Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 14.3s

Precision: binary64

Cost: 20032

Math

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

↓

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

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) (+ (/ a2 (/ (sqrt 2.0) a2)) (/ a1 (/ (sqrt 2.0) a1)))))

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) * ((a2 / (sqrt(2.0) / a2)) + (a1 / (sqrt(2.0) / a1)));
}

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 = cos(th) * ((a2 / (sqrt(2.0d0) / a2)) + (a1 / (sqrt(2.0d0) / a1)))
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.cos(th) * ((a2 / (Math.sqrt(2.0) / a2)) + (a1 / (Math.sqrt(2.0) / a1)));
}

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) * ((a2 / (math.sqrt(2.0) / a2)) + (a1 / (math.sqrt(2.0) / a1)))

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(Float64(a2 / Float64(sqrt(2.0) / a2)) + Float64(a1 / Float64(sqrt(2.0) / a1))))
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) * ((a2 / (sqrt(2.0) / a2)) + (a1 / (sqrt(2.0) / a1)));
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[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / a2), $MachinePrecision]), $MachinePrecision] + N[(a1 / N[(N[Sqrt[2.0], $MachinePrecision] / a1), $MachinePrecision]), $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. Simplified 0.5

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

   [Start] 0.5	\[ \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   distribute-lft-out [=>] 0.5	\[ \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]
   associate-*l/ [=>] 0.5	\[ \color{blue}{\frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\sqrt{2}}} \]
   associate-*r/ [<=] 0.5	\[ \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]
   fma-def [=>] 0.5	\[ \cos th \cdot \frac{\color{blue}{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}{\sqrt{2}} \]

3. Taylor expanded in a1 around 0 0.5

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

4. Simplified 0.5

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

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

5. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.5
Cost	19776

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

Alternative 2
Error	14.5
Cost	13513

\[\begin{array}{l} \mathbf{if}\;th \leq -0.47 \lor \neg \left(th \leq 9.4 \cdot 10^{-5}\right):\\ \;\;\;\;\cos th \cdot \left(a2 \cdot \frac{a2}{\sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \left(\sqrt{0.5} \cdot \left(-0.5 \cdot \left(th \cdot th\right) + 1\right)\right)\\ \end{array} \]

Alternative 3
Error	0.5
Cost	13504

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

Alternative 4
Error	0.5
Cost	13504

\[\frac{a2 \cdot a2 + a1 \cdot a1}{\frac{\sqrt{2}}{\cos th}} \]

Alternative 5
Error	20.1
Cost	13380

\[\begin{array}{l} \mathbf{if}\;a1 \leq -2 \cdot 10^{-105}:\\ \;\;\;\;\cos th \cdot \frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \left(a2 \cdot \frac{a2}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 6
Error	20.1
Cost	13380

\[\begin{array}{l} \mathbf{if}\;a1 \leq -1.3 \cdot 10^{-106}:\\ \;\;\;\;\cos th \cdot \frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot \left(\cos th \cdot a2\right)\right)\\ \end{array} \]

Alternative 7
Error	20.2
Cost	13380

\[\begin{array}{l} \mathbf{if}\;a1 \leq -1.8 \cdot 10^{-107}:\\ \;\;\;\;\frac{a1}{\sqrt{2}} \cdot \left(\cos th \cdot a1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot \left(\cos th \cdot a2\right)\right)\\ \end{array} \]

Alternative 8
Error	25.8
Cost	6976

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

Alternative 9
Error	36.3
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a1 \leq -6.4 \cdot 10^{-105}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]

Alternative 10
Error	40.8
Cost	6720

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

Alternative 11
Error	40.9
Cost	6720

\[a1 \cdot \frac{a1}{\sqrt{2}} \]

Reproduce

herbie shell --seed 2023034 
(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))))