Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 17.2s

Precision: binary64

Cost: 20032

Math

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

↓

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

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

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

3. Taylor expanded in a1 around 0 0.5

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

4. Simplified 0.5

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

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

5. Final simplification 0.5

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

Alternatives

Alternative 1
Error	13.9
Cost	19780

\[\begin{array}{l} \mathbf{if}\;\cos th \leq 0.9964:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \frac{\cos th}{\sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 2
Error	13.9
Cost	19780

\[\begin{array}{l} \mathbf{if}\;\cos th \leq 0.9964:\\ \;\;\;\;\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 3
Error	13.9
Cost	19780

\[\begin{array}{l} \mathbf{if}\;\cos th \leq 0.9964:\\ \;\;\;\;\cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 4
Error	20.8
Cost	13644

\[\begin{array}{l} t_1 := \cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{if}\;a1 \leq -1 \cdot 10^{-109}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a1 \leq -3.4 \cdot 10^{-133}:\\ \;\;\;\;\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \frac{1}{\sqrt{2}}\\ \mathbf{elif}\;a1 \leq -3.1 \cdot 10^{-152}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]

Alternative 5
Error	20.8
Cost	13644

\[\begin{array}{l} t_1 := \sqrt{0.5} \cdot \left(a1 \cdot \left(\cos th \cdot a1\right)\right)\\ \mathbf{if}\;a1 \leq -4.2 \cdot 10^{-108}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a1 \leq -1.75 \cdot 10^{-134}:\\ \;\;\;\;\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \frac{1}{\sqrt{2}}\\ \mathbf{elif}\;a1 \leq -2 \cdot 10^{-153}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]

Alternative 6
Error	20.8
Cost	13644

\[\begin{array}{l} t_1 := \sqrt{0.5} \cdot \left(a1 \cdot \left(\cos th \cdot a1\right)\right)\\ \mathbf{if}\;a1 \leq -3.3 \cdot 10^{-108}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a1 \leq -9.6 \cdot 10^{-135}:\\ \;\;\;\;\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \frac{1}{\sqrt{2}}\\ \mathbf{elif}\;a1 \leq -2.2 \cdot 10^{-152}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\sqrt{2}} \cdot \left(\cos th \cdot a2\right)\\ \end{array} \]

Alternative 7
Error	20.8
Cost	13644

\[\begin{array}{l} t_1 := \cos th \cdot a1\\ \mathbf{if}\;a1 \leq -1.7 \cdot 10^{-111}:\\ \;\;\;\;\frac{t_1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{elif}\;a1 \leq -3.45 \cdot 10^{-133}:\\ \;\;\;\;\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \frac{1}{\sqrt{2}}\\ \mathbf{elif}\;a1 \leq -2.2 \cdot 10^{-152}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\sqrt{2}} \cdot \left(\cos th \cdot a2\right)\\ \end{array} \]

Alternative 8
Error	20.8
Cost	13644

\[\begin{array}{l} t_1 := \cos th \cdot a1\\ \mathbf{if}\;a1 \leq -1.06 \cdot 10^{-113}:\\ \;\;\;\;\frac{t_1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{elif}\;a1 \leq -8.8 \cdot 10^{-135}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt{2}}\\ \mathbf{elif}\;a1 \leq -3.2 \cdot 10^{-152}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\sqrt{2}} \cdot \left(\cos th \cdot a2\right)\\ \end{array} \]

Alternative 9
Error	0.5
Cost	13504

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

Alternative 10
Error	25.7
Cost	6976

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

Alternative 11
Error	36.5
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a1 \leq -1.55 \cdot 10^{-127}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \]

Alternative 12
Error	36.5
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a1 \leq -2.4 \cdot 10^{-127}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]

Alternative 13
Error	36.5
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a1 \leq -2.4 \cdot 10^{-127}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]

Alternative 14
Error	36.5
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a1 \leq -2.4 \cdot 10^{-127}:\\ \;\;\;\;\frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]

Alternative 15
Error	40.5
Cost	6720

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

Reproduce

herbie shell --seed 2023073 
(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))))