Toniolo and Linder, Equation (3a)

Average Error: 1.1 → 0.2

Time: 15.3s

Precision: binary64

Cost: 59144

\[ \begin{array}{c}[kx, ky] = \mathsf{sort}([kx, ky])\\ \end{array} \]

Math

\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]

↓

\[\begin{array}{l} t_0 := \sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)}\\ \mathbf{if}\;\sin ky \leq -6.802879538942135 \cdot 10^{-190}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\sin ky \leq 9.168740307205435 \cdot 10^{-169}:\\ \;\;\;\;\sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\ell}{Om} \cdot \left(2 \cdot \sin kx\right)\right)}\right)}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (sqrt
  (*
   (/ 1.0 2.0)
   (+
    1.0
    (/
     1.0
     (sqrt
      (+
       1.0
       (*
        (pow (/ (* 2.0 l) Om) 2.0)
        (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))))))

↓

(FPCore (l Om kx ky)
 :precision binary64
 (let* ((t_0
         (sqrt
          (*
           0.5
           (+
            1.0
            (/
             1.0
             (sqrt
              (+
               1.0
               (*
                (pow (/ (* 2.0 l) Om) 2.0)
                (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))))))))
   (if (<= (sin ky) -6.802879538942135e-190)
     t_0
     (if (<= (sin ky) 9.168740307205435e-169)
       (cbrt
        (pow (+ 0.5 (/ 0.5 (hypot 1.0 (* (/ l Om) (* 2.0 (sin kx)))))) 1.5))
       t_0))))

C

double code(double l, double Om, double kx, double ky) {
	return sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + (pow(((2.0 * l) / Om), 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))))))));
}

↓

double code(double l, double Om, double kx, double ky) {
	double t_0 = sqrt((0.5 * (1.0 + (1.0 / sqrt((1.0 + (pow(((2.0 * l) / Om), 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))))))));
	double tmp;
	if (sin(ky) <= -6.802879538942135e-190) {
		tmp = t_0;
	} else if (sin(ky) <= 9.168740307205435e-169) {
		tmp = cbrt(pow((0.5 + (0.5 / hypot(1.0, ((l / Om) * (2.0 * sin(kx)))))), 1.5));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	return Math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / Math.sqrt((1.0 + (Math.pow(((2.0 * l) / Om), 2.0) * (Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))))))));
}

↓

public static double code(double l, double Om, double kx, double ky) {
	double t_0 = Math.sqrt((0.5 * (1.0 + (1.0 / Math.sqrt((1.0 + (Math.pow(((2.0 * l) / Om), 2.0) * (Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))))))));
	double tmp;
	if (Math.sin(ky) <= -6.802879538942135e-190) {
		tmp = t_0;
	} else if (Math.sin(ky) <= 9.168740307205435e-169) {
		tmp = Math.cbrt(Math.pow((0.5 + (0.5 / Math.hypot(1.0, ((l / Om) * (2.0 * Math.sin(kx)))))), 1.5));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(l, Om, kx, ky)
	return sqrt(Float64(Float64(1.0 / 2.0) * Float64(1.0 + Float64(1.0 / sqrt(Float64(1.0 + Float64((Float64(Float64(2.0 * l) / Om) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))))))))
end

↓

function code(l, Om, kx, ky)
	t_0 = sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / sqrt(Float64(1.0 + Float64((Float64(Float64(2.0 * l) / Om) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))))))))
	tmp = 0.0
	if (sin(ky) <= -6.802879538942135e-190)
		tmp = t_0;
	elseif (sin(ky) <= 9.168740307205435e-169)
		tmp = cbrt((Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(Float64(l / Om) * Float64(2.0 * sin(kx)))))) ^ 1.5));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[l_, Om_, kx_, ky_] := N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[Sqrt[N[(1.0 + N[(N[Power[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

code[l_, Om_, kx_, ky_] := Block[{t$95$0 = N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[N[(1.0 + N[(N[Power[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Sin[ky], $MachinePrecision], -6.802879538942135e-190], t$95$0, If[LessEqual[N[Sin[ky], $MachinePrecision], 9.168740307205435e-169], N[Power[N[Power[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(N[(l / Om), $MachinePrecision] * N[(2.0 * N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < -6.80287953894213516e-190 or 9.16874030720543468e-169 < (sin.f64 ky)

   1. Initial program 0.1

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]

   if -6.80287953894213516e-190 < (sin.f64 ky) < 9.16874030720543468e-169

   1. Initial program 4.9

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]

   2. Taylor expanded in ky around 0 14.2

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \color{blue}{4 \cdot \frac{{\sin kx}^{2} \cdot {\ell}^{2}}{{Om}^{2}}}}}\right)} \]

   3. Simplified 4.9

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \color{blue}{4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin kx}^{2}\right)}}}\right)} \]
      Proof
      (*.f64 4 (*.f64 (*.f64 (/.f64 l Om) (/.f64 l Om)) (pow.f64 (sin.f64 kx) 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 4 (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 l l) (*.f64 Om Om))) (pow.f64 (sin.f64 kx) 2))): 64 points increase in error, 10 points decrease in error
      (*.f64 4 (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 Om Om)) (pow.f64 (sin.f64 kx) 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 4 (*.f64 (/.f64 (pow.f64 l 2) (Rewrite<= unpow2_binary64 (pow.f64 Om 2))) (pow.f64 (sin.f64 kx) 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 4 (Rewrite<= associate-/r/_binary64 (/.f64 (pow.f64 l 2) (/.f64 (pow.f64 Om 2) (pow.f64 (sin.f64 kx) 2))))): 7 points increase in error, 12 points decrease in error
      (*.f64 4 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 l 2) (pow.f64 (sin.f64 kx) 2)) (pow.f64 Om 2)))): 9 points increase in error, 3 points decrease in error
      (*.f64 4 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 (sin.f64 kx) 2) (pow.f64 l 2))) (pow.f64 Om 2))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 0.4

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\color{blue}{\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \sin kx\right)\right)}}\right)} \]

   5. Applied egg-rr 0.4

      \[\leadsto \color{blue}{\sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\ell}{Om} \cdot \left(\sin kx \cdot 2\right)\right)}\right)}^{1.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -6.802879538942135 \cdot 10^{-190}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)}\\ \mathbf{elif}\;\sin ky \leq 9.168740307205435 \cdot 10^{-169}:\\ \;\;\;\;\sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\ell}{Om} \cdot \left(2 \cdot \sin kx\right)\right)}\right)}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	5.1
Cost	20296

\[\begin{array}{l} t_0 := {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\ell}{Om} \cdot \left(2 \cdot \sin kx\right)\right)}\right)}^{0.5}\\ \mathbf{if}\;Om \leq -1.0156071211543737 \cdot 10^{-216}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;Om \leq 8.78418277384577 \cdot 10^{-103}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	1.0
Cost	20228

\[\begin{array}{l} \mathbf{if}\;ky \leq 2.937334285857123 \cdot 10^{-183}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, 2 \cdot \left(\sin kx \cdot \frac{\ell}{Om}\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, \left(\sin ky \cdot \ell\right) \cdot \frac{2}{Om}\right)}\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022291 
(FPCore (l Om kx ky)
  :name "Toniolo and Linder, Equation (3a)"
  :precision binary64
  (sqrt (* (/ 1.0 2.0) (+ 1.0 (/ 1.0 (sqrt (+ 1.0 (* (pow (/ (* 2.0 l) Om) 2.0) (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))))))