Toniolo and Linder, Equation (3a)

Average Error: 1.1 → 0.0

Time: 15.7s

Precision: binary64

Cost: 32832

Math

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

↓

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

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 (hypot 1.0 (* (hypot (sin ky) (sin kx)) (/ (* 2.0 l) Om)))))))

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 / hypot(1.0, (hypot(sin(ky), sin(kx)) * ((2.0 * l) / Om))))));
}

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.hypot(1.0, (Math.hypot(Math.sin(ky), Math.sin(kx)) * ((2.0 * l) / Om))))));
}

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.hypot(1.0, (math.hypot(math.sin(ky), math.sin(kx)) * ((2.0 * l) / Om))))))

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 / hypot(1.0, Float64(hypot(sin(ky), sin(kx)) * Float64(Float64(2.0 * l) / Om))))))
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 / hypot(1.0, (hypot(sin(ky), sin(kx)) * ((2.0 * l) / Om))))));
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[Sqrt[1.0 ^ 2 + N[(N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] * N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 1.1

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

2. Simplified 1.1

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

   \[\leadsto \sqrt{0.5 + 0.5 \cdot \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)} - 1}}} \]

4. Simplified 0.0

   \[\leadsto \sqrt{0.5 + 0.5 \cdot \frac{1}{\color{blue}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}}} \]
   Proof
   (hypot.f64 1 (*.f64 (hypot.f64 (sin.f64 ky) (sin.f64 kx)) (*.f64 2 (/.f64 l Om)))): 0 points increase in error, 0 points decrease in error
   (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 points increase in error, 6 points decrease in error
   (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)))): 0 points increase in error, 0 points decrease in error
   (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)))): 0 points increase in error, 0 points decrease in error
   (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)))): 0 points increase in error, 0 points decrease in error
   (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)))): 0 points increase in error, 0 points decrease in error
   (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)))): 0 points increase in error, 0 points decrease in error
   (hypot.f64 1 (*.f64 (Rewrite=> hypot-def_binary64 (hypot.f64 (sin.f64 kx) (sin.f64 ky))) (*.f64 2 (/.f64 l Om)))): 6 points increase in error, 2 points decrease in error
   (hypot.f64 1 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 2 (/.f64 l Om)) (hypot.f64 (sin.f64 kx) (sin.f64 ky))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= expm1-log1p_binary64 (expm1.f64 (log1p.f64 (hypot.f64 1 (*.f64 (*.f64 2 (/.f64 l Om)) (hypot.f64 (sin.f64 kx) (sin.f64 ky))))))): 41 points increase in error, 42 points decrease in error
   (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (log1p.f64 (hypot.f64 1 (*.f64 (*.f64 2 (/.f64 l Om)) (hypot.f64 (sin.f64 kx) (sin.f64 ky)))))) 1)): 8 points increase in error, 8 points decrease in error

5. Applied egg-rr 0.0

   \[\leadsto \sqrt{0.5 + \color{blue}{\left(0 + \frac{0.5}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}\right)}} \]

6. Simplified 0.0

   \[\leadsto \sqrt{0.5 + \color{blue}{\frac{0.5}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \frac{2 \cdot \ell}{Om}\right)}}} \]
   Proof
   (/.f64 1/2 (hypot.f64 1 (*.f64 (hypot.f64 (sin.f64 ky) (sin.f64 kx)) (/.f64 (*.f64 2 l) Om)))): 0 points increase in error, 0 points decrease in error
   (/.f64 1/2 (hypot.f64 1 (*.f64 (hypot.f64 (sin.f64 ky) (sin.f64 kx)) (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 l Om)))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-lft-identity_binary64 (+.f64 0 (/.f64 1/2 (hypot.f64 1 (*.f64 (hypot.f64 (sin.f64 ky) (sin.f64 kx)) (*.f64 2 (/.f64 l Om))))))): 0 points increase in error, 0 points decrease in error

7. Final simplification 0.0

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

Alternatives

Alternative 1
Error	3.1
Cost	33032

\[\begin{array}{l} t_0 := \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin kx \cdot \frac{2 \cdot \ell}{Om}\right)}}\\ \mathbf{if}\;\sin kx \leq -5 \cdot 10^{-249}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\sin kx \leq 10^{-127}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, 2 \cdot \frac{ky \cdot \ell}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	3.0
Cost	26756

\[\begin{array}{l} \mathbf{if}\;ky \leq 4 \cdot 10^{-185}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin kx \cdot \frac{2 \cdot \ell}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + {\left(2 \cdot \left(\sin ky \cdot \frac{\ell}{Om}\right)\right)}^{2}}}}\\ \end{array} \]

Alternative 3
Error	3.0
Cost	20228

\[\begin{array}{l} \mathbf{if}\;ky \leq 4 \cdot 10^{-185}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin kx \cdot \frac{2 \cdot \ell}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, \sin ky \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}}\\ \end{array} \]

Alternative 4
Error	9.4
Cost	14096

\[\begin{array}{l} t_0 := \sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, 2 \cdot \frac{ky \cdot \ell}{Om}\right)}}\\ t_1 := \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\ell + \ell}{\frac{Om}{kx}}\right)}}\\ \mathbf{if}\;kx \leq -9.5 \cdot 10^{+54}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;kx \leq -1.18 \cdot 10^{-250}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;kx \leq 10^{-127}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;kx \leq 1.5 \cdot 10^{+171}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	9.5
Cost	13832

\[\begin{array}{l} t_0 := \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\ell + \ell}{\frac{Om}{kx}}\right)}}\\ \mathbf{if}\;\ell \leq -5.2 \cdot 10^{-188}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 8.2 \cdot 10^{-219}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	13.5
Cost	6728

\[\begin{array}{l} \mathbf{if}\;Om \leq -1.75 \cdot 10^{+21}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 6.8 \cdot 10^{-50}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	24.2
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022338 
(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))))))))))