Migdal et al, Equation (64)

Average Error: 99.2% → 99.3%

Time: 32.1s

Precision: binary64

Cost: 20032.00

Math

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

↓

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

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

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) / sqrt(2.0d0)))
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.sqrt(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) * (((a2 * a2) / math.sqrt(2.0)) + ((a1 * a1) / math.sqrt(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(Float64(Float64(a2 * a2) / sqrt(2.0)) + Float64(Float64(a1 * a1) / sqrt(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) * (((a2 * a2) / sqrt(2.0)) + ((a1 * a1) / sqrt(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[(N[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(a1 * a1), $MachinePrecision] / N[Sqrt[2.0], $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 99.3

   \[\leadsto \color{blue}{\cos th \cdot \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}}} \]
   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. Taylor expanded in a1 around 0 99.3

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

4. Simplified 99.3

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

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

5. Final simplification 99.3

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

Alternatives

Alternative 1
Error	99.2%
Cost	19904.00

\[\frac{\frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}}}{\frac{1}{\cos th}} \]

Alternative 2
Error	76.4%
Cost	19780.00

\[\begin{array}{l} \mathbf{if}\;\cos th \leq 0.98:\\ \;\;\;\;\cos th \cdot \left(a1 \cdot \frac{a1}{\sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right)\\ \end{array} \]

Alternative 3
Error	99.3%
Cost	19776.00

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

Alternative 4
Error	68.5%
Cost	13645.00

\[\begin{array}{l} \mathbf{if}\;a2 \leq 3.4 \cdot 10^{-127} \lor \neg \left(a2 \leq 1.85 \cdot 10^{-117}\right) \land a2 \leq 3.1 \cdot 10^{-86}:\\ \;\;\;\;\cos th \cdot \left(a1 \cdot \frac{a1}{\sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\right)\\ \end{array} \]

Alternative 5
Error	68.5%
Cost	13645.00

\[\begin{array}{l} \mathbf{if}\;a2 \leq 5.3 \cdot 10^{-127}:\\ \;\;\;\;\cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 1.8 \cdot 10^{-117} \lor \neg \left(a2 \leq 7.2 \cdot 10^{-86}\right):\\ \;\;\;\;\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \left(a1 \cdot \frac{a1}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 6
Error	68.5%
Cost	13645.00

\[\begin{array}{l} \mathbf{if}\;a2 \leq 5 \cdot 10^{-127}:\\ \;\;\;\;\cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 1.9 \cdot 10^{-117} \lor \neg \left(a2 \leq 2.9 \cdot 10^{-85}\right):\\ \;\;\;\;\cos th \cdot \frac{a2 \cdot a2}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \left(a1 \cdot \frac{a1}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 7
Error	68.5%
Cost	13644.00

\[\begin{array}{l} \mathbf{if}\;a2 \leq 1.85 \cdot 10^{-127}:\\ \;\;\;\;\cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 1.8 \cdot 10^{-117}:\\ \;\;\;\;\cos th \cdot \frac{a2 \cdot a2}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 1.8 \cdot 10^{-85}:\\ \;\;\;\;\cos th \cdot \left(a1 \cdot \frac{a1}{\sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\cos th \cdot \left(a2 \cdot a2\right)\right)\\ \end{array} \]

Alternative 8
Error	68.5%
Cost	13644.00

\[\begin{array}{l} \mathbf{if}\;a2 \leq 4.8 \cdot 10^{-127}:\\ \;\;\;\;\cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 1.8 \cdot 10^{-117}:\\ \;\;\;\;\cos th \cdot \frac{a2 \cdot a2}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 3.3 \cdot 10^{-85}:\\ \;\;\;\;\cos th \cdot \left(a1 \cdot \frac{a1}{\sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot \left(\cos th \cdot a2\right)}{\sqrt{2}}\\ \end{array} \]

Alternative 9
Error	68.5%
Cost	13644.00

\[\begin{array}{l} \mathbf{if}\;a2 \leq 4.9 \cdot 10^{-127}:\\ \;\;\;\;\frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 1.9 \cdot 10^{-117}:\\ \;\;\;\;\cos th \cdot \frac{a2 \cdot a2}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 3.2 \cdot 10^{-86}:\\ \;\;\;\;\cos th \cdot \left(a1 \cdot \frac{a1}{\sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot \left(\cos th \cdot a2\right)}{\sqrt{2}}\\ \end{array} \]

Alternative 10
Error	99.3%
Cost	13504.00

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

Alternative 11
Error	99.3%
Cost	13504.00

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

Alternative 12
Error	59.3%
Cost	6976.00

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

Alternative 13
Error	42.7%
Cost	6852.00

\[\begin{array}{l} \mathbf{if}\;a2 \leq 3.15 \cdot 10^{-127}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]

Alternative 14
Error	42.7%
Cost	6852.00

\[\begin{array}{l} \mathbf{if}\;a2 \leq 5.3 \cdot 10^{-127}:\\ \;\;\;\;\frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]

Alternative 15
Error	42.7%
Cost	6852.00

\[\begin{array}{l} \mathbf{if}\;a2 \leq 4.2 \cdot 10^{-127}:\\ \;\;\;\;\frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \end{array} \]

Alternative 16
Error	36.7%
Cost	6720.00

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

Reproduce

herbie shell --seed 2023093 
(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))))