2-ancestry mixing, positive discriminant

Average Error: 36.1 → 32.6

Time: 12.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ t_1 := g + t_0\\ t_2 := \sqrt[3]{t_1 \cdot \frac{-0.5}{a}}\\ t_3 := t_0 - g\\ t_4 := \sqrt[3]{\frac{1}{2 \cdot a} \cdot t_3} + \sqrt[3]{t_1 \cdot \frac{-1}{2 \cdot a}}\\ t_5 := \frac{\sqrt[3]{0.5 \cdot t_3}}{\sqrt[3]{a}}\\ \mathbf{if}\;t_4 \leq -2 \cdot 10^{-97}:\\ \;\;\;\;t_5 + t_2\\ \mathbf{elif}\;t_4 \leq 0:\\ \;\;\;\;t_2 + \frac{\sqrt[3]{0.5 \cdot \left(g \cdot -2\right)}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;t_5 + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{t_1}\\ \end{array} \]

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 (sqrt (- (* g g) (* h h))))
        (t_1 (+ g t_0))
        (t_2 (cbrt (* t_1 (/ -0.5 a))))
        (t_3 (- t_0 g))
        (t_4
         (+
          (cbrt (* (/ 1.0 (* 2.0 a)) t_3))
          (cbrt (* t_1 (/ -1.0 (* 2.0 a))))))
        (t_5 (/ (cbrt (* 0.5 t_3)) (cbrt a))))
   (if (<= t_4 -2e-97)
     (+ t_5 t_2)
     (if (<= t_4 0.0)
       (+ t_2 (/ (cbrt (* 0.5 (* g -2.0))) (cbrt a)))
       (+ t_5 (* (cbrt (/ -0.5 a)) (cbrt t_1)))))))

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 = sqrt(((g * g) - (h * h)));
	double t_1 = g + t_0;
	double t_2 = cbrt((t_1 * (-0.5 / a)));
	double t_3 = t_0 - g;
	double t_4 = cbrt(((1.0 / (2.0 * a)) * t_3)) + cbrt((t_1 * (-1.0 / (2.0 * a))));
	double t_5 = cbrt((0.5 * t_3)) / cbrt(a);
	double tmp;
	if (t_4 <= -2e-97) {
		tmp = t_5 + t_2;
	} else if (t_4 <= 0.0) {
		tmp = t_2 + (cbrt((0.5 * (g * -2.0))) / cbrt(a));
	} else {
		tmp = t_5 + (cbrt((-0.5 / a)) * cbrt(t_1));
	}
	return tmp;
}

Java

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

↓

public static double code(double g, double h, double a) {
	double t_0 = Math.sqrt(((g * g) - (h * h)));
	double t_1 = g + t_0;
	double t_2 = Math.cbrt((t_1 * (-0.5 / a)));
	double t_3 = t_0 - g;
	double t_4 = Math.cbrt(((1.0 / (2.0 * a)) * t_3)) + Math.cbrt((t_1 * (-1.0 / (2.0 * a))));
	double t_5 = Math.cbrt((0.5 * t_3)) / Math.cbrt(a);
	double tmp;
	if (t_4 <= -2e-97) {
		tmp = t_5 + t_2;
	} else if (t_4 <= 0.0) {
		tmp = t_2 + (Math.cbrt((0.5 * (g * -2.0))) / Math.cbrt(a));
	} else {
		tmp = t_5 + (Math.cbrt((-0.5 / a)) * Math.cbrt(t_1));
	}
	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 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	t_1 = Float64(g + t_0)
	t_2 = cbrt(Float64(t_1 * Float64(-0.5 / a)))
	t_3 = Float64(t_0 - g)
	t_4 = Float64(cbrt(Float64(Float64(1.0 / Float64(2.0 * a)) * t_3)) + cbrt(Float64(t_1 * Float64(-1.0 / Float64(2.0 * a)))))
	t_5 = Float64(cbrt(Float64(0.5 * t_3)) / cbrt(a))
	tmp = 0.0
	if (t_4 <= -2e-97)
		tmp = Float64(t_5 + t_2);
	elseif (t_4 <= 0.0)
		tmp = Float64(t_2 + Float64(cbrt(Float64(0.5 * Float64(g * -2.0))) / cbrt(a)));
	else
		tmp = Float64(t_5 + Float64(cbrt(Float64(-0.5 / a)) * cbrt(t_1)));
	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[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(g + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(t$95$1 * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 - g), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[N[(N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(t$95$1 * N[(-1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Power[N[(0.5 * t$95$3), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -2e-97], N[(t$95$5 + t$95$2), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(t$95$2 + N[(N[Power[N[(0.5 * N[(g * -2.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$5 + N[(N[Power[N[(-0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[t$95$1, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Error

Bits error versus g

Bits error versus h

Bits error versus a

Derivation

1. Split input into 3 regimes

2. if (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < -2.00000000000000007e-97

   1. Initial program 10.3

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

   2. Simplified 10.3

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

   3. Applied egg-rr 8.4

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

   if -2.00000000000000007e-97 < (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < -0.0

   1. Initial program 54.4

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

   2. Simplified 54.4

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

   3. Applied egg-rr 38.0

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

   4. Taylor expanded in g around -inf 10.5

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

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

   1. Initial program 44.2

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

   2. Simplified 44.2

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

   3. Applied egg-rr 43.4

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

   4. Applied egg-rr 42.4

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

4. Final simplification 32.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} \leq -2 \cdot 10^{-97}:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)}}{\sqrt[3]{a}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}\\ \mathbf{elif}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} \leq 0:\\ \;\;\;\;\sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} + \frac{\sqrt[3]{0.5 \cdot \left(g \cdot -2\right)}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)}}{\sqrt[3]{a}} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + \sqrt{g \cdot g - h \cdot h}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022160 
(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))))))))