Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 17.2s

Precision: binary64

Cost: 13504

Math

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

↓

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

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
 (/ (+ (* a2 a2) (* a1 a1)) (/ (sqrt 2.0) (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 ((a2 * a2) + (a1 * a1)) / (sqrt(2.0) / 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 = ((a2 * a2) + (a1 * a1)) / (sqrt(2.0d0) / 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 ((a2 * a2) + (a1 * a1)) / (Math.sqrt(2.0) / 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 ((a2 * a2) + (a1 * a1)) / (math.sqrt(2.0) / 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(Float64(Float64(a2 * a2) + Float64(a1 * a1)) / Float64(sqrt(2.0) / 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 = ((a2 * a2) + (a1 * a1)) / (sqrt(2.0) / 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[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] / N[Cos[th], $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 th around inf 0.5

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

4. Simplified 0.5

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

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

5. Final simplification 0.5

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

Alternatives

Alternative 1
Error	20.3
Cost	13708

\[\begin{array}{l} t_1 := \left(a2 \cdot a2\right) \cdot \cos th\\ \mathbf{if}\;a2 \leq 2.5 \cdot 10^{-137}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot \left(a1 \cdot \cos th\right)\right)\\ \mathbf{elif}\;a2 \leq 6.2 \cdot 10^{-94}:\\ \;\;\;\;\frac{t_1}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 2.7 \cdot 10^{-80}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \frac{\cos th}{\sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot {2}^{-0.5}\\ \end{array} \]

Alternative 2
Error	20.3
Cost	13645

\[\begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;a2 \leq 2.7 \cdot 10^{-137} \lor \neg \left(a2 \leq 5.5 \cdot 10^{-94}\right) \land a2 \leq 2.65 \cdot 10^{-80}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot t_1\right)\\ \end{array} \]

Alternative 3
Error	20.3
Cost	13645

\[\begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;a2 \leq 1.7 \cdot 10^{-137}:\\ \;\;\;\;\cos th \cdot \frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{elif}\;a2 \leq 4.7 \cdot 10^{-94} \lor \neg \left(a2 \leq 2 \cdot 10^{-80}\right):\\ \;\;\;\;a2 \cdot \left(a2 \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot t_1\right)\\ \end{array} \]

Alternative 4
Error	20.4
Cost	13645

\[\begin{array}{l} \mathbf{if}\;a2 \leq 4.5 \cdot 10^{-139}:\\ \;\;\;\;\cos th \cdot \frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{elif}\;a2 \leq 4.8 \cdot 10^{-94} \lor \neg \left(a2 \leq 1.95 \cdot 10^{-80}\right):\\ \;\;\;\;\cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \frac{\cos th}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 5
Error	20.3
Cost	13645

\[\begin{array}{l} \mathbf{if}\;a2 \leq 2.2 \cdot 10^{-137}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot \left(a1 \cdot \cos th\right)\right)\\ \mathbf{elif}\;a2 \leq 4.7 \cdot 10^{-94} \lor \neg \left(a2 \leq 1.62 \cdot 10^{-80}\right):\\ \;\;\;\;\cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \frac{\cos th}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 6
Error	20.3
Cost	13644

\[\begin{array}{l} \mathbf{if}\;a2 \leq 5.1 \cdot 10^{-139}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot \left(a1 \cdot \cos th\right)\right)\\ \mathbf{elif}\;a2 \leq 4.8 \cdot 10^{-94}:\\ \;\;\;\;\cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}\\ \mathbf{elif}\;a2 \leq 1.35 \cdot 10^{-80}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \frac{\cos th}{\sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\sqrt{2}} \cdot \left(a2 \cdot \cos th\right)\\ \end{array} \]

Alternative 7
Error	20.3
Cost	13644

\[\begin{array}{l} \mathbf{if}\;a2 \leq 6.5 \cdot 10^{-138}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot \left(a1 \cdot \cos th\right)\right)\\ \mathbf{elif}\;a2 \leq 5 \cdot 10^{-94}:\\ \;\;\;\;\frac{\left(a2 \cdot a2\right) \cdot \cos th}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 1.65 \cdot 10^{-80}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \frac{\cos th}{\sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\sqrt{2}} \cdot \left(a2 \cdot \cos th\right)\\ \end{array} \]

Alternative 8
Error	14.6
Cost	13513

\[\begin{array}{l} t_1 := a2 \cdot a2 + a1 \cdot a1\\ \mathbf{if}\;th \leq -57000000000000 \lor \neg \left(th \leq 86000000\right):\\ \;\;\;\;a1 \cdot \left(a1 \cdot \frac{\cos th}{\sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;{0.25}^{0.25} \cdot \left(t_1 + -0.5 \cdot \left(t_1 \cdot \left(th \cdot th\right)\right)\right)\\ \end{array} \]

Alternative 9
Error	0.5
Cost	13504

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

Alternative 10
Error	36.4
Cost	7117

\[\begin{array}{l} \mathbf{if}\;a1 \leq -4.7 \cdot 10^{-74} \lor \neg \left(a1 \leq -5.3 \cdot 10^{-86}\right) \land a1 \leq -1.5 \cdot 10^{-138}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 11
Error	36.5
Cost	7117

\[\begin{array}{l} \mathbf{if}\;a1 \leq -1.85 \cdot 10^{-76}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{elif}\;a1 \leq -5.3 \cdot 10^{-86} \lor \neg \left(a1 \leq -1.75 \cdot 10^{-141}\right):\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 12
Error	36.5
Cost	7116

\[\begin{array}{l} \mathbf{if}\;a1 \leq -3.05 \cdot 10^{-74}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{elif}\;a1 \leq -5.6 \cdot 10^{-86}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \mathbf{elif}\;a1 \leq -5.2 \cdot 10^{-137}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]

Alternative 13
Error	36.4
Cost	7116

\[\begin{array}{l} \mathbf{if}\;a1 \leq -7.8 \cdot 10^{-75}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{elif}\;a1 \leq -5.5 \cdot 10^{-86}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \mathbf{elif}\;a1 \leq -3.3 \cdot 10^{-137}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]

Alternative 14
Error	25.5
Cost	6976

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

Alternative 15
Error	25.5
Cost	6976

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

Alternative 16
Error	40.5
Cost	6720

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

Reproduce

herbie shell --seed 2023053 
(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))))