Migdal et al, Equation (64)

Average Accuracy: 99.2% → 99.3%

Time: 15.8s

Precision: binary64

Cost: 20096

Math

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

↓

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

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 a2) (sqrt 2.0)) (* a1 (* a1 (pow 2.0 -0.5))))))

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 * a2) / sqrt(2.0)) + (a1 * (a1 * pow(2.0, -0.5))));
}

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 * a2) / sqrt(2.0d0)) + (a1 * (a1 * (2.0d0 ** (-0.5d0)))))
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 * a2) / Math.sqrt(2.0)) + (a1 * (a1 * Math.pow(2.0, -0.5))));
}

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 * a2) / math.sqrt(2.0)) + (a1 * (a1 * math.pow(2.0, -0.5))))

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(Float64(a2 * a2) / sqrt(2.0)) + Float64(a1 * Float64(a1 * (2.0 ^ -0.5)))))
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 * a2) / sqrt(2.0)) + (a1 * (a1 * (2.0 ^ -0.5))));
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[(N[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + N[(a1 * N[(a1 * N[Power[2.0, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

2. Simplified 99.2%

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

   [Start] 99.2	\[ \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   distribute-lft-out [=>] 99.2	\[ \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]

3. Applied egg-rr 99.2%

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

   [Start] 99.2	\[ \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   distribute-lft-in [=>] 99.2	\[ \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)} \]
   div-inv [=>] 99.1	\[ \color{blue}{\left(\cos th \cdot \frac{1}{\sqrt{2}}\right)} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   associate-*l* [=>] 99.1	\[ \color{blue}{\cos th \cdot \left(\frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right)\right)} + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   fma-def [=>] 99.1	\[ \color{blue}{\mathsf{fma}\left(\cos th, \frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right), \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\right)} \]
   pow1/2 [=>] 99.1	\[ \mathsf{fma}\left(\cos th, \frac{1}{\color{blue}{{2}^{0.5}}} \cdot \left(a1 \cdot a1\right), \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\right) \]
   pow-flip [=>] 99.2	\[ \mathsf{fma}\left(\cos th, \color{blue}{{2}^{\left(-0.5\right)}} \cdot \left(a1 \cdot a1\right), \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\right) \]
   metadata-eval [=>] 99.2	\[ \mathsf{fma}\left(\cos th, {2}^{\color{blue}{-0.5}} \cdot \left(a1 \cdot a1\right), \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\right) \]
   associate-*l/ [=>] 99.2	\[ \mathsf{fma}\left(\cos th, {2}^{-0.5} \cdot \left(a1 \cdot a1\right), \color{blue}{\frac{\cos th \cdot \left(a2 \cdot a2\right)}{\sqrt{2}}}\right) \]

