Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 17.1s

Precision: binary64

Cost: 26112

Math

\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]

↓

\[\cos th \cdot \left({2}^{-0.5} \cdot {\left(\mathsf{hypot}\left(a2, a1\right)\right)}^{2}\right) \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (+
  (* (/ (cos th) (sqrt 2.0)) (* a1 a1))
  (* (/ (cos th) (sqrt 2.0)) (* a2 a2))))

↓

(FPCore (a1 a2 th)
 :precision binary64
 (* (cos th) (* (pow 2.0 -0.5) (pow (hypot a2 a1) 2.0))))

C

double code(double a1, double a2, double th) {
	return ((cos(th) / sqrt(2.0)) * (a1 * a1)) + ((cos(th) / sqrt(2.0)) * (a2 * a2));
}

↓

double code(double a1, double a2, double th) {
	return cos(th) * (pow(2.0, -0.5) * pow(hypot(a2, a1), 2.0));
}

Java

public static double code(double a1, double a2, double th) {
	return ((Math.cos(th) / Math.sqrt(2.0)) * (a1 * a1)) + ((Math.cos(th) / Math.sqrt(2.0)) * (a2 * a2));
}

↓

public static double code(double a1, double a2, double th) {
	return Math.cos(th) * (Math.pow(2.0, -0.5) * Math.pow(Math.hypot(a2, a1), 2.0));
}

Python

def code(a1, a2, th):
	return ((math.cos(th) / math.sqrt(2.0)) * (a1 * a1)) + ((math.cos(th) / math.sqrt(2.0)) * (a2 * a2))

↓

def code(a1, a2, th):
	return math.cos(th) * (math.pow(2.0, -0.5) * math.pow(math.hypot(a2, a1), 2.0))

Julia

function code(a1, a2, th)
	return Float64(Float64(Float64(cos(th) / sqrt(2.0)) * Float64(a1 * a1)) + Float64(Float64(cos(th) / sqrt(2.0)) * Float64(a2 * a2)))
end

↓

function code(a1, a2, th)
	return Float64(cos(th) * Float64((2.0 ^ -0.5) * (hypot(a2, a1) ^ 2.0)))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = ((cos(th) / sqrt(2.0)) * (a1 * a1)) + ((cos(th) / sqrt(2.0)) * (a2 * a2));
end

↓

function tmp = code(a1, a2, th)
	tmp = cos(th) * ((2.0 ^ -0.5) * (hypot(a2, a1) ^ 2.0));
end

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(N[Power[2.0, -0.5], $MachinePrecision] * N[Power[N[Sqrt[a2 ^ 2 + a1 ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.5

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]

2. Applied egg-rr 0.5

   \[\leadsto \color{blue}{\cos th \cdot \left({2}^{-0.5} \cdot {\left(\mathsf{hypot}\left(a2, a1\right)\right)}^{2}\right)} \]

3. Final simplification 0.5

   \[\leadsto \cos th \cdot \left({2}^{-0.5} \cdot {\left(\mathsf{hypot}\left(a2, a1\right)\right)}^{2}\right) \]

Alternatives

Alternative 1
Error	0.5
Cost	26048

\[\frac{{\left(\mathsf{hypot}\left(a2, a1\right)\right)}^{2}}{\frac{\sqrt{2}}{\cos th}} \]

Alternative 2
Error	0.5
Cost	13568

\[\cos th \cdot \left({2}^{-0.5} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right)\right) \]

Alternative 3
Error	14.4
Cost	13512

\[\begin{array}{l} t_1 := \left(a1 \cdot a1\right) \cdot \left(\cos th \cdot \sqrt{0.5}\right)\\ \mathbf{if}\;th \leq -9.005873630663595 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;th \leq 0.898846679155726:\\ \;\;\;\;\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	14.4
Cost	13512

\[\begin{array}{l} \mathbf{if}\;th \leq -9.005873630663595 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot \left(\cos th \cdot a1\right)\right)\\ \mathbf{elif}\;th \leq 0.898846679155726:\\ \;\;\;\;\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \left(\cos th \cdot \sqrt{0.5}\right)\\ \end{array} \]

Alternative 5
Error	0.5
Cost	13504

\[\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \left(\cos th \cdot \sqrt{0.5}\right) \]

Alternative 6
Error	20.7
Cost	13380

\[\begin{array}{l} \mathbf{if}\;a1 \leq -5.054766357456002 \cdot 10^{-114}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot \left(\cos th \cdot a1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\right)\\ \end{array} \]

Alternative 7
Error	20.7
Cost	13380

\[\begin{array}{l} \mathbf{if}\;a1 \leq -5.054766357456002 \cdot 10^{-114}:\\ \;\;\;\;\cos th \cdot \left(\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\right)\\ \end{array} \]

Alternative 8
Error	20.7
Cost	13380

\[\begin{array}{l} \mathbf{if}\;a1 \leq -5.054766357456002 \cdot 10^{-114}:\\ \;\;\;\;\cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\right)\\ \end{array} \]

Alternative 9
Error	26.1
Cost	6976

\[\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \sqrt{0.5} \]

Alternative 10
Error	36.9
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a1 \leq -6.574657707543116 \cdot 10^{-133}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \end{array} \]

Alternative 11
Error	36.9
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a1 \leq -6.574657707543116 \cdot 10^{-133}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \]

Alternative 12
Error	36.9
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a1 \leq -6.574657707543116 \cdot 10^{-133}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \]

Alternative 13
Error	40.5
Cost	6720

\[a2 \cdot \frac{a2}{\sqrt{2}} \]

Alternative 14
Error	40.5
Cost	6720

\[\frac{a2 \cdot a2}{\sqrt{2}} \]

Reproduce

herbie shell --seed 2022294 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2.0)) (* a1 a1)) (* (/ (cos th) (sqrt 2.0)) (* a2 a2))))