Toniolo and Linder, Equation (3a)

Average Error: 0.9 → 0.0

Time: 14.5s

Precision: binary64

Cost: 45696

Math

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

↓

\[\sqrt{0.5 + 0.5 \cdot {\left(\sqrt{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}\right)}^{-2}} \]

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
 (sqrt
  (+
   0.5
   (*
    0.5
    (pow
     (sqrt (hypot 1.0 (* (hypot (sin ky) (sin kx)) (* 2.0 (/ l Om)))))
     -2.0)))))

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) {
	return sqrt((0.5 + (0.5 * pow(sqrt(hypot(1.0, (hypot(sin(ky), sin(kx)) * (2.0 * (l / Om))))), -2.0))));
}

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) {
	return Math.sqrt((0.5 + (0.5 * Math.pow(Math.sqrt(Math.hypot(1.0, (Math.hypot(Math.sin(ky), Math.sin(kx)) * (2.0 * (l / Om))))), -2.0))));
}

Python

def code(l, Om, kx, 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)))))))))

↓

def code(l, Om, kx, ky):
	return math.sqrt((0.5 + (0.5 * math.pow(math.sqrt(math.hypot(1.0, (math.hypot(math.sin(ky), math.sin(kx)) * (2.0 * (l / Om))))), -2.0))))

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)
	return sqrt(Float64(0.5 + Float64(0.5 * (sqrt(hypot(1.0, Float64(hypot(sin(ky), sin(kx)) * Float64(2.0 * Float64(l / Om))))) ^ -2.0))))
end

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + ((((2.0 * l) / Om) ^ 2.0) * ((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))))))));
end

↓

function tmp = code(l, Om, kx, ky)
	tmp = sqrt((0.5 + (0.5 * (sqrt(hypot(1.0, (hypot(sin(ky), sin(kx)) * (2.0 * (l / Om))))) ^ -2.0))));
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_] := N[Sqrt[N[(0.5 + N[(0.5 * N[Power[N[Sqrt[N[Sqrt[1.0 ^ 2 + N[(N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] * N[(2.0 * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 0.9

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

2. Simplified 0.9

   \[\leadsto \color{blue}{\sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}}} \]
   Proof
   (sqrt.f64 (+.f64 1/2 (*.f64 1/2 (/.f64 1 (sqrt.f64 (+.f64 1 (*.f64 (pow.f64 (/.f64 2 (/.f64 Om l)) 2) (+.f64 (pow.f64 (sin.f64 kx) 2) (pow.f64 (sin.f64 ky) 2))))))))): 0 points increase in error, 0 points decrease in error
   (sqrt.f64 (+.f64 (Rewrite<= metadata-eval (*.f64 1/2 1)) (*.f64 1/2 (/.f64 1 (sqrt.f64 (+.f64 1 (*.f64 (pow.f64 (/.f64 2 (/.f64 Om l)) 2) (+.f64 (pow.f64 (sin.f64 kx) 2) (pow.f64 (sin.f64 ky) 2))))))))): 0 points increase in error, 0 points decrease in error
   (sqrt.f64 (+.f64 (*.f64 (Rewrite<= metadata-eval (/.f64 1 2)) 1) (*.f64 1/2 (/.f64 1 (sqrt.f64 (+.f64 1 (*.f64 (pow.f64 (/.f64 2 (/.f64 Om l)) 2) (+.f64 (pow.f64 (sin.f64 kx) 2) (pow.f64 (sin.f64 ky) 2))))))))): 0 points increase in error, 0 points decrease in error
   (sqrt.f64 (+.f64 (*.f64 (/.f64 1 2) 1) (*.f64 (Rewrite<= metadata-eval (/.f64 1 2)) (/.f64 1 (sqrt.f64 (+.f64 1 (*.f64 (pow.f64 (/.f64 2 (/.f64 Om l)) 2) (+.f64 (pow.f64 (sin.f64 kx) 2) (pow.f64 (sin.f64 ky) 2))))))))): 0 points increase in error, 0 points decrease in error
   (sqrt.f64 (+.f64 (*.f64 (/.f64 1 2) 1) (*.f64 (/.f64 1 2) (/.f64 1 (sqrt.f64 (+.f64 1 (*.f64 (pow.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 2 l) Om)) 2) (+.f64 (pow.f64 (sin.f64 kx) 2) (pow.f64 (sin.f64 ky) 2))))))))): 0 points increase in error, 0 points decrease in error
   (sqrt.f64 (Rewrite<= distribute-lft-in_binary64 (*.f64 (/.f64 1 2) (+.f64 1 (/.f64 1 (sqrt.f64 (+.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 2 l) Om) 2) (+.f64 (pow.f64 (sin.f64 kx) 2) (pow.f64 (sin.f64 ky) 2)))))))))): 0 points increase in error, 0 points decrease in error

3. Applied egg-rr 0.0

   \[\leadsto \sqrt{0.5 + 0.5 \cdot \color{blue}{\left({\left(\sqrt{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}\right)}^{-1} \cdot {\left(\sqrt{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}\right)}^{-1}\right)}} \]

