74.7%

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

   9. Applied rewrites 96.0%

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

   if 5.0000000000000002e-169 < h

   1. Initial program 42.6%

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

   4. Applied rewrites 9.7%

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

   5. Taylor expanded in h around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 20.9%

      \[\leadsto \frac{\sqrt[3]{\color{blue}{h \cdot \mathsf{fma}\left(-0.5, h \cdot \frac{\mathsf{fma}\left(0.25, \frac{{\left(0 \cdot g\right)}^{2}}{g \cdot g}, 1\right)}{g}, 0.5 \cdot \frac{0 \cdot g}{g}\right)}} + \sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}}{\sqrt[3]{a \cdot 2}} \]

   8. Taylor expanded in g around -inf

      \[\leadsto \frac{\sqrt[3]{h \cdot \mathsf{fma}\left(\frac{-1}{2}, h \cdot \frac{\mathsf{fma}\left(\frac{1}{4}, \frac{{\left(0 \cdot g\right)}^{2}}{g \cdot g}, 1\right)}{g}, \frac{1}{2} \cdot \frac{0 \cdot g}{g}\right)} + \sqrt[3]{\left(-g\right) - \color{blue}{-1 \cdot \left(g \cdot {\left(\sqrt{-1}\right)}^{2}\right)}}}{\sqrt[3]{a \cdot 2}} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\sqrt[3]{h \cdot \mathsf{fma}\left(\frac{-1}{2}, h \cdot \frac{\mathsf{fma}\left(\frac{1}{4}, \frac{{\left(0 \cdot g\right)}^{2}}{g \cdot g}, 1\right)}{g}, \frac{1}{2} \cdot \frac{0 \cdot g}{g}\right)} + \sqrt[3]{\left(-g\right) - \color{blue}{\left(-1 \cdot g\right) \cdot {\left(\sqrt{-1}\right)}^{2}}}}{\sqrt[3]{a \cdot 2}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\sqrt[3]{h \cdot \mathsf{fma}\left(\frac{-1}{2}, h \cdot \frac{\mathsf{fma}\left(\frac{1}{4}, \frac{{\left(0 \cdot g\right)}^{2}}{g \cdot g}, 1\right)}{g}, \frac{1}{2} \cdot \frac{0 \cdot g}{g}\right)} + \sqrt[3]{\left(-g\right) - \left(-1 \cdot g\right) \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}}}{\sqrt[3]{a \cdot 2}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \frac{\sqrt[3]{h \cdot \mathsf{fma}\left(\frac{-1}{2}, h \cdot \frac{\mathsf{fma}\left(\frac{1}{4}, \frac{{\left(0 \cdot g\right)}^{2}}{g \cdot g}, 1\right)}{g}, \frac{1}{2} \cdot \frac{0 \cdot g}{g}\right)} + \sqrt[3]{\left(-g\right) - \left(-1 \cdot g\right) \cdot \color{blue}{-1}}}{\sqrt[3]{a \cdot 2}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt[3]{h \cdot \mathsf{fma}\left(\frac{-1}{2}, h \cdot \frac{\mathsf{fma}\left(\frac{1}{4}, \frac{{\left(0 \cdot g\right)}^{2}}{g \cdot g}, 1\right)}{g}, \frac{1}{2} \cdot \frac{0 \cdot g}{g}\right)} + \sqrt[3]{\left(-g\right) - \color{blue}{\left(-1 \cdot g\right) \cdot -1}}}{\sqrt[3]{a \cdot 2}} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{\sqrt[3]{h \cdot \mathsf{fma}\left(\frac{-1}{2}, h \cdot \frac{\mathsf{fma}\left(\frac{1}{4}, \frac{{\left(0 \cdot g\right)}^{2}}{g \cdot g}, 1\right)}{g}, \frac{1}{2} \cdot \frac{0 \cdot g}{g}\right)} + \sqrt[3]{\left(-g\right) - \color{blue}{\left(\mathsf{neg}\left(g\right)\right)} \cdot -1}}{\sqrt[3]{a \cdot 2}} \]

      6. lower-neg.f64 98.8

         \[\leadsto \frac{\sqrt[3]{h \cdot \mathsf{fma}\left(-0.5, h \cdot \frac{\mathsf{fma}\left(0.25, \frac{{\left(0 \cdot g\right)}^{2}}{g \cdot g}, 1\right)}{g}, 0.5 \cdot \frac{0 \cdot g}{g}\right)} + \sqrt[3]{\left(-g\right) - \color{blue}{\left(-g\right)} \cdot -1}}{\sqrt[3]{a \cdot 2}} \]

   10. Applied rewrites 98.8%

       \[\leadsto \frac{\sqrt[3]{h \cdot \mathsf{fma}\left(-0.5, h \cdot \frac{\mathsf{fma}\left(0.25, \frac{{\left(0 \cdot g\right)}^{2}}{g \cdot g}, 1\right)}{g}, 0.5 \cdot \frac{0 \cdot g}{g}\right)} + \sqrt[3]{\left(-g\right) - \color{blue}{\left(-g\right) \cdot -1}}}{\sqrt[3]{a \cdot 2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 95.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} h_m = \left|h\right| \\ \frac{\sqrt[3]{g \cdot -1}}{\sqrt[3]{a}} \end{array} \]

FPCore

h_m = (fabs.f64 h)
(FPCore (g h_m a) :precision binary64 (/ (cbrt (* g -1.0)) (cbrt a)))

C

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

Java

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

Julia

h_m = abs(h)
function code(g, h_m, a)
	return Float64(cbrt(Float64(g * -1.0)) / cbrt(a))
end

Wolfram

h_m = N[Abs[h], $MachinePrecision]
code[g_, h$95$m_, a_] := N[(N[Power[N[(g * -1.0), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.8%

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

4. Applied rewrites 10.5%

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

5. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-cbrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-cbrt.f64 72.7

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

7. Applied rewrites 72.7%

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

9. Applied rewrites 96.0%

   \[\leadsto \frac{\sqrt[3]{g \cdot -1}}{\color{blue}{\sqrt[3]{a}}} \]
10. Add Preprocessing

Alternative 5: 74.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} h_m = \left|h\right| \\ \sqrt[3]{\frac{g}{a} \cdot -1} \end{array} \]

FPCore

h_m = (fabs.f64 h)
(FPCore (g h_m a) :precision binary64 (cbrt (* (/ g a) -1.0)))

C

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

Java

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

Julia

h_m = abs(h)
function code(g, h_m, a)
	return cbrt(Float64(Float64(g / a) * -1.0))
end

Wolfram

h_m = N[Abs[h], $MachinePrecision]
code[g_, h$95$m_, a_] := N[Power[N[(N[(g / a), $MachinePrecision] * -1.0), $MachinePrecision], 1/3], $MachinePrecision]

Derivation

1. Initial program 46.8%

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

4. Applied rewrites 10.5%

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

5. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-cbrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-cbrt.f64 72.7

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

7. Applied rewrites 72.7%

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

9. Applied rewrites 72.7%

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

Reproduce

herbie shell --seed 2025003 
(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))))))))