2-ancestry mixing, positive discriminant

Average Error: 36.1 → 33.8

Time: 13.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := h \cdot \left(-h\right)\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ t_2 := \sqrt[3]{\frac{0.5}{a} \cdot \left(t_1 - g\right)} + \frac{\sqrt[3]{\left(g + \sqrt{\mathsf{fma}\left(g, g, t_0\right)}\right) \cdot -0.5}}{\sqrt[3]{a}}\\ \mathbf{if}\;h \leq -4.3708065047631245 \cdot 10^{-179}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;h \leq 6.613043387155913 \cdot 10^{-258}:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot \left(\mathsf{hypot}\left(g, \sqrt{t_0}\right) - g\right)}}{\sqrt[3]{a}} + \sqrt[3]{\left(g + t_1\right) \cdot \frac{-0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

FPCore

(FPCore (g h a)
 :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))))))))

↓

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (* h (- h)))
        (t_1 (sqrt (- (* g g) (* h h))))
        (t_2
         (+
          (cbrt (* (/ 0.5 a) (- t_1 g)))
          (/ (cbrt (* (+ g (sqrt (fma g g t_0))) -0.5)) (cbrt a)))))
   (if (<= h -4.3708065047631245e-179)
     t_2
     (if (<= h 6.613043387155913e-258)
       (+
        (/ (cbrt (* 0.5 (- (hypot g (sqrt t_0)) g))) (cbrt a))
        (cbrt (* (+ g t_1) (/ -0.5 a))))
       t_2))))

C

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

↓

double code(double g, double h, double a) {
	double t_0 = h * -h;
	double t_1 = sqrt(((g * g) - (h * h)));
	double t_2 = cbrt(((0.5 / a) * (t_1 - g))) + (cbrt(((g + sqrt(fma(g, g, t_0))) * -0.5)) / cbrt(a));
	double tmp;
	if (h <= -4.3708065047631245e-179) {
		tmp = t_2;
	} else if (h <= 6.613043387155913e-258) {
		tmp = (cbrt((0.5 * (hypot(g, sqrt(t_0)) - g))) / cbrt(a)) + cbrt(((g + t_1) * (-0.5 / a)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

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

↓

function code(g, h, a)
	t_0 = Float64(h * Float64(-h))
	t_1 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	t_2 = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(t_1 - g))) + Float64(cbrt(Float64(Float64(g + sqrt(fma(g, g, t_0))) * -0.5)) / cbrt(a)))
	tmp = 0.0
	if (h <= -4.3708065047631245e-179)
		tmp = t_2;
	elseif (h <= 6.613043387155913e-258)
		tmp = Float64(Float64(cbrt(Float64(0.5 * Float64(hypot(g, sqrt(t_0)) - g))) / cbrt(a)) + cbrt(Float64(Float64(g + t_1) * Float64(-0.5 / a))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

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

↓

code[g_, h_, a_] := Block[{t$95$0 = N[(h * (-h)), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(t$95$1 - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(N[(g + N[Sqrt[N[(g * g + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -4.3708065047631245e-179], t$95$2, If[LessEqual[h, 6.613043387155913e-258], N[(N[(N[Power[N[(0.5 * N[(N[Sqrt[g ^ 2 + N[Sqrt[t$95$0], $MachinePrecision] ^ 2], $MachinePrecision] - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(g + t$95$1), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if h < -4.37080650476312451e-179 or 6.6130433871559132e-258 < h

   1. Initial program 37.8

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

   2. Simplified 37.8

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

   3. Applied egg-rr 42.8

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(g \cdot g - h \cdot h\right)\right)}}\right) \cdot \frac{-0.5}{a}} \]

   4. Applied egg-rr 41.1

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

   5. Applied egg-rr 35.9

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

   if -4.37080650476312451e-179 < h < 6.6130433871559132e-258

   1. Initial program 31.3

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

   2. Simplified 31.3

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

   3. Applied egg-rr 28.0

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

4. Final simplification 33.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -4.3708065047631245 \cdot 10^{-179}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \frac{\sqrt[3]{\left(g + \sqrt{\mathsf{fma}\left(g, g, h \cdot \left(-h\right)\right)}\right) \cdot -0.5}}{\sqrt[3]{a}}\\ \mathbf{elif}\;h \leq 6.613043387155913 \cdot 10^{-258}:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot \left(\mathsf{hypot}\left(g, \sqrt{h \cdot \left(-h\right)}\right) - g\right)}}{\sqrt[3]{a}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \frac{\sqrt[3]{\left(g + \sqrt{\mathsf{fma}\left(g, g, h \cdot \left(-h\right)\right)}\right) \cdot -0.5}}{\sqrt[3]{a}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022210 
(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))))))))