Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 16.7s

Precision: binary64

Cost: 13632

Math

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

↓

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

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

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

   [Start] 0.5	\[ \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   rational_best-simplify-47 [=>] 0.5	\[ \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right)} \]

3. Applied egg-rr 49.5

   \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \color{blue}{\left(\frac{\frac{1}{a2 \cdot a2 + a1 \cdot a1}}{\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \left(a2 \cdot a2 + a1 \cdot a1\right)} \cdot \left(\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \left(\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \left(\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \left(a2 \cdot a2 + a1 \cdot a1\right)\right)\right)\right)\right)} \]

4. Applied egg-rr 0.5

   \[\leadsto \color{blue}{\frac{\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \frac{\sqrt{2} \cdot \cos th}{1}}{2}} \]

5. Simplified 0.5

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

   [Start] 0.5	\[ \frac{\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \frac{\sqrt{2} \cdot \cos th}{1}}{2} \]
   rational_best-simplify-7 [=>] 0.5	\[ \frac{\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \cos th\right)}}{2} \]

6. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.5
Cost	13632

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

Alternative 2
Error	0.5
Cost	13504

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

Alternative 3
Error	25.5
Cost	7104

\[\frac{1}{\sqrt{2}} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

Alternative 4
Error	25.5
Cost	7104

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

Reproduce

herbie shell --seed 2023094 
(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))))