Migdal et al, Equation (64)

Average Error: 0.6 → 0.5

Time: 11.8s

Precision: binary64

Cost: 13504

Math

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

↓

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

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
 (* (sqrt 0.5) (* (+ (* a1 a1) (* a2 a2)) (cos th))))

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 sqrt(0.5) * (((a1 * a1) + (a2 * a2)) * cos(th));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = ((cos(th) / sqrt(2.0d0)) * (a1 * a1)) + ((cos(th) / sqrt(2.0d0)) * (a2 * a2))
end function

↓

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = sqrt(0.5d0) * (((a1 * a1) + (a2 * a2)) * cos(th))
end function

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.sqrt(0.5) * (((a1 * a1) + (a2 * a2)) * Math.cos(th));
}

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.sqrt(0.5) * (((a1 * a1) + (a2 * a2)) * math.cos(th))

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(sqrt(0.5) * Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * cos(th)))
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 = sqrt(0.5) * (((a1 * a1) + (a2 * a2)) * cos(th));
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[Sqrt[0.5], $MachinePrecision] * N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.6

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

2. Simplified 0.6

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

   [Start] 0.6	\[ \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   distribute-lft-out [=>] 0.6	\[ \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]

3. Applied egg-rr 0.5

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

4. Taylor expanded in th around inf 0.5

   \[\leadsto \color{blue}{\sqrt{0.5} \cdot \left(\left({a2}^{2} + {a1}^{2}\right) \cdot \cos th\right)} \]

5. Simplified 0.5

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

   [Start] 0.5	\[ \sqrt{0.5} \cdot \left(\left({a2}^{2} + {a1}^{2}\right) \cdot \cos th\right) \]
   +-commutative [=>] 0.5	\[ \sqrt{0.5} \cdot \left(\color{blue}{\left({a1}^{2} + {a2}^{2}\right)} \cdot \cos th\right) \]
   unpow2 [=>] 0.5	\[ \sqrt{0.5} \cdot \left(\left(\color{blue}{a1 \cdot a1} + {a2}^{2}\right) \cdot \cos th\right) \]
   unpow2 [=>] 0.5	\[ \sqrt{0.5} \cdot \left(\left(a1 \cdot a1 + \color{blue}{a2 \cdot a2}\right) \cdot \cos th\right) \]

6. Final simplification 0.5

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

Alternatives

Alternative 1
Error	27.9
Cost	19780

\[\begin{array}{l} \mathbf{if}\;\cos th \leq 1:\\ \;\;\;\;\cos th \cdot \frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \end{array} \]

Alternative 2
Error	21.7
Cost	13380

\[\begin{array}{l} \mathbf{if}\;a1 \leq -5.8 \cdot 10^{-148}:\\ \;\;\;\;\cos th \cdot \frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]

Alternative 3
Error	21.7
Cost	13380

\[\begin{array}{l} \mathbf{if}\;a1 \leq -6.2 \cdot 10^{-148}:\\ \;\;\;\;\cos th \cdot \frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot \left(a2 \cdot \cos th\right)\right)\\ \end{array} \]

Alternative 4
Error	21.7
Cost	13380

\[\begin{array}{l} \mathbf{if}\;a1 \leq -8 \cdot 10^{-148}:\\ \;\;\;\;\cos th \cdot \frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\left(a2 \cdot a2\right) \cdot \cos th\right)\\ \end{array} \]

Alternative 5
Error	25.6
Cost	6976

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

Alternative 6
Error	36.8
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a1 \leq -5.8 \cdot 10^{-138}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot a1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot a2\right)\\ \end{array} \]

Alternative 7
Error	36.8
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a1 \leq -5.2 \cdot 10^{-138}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot a2\right)\\ \end{array} \]

Alternative 8
Error	36.8
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a1 \leq -5.8 \cdot 10^{-138}:\\ \;\;\;\;\frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot a2\right)\\ \end{array} \]

Alternative 9
Error	40.5
Cost	6720

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

Reproduce

herbie shell --seed 2023056 
(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))))