Migdal et al, Equation (64)

Average Accuracy: 99.2% → 99.3%

Time: 21.3s

Precision: binary64

Cost: 13504

Math

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

↓

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

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) (* (+ (* a2 a2) (* a1 a1)) (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) * (((a2 * a2) + (a1 * a1)) * 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) * (((a2 * a2) + (a1 * a1)) * 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) * (((a2 * a2) + (a1 * a1)) * 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) * (((a2 * a2) + (a1 * a1)) * 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(a2 * a2) + Float64(a1 * a1)) * 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) * (((a2 * a2) + (a1 * a1)) * 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[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $MachinePrecision]), $MachinePrecision] * N[Cos[th], $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 99.3%

   \[\leadsto \color{blue}{\cos th \cdot \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}}} \]
   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	\[ \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]
   associate-*l/ [=>] 99.3	\[ \color{blue}{\frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\sqrt{2}}} \]
   associate-*r/ [<=] 99.3	\[ \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]
   fma-def [=>] 99.3	\[ \cos th \cdot \frac{\color{blue}{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}{\sqrt{2}} \]

3. Applied egg-rr 99.3%

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

   [Start] 99.3	\[ \cos th \cdot \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}} \]
   div-inv [=>] 99.1	\[ \cos th \cdot \color{blue}{\left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \frac{1}{\sqrt{2}}\right)} \]
   *-commutative [=>] 99.1	\[ \cos th \cdot \color{blue}{\left(\frac{1}{\sqrt{2}} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right)} \]
   fma-udef [=>] 99.1	\[ \cos th \cdot \left(\frac{1}{\sqrt{2}} \cdot \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)}\right) \]
   +-commutative [=>] 99.1	\[ \cos th \cdot \left(\frac{1}{\sqrt{2}} \cdot \color{blue}{\left(a2 \cdot a2 + a1 \cdot a1\right)}\right) \]
   distribute-lft-in [=>] 99.1	\[ \cos th \cdot \color{blue}{\left(\frac{1}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) + \frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right)\right)} \]
   pow1/2 [=>] 99.1	\[ \cos th \cdot \left(\frac{1}{\color{blue}{{2}^{0.5}}} \cdot \left(a2 \cdot a2\right) + \frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right)\right) \]
   pow-flip [=>] 99.2	\[ \cos th \cdot \left(\color{blue}{{2}^{\left(-0.5\right)}} \cdot \left(a2 \cdot a2\right) + \frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right)\right) \]
   metadata-eval [=>] 99.2	\[ \cos th \cdot \left({2}^{\color{blue}{-0.5}} \cdot \left(a2 \cdot a2\right) + \frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right)\right) \]
   pow1/2 [=>] 99.2	\[ \cos th \cdot \left({2}^{-0.5} \cdot \left(a2 \cdot a2\right) + \frac{1}{\color{blue}{{2}^{0.5}}} \cdot \left(a1 \cdot a1\right)\right) \]
   pow-flip [=>] 99.3	\[ \cos th \cdot \left({2}^{-0.5} \cdot \left(a2 \cdot a2\right) + \color{blue}{{2}^{\left(-0.5\right)}} \cdot \left(a1 \cdot a1\right)\right) \]
   metadata-eval [=>] 99.3	\[ \cos th \cdot \left({2}^{-0.5} \cdot \left(a2 \cdot a2\right) + {2}^{\color{blue}{-0.5}} \cdot \left(a1 \cdot a1\right)\right) \]

4. Simplified 99.3%

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

   [Start] 99.3	\[ \cos th \cdot \left({2}^{-0.5} \cdot \left(a2 \cdot a2\right) + {2}^{-0.5} \cdot \left(a1 \cdot a1\right)\right) \]
   distribute-lft-out [=>] 99.3	\[ \cos th \cdot \color{blue}{\left({2}^{-0.5} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right)\right)} \]

5. Taylor expanded in th around inf 99.3%

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

6. Simplified 99.3%

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

   [Start] 99.3	\[ \sqrt{0.5} \cdot \left(\left({a2}^{2} + {a1}^{2}\right) \cdot \cos th\right) \]
   unpow2 [=>] 99.3	\[ \sqrt{0.5} \cdot \left(\left(\color{blue}{a2 \cdot a2} + {a1}^{2}\right) \cdot \cos th\right) \]
   unpow2 [=>] 99.3	\[ \sqrt{0.5} \cdot \left(\left(a2 \cdot a2 + \color{blue}{a1 \cdot a1}\right) \cdot \cos th\right) \]

