Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 19.3s

Precision: binary64

Cost: 13504

Math

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

↓

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

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

   \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]
   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 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	14.5
Cost	19780

\[\begin{array}{l} \mathbf{if}\;\cos th \leq 0.94:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \frac{\cos th}{\sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2 + a1 \cdot a1}{\sqrt{2}}\\ \end{array} \]

Alternative 2
Error	14.6
Cost	19780

\[\begin{array}{l} \mathbf{if}\;\cos th \leq 0.94:\\ \;\;\;\;\cos th \cdot \left(\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2 + a1 \cdot a1}{\sqrt{2}}\\ \end{array} \]

Alternative 3
Error	20.7
Cost	13645

\[\begin{array}{l} \mathbf{if}\;a2 \leq 2.8 \cdot 10^{-154} \lor \neg \left(a2 \leq 2.15 \cdot 10^{-110}\right) \land a2 \leq 3.4 \cdot 10^{-80}:\\ \;\;\;\;\cos th \cdot \left(\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\right)\\ \end{array} \]

Alternative 4
Error	20.7
Cost	13644

\[\begin{array}{l} t_1 := \cos th \cdot \left(\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\right)\\ \mathbf{if}\;a2 \leq 2.35 \cdot 10^{-151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a2 \leq 2.7 \cdot 10^{-110}:\\ \;\;\;\;\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;a2 \leq 2.35 \cdot 10^{-80}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \left(a2 \cdot \frac{a2}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 5
Error	20.7
Cost	13644

\[\begin{array}{l} \mathbf{if}\;a2 \leq 2.35 \cdot 10^{-151}:\\ \;\;\;\;\cos th \cdot \frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{elif}\;a2 \leq 2.1 \cdot 10^{-110}:\\ \;\;\;\;\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;a2 \leq 2.5 \cdot 10^{-80}:\\ \;\;\;\;\cos th \cdot \left(\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \left(a2 \cdot \frac{a2}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 6
Error	20.7
Cost	13644

\[\begin{array}{l} \mathbf{if}\;a2 \leq 2.35 \cdot 10^{-151}:\\ \;\;\;\;\cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 2 \cdot 10^{-110}:\\ \;\;\;\;\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;a2 \leq 3.5 \cdot 10^{-80}:\\ \;\;\;\;\cos th \cdot \left(\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \left(a2 \cdot \frac{a2}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 7
Error	20.7
Cost	13644

\[\begin{array}{l} \mathbf{if}\;a2 \leq 2.35 \cdot 10^{-151}:\\ \;\;\;\;\cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 1.2 \cdot 10^{-109}:\\ \;\;\;\;\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;a2 \leq 2.6 \cdot 10^{-80}:\\ \;\;\;\;\cos th \cdot \left(\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \frac{a2 \cdot a2}{\sqrt{2}}\\ \end{array} \]

Alternative 8
Error	20.8
Cost	13644

\[\begin{array}{l} \mathbf{if}\;a2 \leq 1.85 \cdot 10^{-151}:\\ \;\;\;\;\cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 4 \cdot 10^{-110}:\\ \;\;\;\;\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;a2 \leq 2.2 \cdot 10^{-80}:\\ \;\;\;\;\cos th \cdot \left(\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos th}{\frac{\sqrt{2}}{a2 \cdot a2}}\\ \end{array} \]

Alternative 9
Error	20.7
Cost	13644

\[\begin{array}{l} \mathbf{if}\;a2 \leq 2.35 \cdot 10^{-151}:\\ \;\;\;\;\frac{\left(a1 \cdot a1\right) \cdot \cos th}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 1.02 \cdot 10^{-109}:\\ \;\;\;\;\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;a2 \leq 1.6 \cdot 10^{-80}:\\ \;\;\;\;\cos th \cdot \left(\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos th}{\frac{\sqrt{2}}{a2 \cdot a2}}\\ \end{array} \]

Alternative 10
Error	20.7
Cost	13644

\[\begin{array}{l} \mathbf{if}\;a2 \leq 2.2 \cdot 10^{-151}:\\ \;\;\;\;\frac{\left(a1 \cdot a1\right) \cdot \cos th}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 2 \cdot 10^{-110}:\\ \;\;\;\;\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;a2 \leq 2 \cdot 10^{-80}:\\ \;\;\;\;\cos th \cdot \left(\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a2 \cdot a2\right) \cdot \cos th}{\sqrt{2}}\\ \end{array} \]

Alternative 11
Error	0.5
Cost	13504

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

Alternative 12
Error	0.5
Cost	13504

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

Alternative 13
Error	36.2
Cost	7244

\[\begin{array}{l} \mathbf{if}\;a1 \leq -1.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{elif}\;a1 \leq -1.1 \cdot 10^{-101}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \mathbf{elif}\;a1 \leq -1.05 \cdot 10^{-125}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{2}}{a1 \cdot a1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \end{array} \]

Alternative 14
Error	36.2
Cost	7117

\[\begin{array}{l} \mathbf{if}\;a1 \leq -1.8 \cdot 10^{-18} \lor \neg \left(a1 \leq -4.2 \cdot 10^{-101}\right) \land a1 \leq -3.2 \cdot 10^{-127}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 15
Error	36.2
Cost	7117

\[\begin{array}{l} \mathbf{if}\;a1 \leq -5 \cdot 10^{-18} \lor \neg \left(a1 \leq -1.4 \cdot 10^{-101}\right) \land a1 \leq -1.55 \cdot 10^{-125}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 16
Error	36.2
Cost	7116

\[\begin{array}{l} t_1 := a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{if}\;a1 \leq -9.6 \cdot 10^{-18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a1 \leq -1 \cdot 10^{-101}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \mathbf{elif}\;a1 \leq -1.55 \cdot 10^{-130}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]

Alternative 17
Error	36.2
Cost	7116

\[\begin{array}{l} t_1 := \frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{if}\;a1 \leq -1.8 \cdot 10^{-18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a1 \leq -7.8 \cdot 10^{-101}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \mathbf{elif}\;a1 \leq -1 \cdot 10^{-125}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]

Alternative 18
Error	36.2
Cost	7116

\[\begin{array}{l} \mathbf{if}\;a1 \leq -1.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{elif}\;a1 \leq -7.6 \cdot 10^{-102}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \mathbf{elif}\;a1 \leq -2.9 \cdot 10^{-125}:\\ \;\;\;\;\frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]

Alternative 19
Error	36.2
Cost	7116

\[\begin{array}{l} \mathbf{if}\;a1 \leq -1.35 \cdot 10^{-18}:\\ \;\;\;\;\frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{elif}\;a1 \leq -7.2 \cdot 10^{-102}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \mathbf{elif}\;a1 \leq -4 \cdot 10^{-126}:\\ \;\;\;\;\frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \end{array} \]

Alternative 20
Error	25.2
Cost	6976

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

Alternative 21
Error	25.2
Cost	6976

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

Alternative 22
Error	40.0
Cost	6720

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

Reproduce

herbie shell --seed 2023045 
(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))))