4. Simplified 99.3%

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

   [Start] 99.2	\[ \mathsf{fma}\left(\cos th, {2}^{-0.5} \cdot \left(a1 \cdot a1\right), \frac{\cos th \cdot \left(a2 \cdot a2\right)}{\sqrt{2}}\right) \]
   fma-udef [=>] 99.2	\[ \color{blue}{\cos th \cdot \left({2}^{-0.5} \cdot \left(a1 \cdot a1\right)\right) + \frac{\cos th \cdot \left(a2 \cdot a2\right)}{\sqrt{2}}} \]
   +-commutative [=>] 99.2	\[ \color{blue}{\frac{\cos th \cdot \left(a2 \cdot a2\right)}{\sqrt{2}} + \cos th \cdot \left({2}^{-0.5} \cdot \left(a1 \cdot a1\right)\right)} \]
   *-commutative [=>] 99.2	\[ \frac{\color{blue}{\left(a2 \cdot a2\right) \cdot \cos th}}{\sqrt{2}} + \cos th \cdot \left({2}^{-0.5} \cdot \left(a1 \cdot a1\right)\right) \]
   associate-/l* [=>] 99.2	\[ \color{blue}{\frac{a2 \cdot a2}{\frac{\sqrt{2}}{\cos th}}} + \cos th \cdot \left({2}^{-0.5} \cdot \left(a1 \cdot a1\right)\right) \]
   associate-/r/ [=>] 99.2	\[ \color{blue}{\frac{a2 \cdot a2}{\sqrt{2}} \cdot \cos th} + \cos th \cdot \left({2}^{-0.5} \cdot \left(a1 \cdot a1\right)\right) \]
   unpow2 [<=] 99.2	\[ \frac{\color{blue}{{a2}^{2}}}{\sqrt{2}} \cdot \cos th + \cos th \cdot \left({2}^{-0.5} \cdot \left(a1 \cdot a1\right)\right) \]
   *-commutative [=>] 99.2	\[ \frac{{a2}^{2}}{\sqrt{2}} \cdot \cos th + \color{blue}{\left({2}^{-0.5} \cdot \left(a1 \cdot a1\right)\right) \cdot \cos th} \]
   distribute-rgt-out [=>] 99.2	\[ \color{blue}{\cos th \cdot \left(\frac{{a2}^{2}}{\sqrt{2}} + {2}^{-0.5} \cdot \left(a1 \cdot a1\right)\right)} \]
   unpow2 [=>] 99.2	\[ \cos th \cdot \left(\frac{\color{blue}{a2 \cdot a2}}{\sqrt{2}} + {2}^{-0.5} \cdot \left(a1 \cdot a1\right)\right) \]
   *-commutative [=>] 99.2	\[ \cos th \cdot \left(\frac{a2 \cdot a2}{\sqrt{2}} + \color{blue}{\left(a1 \cdot a1\right) \cdot {2}^{-0.5}}\right) \]
   associate-*l* [=>] 99.3	\[ \cos th \cdot \left(\frac{a2 \cdot a2}{\sqrt{2}} + \color{blue}{a1 \cdot \left(a1 \cdot {2}^{-0.5}\right)}\right) \]

5. Final simplification 99.3%

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

Alternatives

Alternative 1
Accuracy	99.3%
Cost	20096

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

Alternative 2
Accuracy	99.2%
Cost	19776

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

Alternative 3
Accuracy	77.9%
Cost	13513

\[\begin{array}{l} \mathbf{if}\;th \leq -0.0029 \lor \neg \left(th \leq 5 \cdot 10^{-8}\right):\\ \;\;\;\;\cos th \cdot \left(\sqrt{0.5} \cdot \left(a1 \cdot a1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2 + a1 \cdot a1}{\sqrt{2}}\\ \end{array} \]

Alternative 4
Accuracy	99.2%
Cost	13504

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

Alternative 5
Accuracy	68.4%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;a2 \leq 1.25 \cdot 10^{-111}:\\ \;\;\;\;\cos th \cdot \left(\sqrt{0.5} \cdot \left(a1 \cdot a1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]

Alternative 6
Accuracy	68.4%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;a2 \leq 1.25 \cdot 10^{-111}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\cos th \cdot \left(a1 \cdot a1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]

Alternative 7
Accuracy	68.4%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;a2 \leq 1.45 \cdot 10^{-111}:\\ \;\;\;\;\left(\cos th \cdot a1\right) \cdot \left(a1 \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]

Alternative 8
Accuracy	68.3%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;a2 \leq 2.6 \cdot 10^{-114}:\\ \;\;\;\;\cos th \cdot \frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]

Alternative 9
Accuracy	59.4%
Cost	6976

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

Alternative 10
Accuracy	42.0%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a2 \leq 1.1 \cdot 10^{-157}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot a1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 11
Accuracy	42.0%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a2 \leq 1.1 \cdot 10^{-157}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 12
Accuracy	42.0%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a2 \leq 1.1 \cdot 10^{-157}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 13
Accuracy	42.0%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a2 \leq 1.1 \cdot 10^{-157}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]

Alternative 14
Accuracy	36.1%
Cost	6720

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

Alternative 15
Accuracy	13.3%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023147 
(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))))