2-ancestry mixing, positive discriminant

Average Error: 36.4 → 31.9

Time: 12.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h} - g\\ \mathbf{if}\;g \leq -1.1010205556891214 \cdot 10^{-160}:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot t_0}}{\sqrt[3]{a}} + \sqrt[3]{\left(0.5 \cdot \left(h \cdot \frac{h}{g}\right)\right) \cdot \frac{-0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{t_0 \cdot \frac{0.5}{a}} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + \left(g + \frac{\left(h \cdot h\right) \cdot -0.5}{g}\right)}\\ \end{array} \]

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))) g)))
   (if (<= g -1.1010205556891214e-160)
     (+
      (/ (cbrt (* 0.5 t_0)) (cbrt a))
      (cbrt (* (* 0.5 (* h (/ h g))) (/ -0.5 a))))
     (+
      (cbrt (* t_0 (/ 0.5 a)))
      (* (cbrt (/ -0.5 a)) (cbrt (+ g (+ g (/ (* (* h h) -0.5) g)))))))))

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))) - g;
	double tmp;
	if (g <= -1.1010205556891214e-160) {
		tmp = (cbrt((0.5 * t_0)) / cbrt(a)) + cbrt(((0.5 * (h * (h / g))) * (-0.5 / a)));
	} else {
		tmp = cbrt((t_0 * (0.5 / a))) + (cbrt((-0.5 / a)) * cbrt((g + (g + (((h * h) * -0.5) / g)))));
	}
	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))) - g;
	double tmp;
	if (g <= -1.1010205556891214e-160) {
		tmp = (Math.cbrt((0.5 * t_0)) / Math.cbrt(a)) + Math.cbrt(((0.5 * (h * (h / g))) * (-0.5 / a)));
	} else {
		tmp = Math.cbrt((t_0 * (0.5 / a))) + (Math.cbrt((-0.5 / a)) * Math.cbrt((g + (g + (((h * h) * -0.5) / g)))));
	}
	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(sqrt(Float64(Float64(g * g) - Float64(h * h))) - g)
	tmp = 0.0
	if (g <= -1.1010205556891214e-160)
		tmp = Float64(Float64(cbrt(Float64(0.5 * t_0)) / cbrt(a)) + cbrt(Float64(Float64(0.5 * Float64(h * Float64(h / g))) * Float64(-0.5 / a))));
	else
		tmp = Float64(cbrt(Float64(t_0 * Float64(0.5 / a))) + Float64(cbrt(Float64(-0.5 / a)) * cbrt(Float64(g + Float64(g + Float64(Float64(Float64(h * h) * -0.5) / g))))));
	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[(N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - g), $MachinePrecision]}, If[LessEqual[g, -1.1010205556891214e-160], N[(N[(N[Power[N[(0.5 * t$95$0), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(0.5 * N[(h * N[(h / g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(t$95$0 * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(-0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(g + N[(g + N[(N[(N[(h * h), $MachinePrecision] * -0.5), $MachinePrecision] / g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Error

Bits error versus g

Bits error versus h

Bits error versus a

Derivation

1. Split input into 2 regimes

2. if g < -1.10102055568912144e-160

   1. Initial program 35.1

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

   2. Simplified 35.1

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

   3. Applied egg-rr 31.7

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

   4. Taylor expanded in g around -inf 31.4

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

   5. Simplified 31.4

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

   if -1.10102055568912144e-160 < g

   1. Initial program 37.5

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

   2. Simplified 37.4

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

   3. Applied egg-rr 33.7

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

   4. Taylor expanded in g around inf 32.3

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

   5. Simplified 32.3

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

4. Final simplification 31.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -1.1010205556891214 \cdot 10^{-160}:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)}}{\sqrt[3]{a}} + \sqrt[3]{\left(0.5 \cdot \left(h \cdot \frac{h}{g}\right)\right) \cdot \frac{-0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} - g\right) \cdot \frac{0.5}{a}} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + \left(g + \frac{\left(h \cdot h\right) \cdot -0.5}{g}\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022150 
(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))))))))