2-ancestry mixing, positive discriminant

Average Error: 36.4 → 35.8

Time: 14.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{2 \cdot a}\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ t_2 := \sqrt[3]{\left(g + t_1\right) \cdot \frac{-1}{2 \cdot a}}\\ t_3 := \sqrt[3]{t_0 \cdot \left(t_1 - g\right)}\\ \mathbf{if}\;t_3 + t_2 \leq 3.7663600711920855 \cdot 10^{+98}:\\ \;\;\;\;t_3 + \sqrt[3]{t_0 \cdot \mathsf{fma}\left(-1, g, -t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2 + e^{0.3333333333333333 \cdot \left(\log \left(\sqrt{{g}^{2} - {h}^{2}} - g\right) - \log a\right)} \cdot \sqrt[3]{0.5}\\ \end{array} \]

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 (/ 1.0 (* 2.0 a)))
        (t_1 (sqrt (- (* g g) (* h h))))
        (t_2 (cbrt (* (+ g t_1) (/ -1.0 (* 2.0 a)))))
        (t_3 (cbrt (* t_0 (- t_1 g)))))
   (if (<= (+ t_3 t_2) 3.7663600711920855e+98)
     (+ t_3 (cbrt (* t_0 (fma -1.0 g (- t_1)))))
     (+
      t_2
      (*
       (exp
        (*
         0.3333333333333333
         (- (log (- (sqrt (- (pow g 2.0) (pow h 2.0))) g)) (log a))))
       (cbrt 0.5))))))

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 = 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 t_3 = cbrt((t_0 * (t_1 - g)));
	double tmp;
	if ((t_3 + t_2) <= 3.7663600711920855e+98) {
		tmp = t_3 + cbrt((t_0 * fma(-1.0, g, -t_1)));
	} else {
		tmp = t_2 + (exp((0.3333333333333333 * (log((sqrt((pow(g, 2.0) - pow(h, 2.0))) - g)) - log(a)))) * cbrt(0.5));
	}
	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(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))))
	t_3 = cbrt(Float64(t_0 * Float64(t_1 - g)))
	tmp = 0.0
	if (Float64(t_3 + t_2) <= 3.7663600711920855e+98)
		tmp = Float64(t_3 + cbrt(Float64(t_0 * fma(-1.0, g, Float64(-t_1)))));
	else
		tmp = Float64(t_2 + Float64(exp(Float64(0.3333333333333333 * Float64(log(Float64(sqrt(Float64((g ^ 2.0) - (h ^ 2.0))) - g)) - log(a)))) * cbrt(0.5)));
	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[(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]}, Block[{t$95$3 = N[Power[N[(t$95$0 * N[(t$95$1 - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(t$95$3 + t$95$2), $MachinePrecision], 3.7663600711920855e+98], N[(t$95$3 + N[Power[N[(t$95$0 * N[(-1.0 * g + (-t$95$1)), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(N[Exp[N[(0.3333333333333333 * N[(N[Log[N[(N[Sqrt[N[(N[Power[g, 2.0], $MachinePrecision] - N[Power[h, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - g), $MachinePrecision]], $MachinePrecision] - N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[0.5, 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 (+.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))))))) < 3.76636007119208554e98

   1. Initial program 12.1

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

   2. Applied egg-rr 18.5

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

   3. Applied egg-rr 12.1

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

   if 3.76636007119208554e98 < (+.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 63.5

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

   2. Taylor expanded in a around 0 62.2

      \[\leadsto \color{blue}{e^{0.3333333333333333 \cdot \left(\log \left(\sqrt{{g}^{2} - {h}^{2}} - g\right) - \log a\right)} \cdot \sqrt[3]{0.5}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} \leq 3.7663600711920855 \cdot 10^{+98}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \mathsf{fma}\left(-1, g, -\sqrt{g \cdot g - h \cdot h}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} + e^{0.3333333333333333 \cdot \left(\log \left(\sqrt{{g}^{2} - {h}^{2}} - g\right) - \log a\right)} \cdot \sqrt[3]{0.5}\\ \end{array} \]

Reproduce

herbie shell --seed 2022140 
(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))))))))