7. Final simplification 99.3%

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

Alternatives

Alternative 1
Accuracy	76.8%
Cost	13513

\[\begin{array}{l} t_1 := a2 \cdot a2 + a1 \cdot a1\\ \mathbf{if}\;th \leq -0.022 \lor \neg \left(th \leq 0.115\right):\\ \;\;\;\;a1 \cdot \left(a1 \cdot \left(\sqrt{0.5} \cdot \cos th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(t_1 + -0.5 \cdot \left(t_1 \cdot \left(th \cdot th\right)\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	76.8%
Cost	13512

\[\begin{array}{l} t_1 := a2 \cdot a2 + a1 \cdot a1\\ \mathbf{if}\;th \leq -0.02:\\ \;\;\;\;\cos th \cdot \left(\sqrt{0.5} \cdot \left(a1 \cdot a1\right)\right)\\ \mathbf{elif}\;th \leq 0.215:\\ \;\;\;\;\sqrt{0.5} \cdot \left(t_1 + -0.5 \cdot \left(t_1 \cdot \left(th \cdot th\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \left(\sqrt{0.5} \cdot \cos th\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	67.5%
Cost	13512

\[\begin{array}{l} t_1 := a2 \cdot a2 + a1 \cdot a1\\ \mathbf{if}\;a2 \leq 2.5 \cdot 10^{-147}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \left(\sqrt{0.5} \cdot \cos th\right)\right)\\ \mathbf{elif}\;a2 \leq 8.6 \cdot 10^{-83}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(t_1 + -0.5 \cdot \left(t_1 \cdot \left(th \cdot th\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \left(\sqrt{0.5} \cdot \left(a2 \cdot a2\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	67.5%
Cost	13512

\[\begin{array}{l} t_1 := a2 \cdot a2 + a1 \cdot a1\\ \mathbf{if}\;a2 \leq 7.2 \cdot 10^{-146}:\\ \;\;\;\;\cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 1.65 \cdot 10^{-82}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(t_1 + -0.5 \cdot \left(t_1 \cdot \left(th \cdot th\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \left(\sqrt{0.5} \cdot \left(a2 \cdot a2\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	67.5%
Cost	13512

\[\begin{array}{l} t_1 := a2 \cdot a2 + a1 \cdot a1\\ \mathbf{if}\;a2 \leq 3.9 \cdot 10^{-146}:\\ \;\;\;\;\cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 9.2 \cdot 10^{-83}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(t_1 + -0.5 \cdot \left(t_1 \cdot \left(th \cdot th\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\left(a2 \cdot a2\right) \cdot \cos th\right)\\ \end{array} \]

Alternative 6
Accuracy	67.6%
Cost	13512

\[\begin{array}{l} t_1 := a2 \cdot a2 + a1 \cdot a1\\ \mathbf{if}\;a2 \leq 6.2 \cdot 10^{-145}:\\ \;\;\;\;\frac{a1}{\sqrt{2}} \cdot \left(a1 \cdot \cos th\right)\\ \mathbf{elif}\;a2 \leq 1.1 \cdot 10^{-82}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(t_1 + -0.5 \cdot \left(t_1 \cdot \left(th \cdot th\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\left(a2 \cdot a2\right) \cdot \cos th\right)\\ \end{array} \]

Alternative 7
Accuracy	59.0%
Cost	6976

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

Alternative 8
Accuracy	42.4%
Cost	6916

\[\begin{array}{l} \mathbf{if}\;a1 \leq -1.65 \cdot 10^{-76}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot {2}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot a2\right)\\ \end{array} \]

Alternative 9
Accuracy	42.4%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a1 \leq -4.2 \cdot 10^{-77}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot a1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot a2\right)\\ \end{array} \]

Alternative 10
Accuracy	42.4%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a1 \leq -1.5 \cdot 10^{-75}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot a2\right)\\ \end{array} \]

Alternative 11
Accuracy	36.4%
Cost	6720

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

Reproduce

herbie shell --seed 2023129 
(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))))