4. Simplified 0.0

   \[\leadsto \sqrt{0.5 + 0.5 \cdot \color{blue}{{\left(\sqrt{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}\right)}^{-2}}} \]
   Proof
   (pow.f64 (sqrt.f64 (hypot.f64 1 (*.f64 (hypot.f64 (sin.f64 ky) (sin.f64 kx)) (*.f64 2 (/.f64 l Om))))) -2): 0 points increase in error, 0 points decrease in error
   (pow.f64 (sqrt.f64 (hypot.f64 1 (*.f64 (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 (sin.f64 ky) (sin.f64 ky)) (*.f64 (sin.f64 kx) (sin.f64 kx))))) (*.f64 2 (/.f64 l Om))))) -2): 4 points increase in error, 4 points decrease in error
   (pow.f64 (sqrt.f64 (hypot.f64 1 (*.f64 (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 (sin.f64 ky) 2)) (*.f64 (sin.f64 kx) (sin.f64 kx)))) (*.f64 2 (/.f64 l Om))))) -2): 0 points increase in error, 0 points decrease in error
   (pow.f64 (sqrt.f64 (hypot.f64 1 (*.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 ky) 2) (Rewrite<= unpow2_binary64 (pow.f64 (sin.f64 kx) 2)))) (*.f64 2 (/.f64 l Om))))) -2): 0 points increase in error, 0 points decrease in error
   (pow.f64 (sqrt.f64 (hypot.f64 1 (*.f64 (sqrt.f64 (Rewrite=> +-commutative_binary64 (+.f64 (pow.f64 (sin.f64 kx) 2) (pow.f64 (sin.f64 ky) 2)))) (*.f64 2 (/.f64 l Om))))) -2): 0 points increase in error, 0 points decrease in error
   (pow.f64 (sqrt.f64 (hypot.f64 1 (*.f64 (sqrt.f64 (+.f64 (Rewrite=> unpow2_binary64 (*.f64 (sin.f64 kx) (sin.f64 kx))) (pow.f64 (sin.f64 ky) 2))) (*.f64 2 (/.f64 l Om))))) -2): 0 points increase in error, 0 points decrease in error
   (pow.f64 (sqrt.f64 (hypot.f64 1 (*.f64 (sqrt.f64 (+.f64 (*.f64 (sin.f64 kx) (sin.f64 kx)) (Rewrite=> unpow2_binary64 (*.f64 (sin.f64 ky) (sin.f64 ky))))) (*.f64 2 (/.f64 l Om))))) -2): 0 points increase in error, 0 points decrease in error
   (pow.f64 (sqrt.f64 (hypot.f64 1 (*.f64 (Rewrite=> hypot-def_binary64 (hypot.f64 (sin.f64 kx) (sin.f64 ky))) (*.f64 2 (/.f64 l Om))))) -2): 4 points increase in error, 4 points decrease in error
   (pow.f64 (sqrt.f64 (hypot.f64 1 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 2 (/.f64 l Om)) (hypot.f64 (sin.f64 kx) (sin.f64 ky)))))) -2): 0 points increase in error, 0 points decrease in error
   (pow.f64 (sqrt.f64 (hypot.f64 1 (*.f64 (*.f64 2 (/.f64 l Om)) (hypot.f64 (sin.f64 kx) (sin.f64 ky))))) (Rewrite<= metadata-eval (*.f64 2 -1))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 (sqrt.f64 (hypot.f64 1 (*.f64 (*.f64 2 (/.f64 l Om)) (hypot.f64 (sin.f64 kx) (sin.f64 ky))))) -1) (pow.f64 (sqrt.f64 (hypot.f64 1 (*.f64 (*.f64 2 (/.f64 l Om)) (hypot.f64 (sin.f64 kx) (sin.f64 ky))))) -1))): 18 points increase in error, 21 points decrease in error

5. Final simplification 0.0

   \[\leadsto \sqrt{0.5 + 0.5 \cdot {\left(\sqrt{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}\right)}^{-2}} \]

Alternatives

Alternative 1
Error	2.5
Cost	27272

\[\begin{array}{l} t_0 := \frac{2 \cdot \ell}{Om}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+72}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;t_0 \leq 10^{+33}:\\ \;\;\;\;{\left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin kx \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}\right)}^{1.5}\right)}^{0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 2
Error	2.5
Cost	20744

\[\begin{array}{l} t_0 := \frac{2 \cdot \ell}{Om}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+72}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;t_0 \leq 10^{+33}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin kx \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 3
Error	3.4
Cost	7240

\[\begin{array}{l} t_0 := \frac{2 \cdot \ell}{Om}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;t_0 \leq 118000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 4
Error	23.7
Cost	6728

\[\begin{array}{l} \mathbf{if}\;Om \leq -2.5 \cdot 10^{+48}:\\ \;\;\;\;1 + \frac{kx \cdot kx}{\frac{Om \cdot Om}{\ell \cdot \ell}} \cdot -0.5\\ \mathbf{elif}\;Om \leq 115000000:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{ky \cdot ky}}\\ \end{array} \]

Alternative 5
Error	39.6
Cost	1092

\[\begin{array}{l} \mathbf{if}\;kx \leq -1.45 \cdot 10^{+151}:\\ \;\;\;\;1 + -0.5 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{ky \cdot ky}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{kx \cdot kx}{\frac{Om \cdot Om}{\ell \cdot \ell}} \cdot -0.5\\ \end{array} \]

Alternative 6
Error	42.0
Cost	960

\[1 + \frac{kx \cdot kx}{\frac{Om \cdot Om}{\ell \cdot \ell}} \cdot -0.5 \]

Reproduce

herbie shell --seed 2022329 
(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))))))))))