Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 11.4s

Precision: binary64

Cost: 13504

Math

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

↓

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

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
 (* (* (sqrt 0.5) (cos th)) (+ (* a2 a2) (* a1 a1))))

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 (sqrt(0.5) * cos(th)) * ((a2 * a2) + (a1 * a1));
}

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 = (sqrt(0.5d0) * cos(th)) * ((a2 * a2) + (a1 * a1))
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.sqrt(0.5) * Math.cos(th)) * ((a2 * a2) + (a1 * a1));
}

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.sqrt(0.5) * math.cos(th)) * ((a2 * a2) + (a1 * a1))

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(sqrt(0.5) * cos(th)) * Float64(Float64(a2 * a2) + Float64(a1 * a1)))
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 = (sqrt(0.5) * cos(th)) * ((a2 * a2) + (a1 * a1));
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[Sqrt[0.5], $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $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. Taylor expanded in th around inf 0.4

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

3. Simplified 0.5

   \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right)} \]
   Proof
   (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 2)) (+.f64 (*.f64 a2 a2) (*.f64 a1 a1))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 2)) (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 a2 2)) (*.f64 a1 a1))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 2)) (+.f64 (pow.f64 a2 2) (Rewrite<= unpow2_binary64 (pow.f64 a1 2)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 2)) (pow.f64 a2 2)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 2)) (pow.f64 a1 2)))): 0 points increase in error, 2 points decrease in error
   (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 a2 2) (/.f64 (cos.f64 th) (sqrt.f64 2)))) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 2)) (pow.f64 a1 2))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (pow.f64 a2 2) (cos.f64 th)) (sqrt.f64 2))) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 2)) (pow.f64 a1 2))): 9 points increase in error, 18 points decrease in error
   (+.f64 (/.f64 (*.f64 (pow.f64 a2 2) (cos.f64 th)) (sqrt.f64 2)) (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (cos.f64 th) (pow.f64 a1 2)) (sqrt.f64 2)))): 15 points increase in error, 14 points decrease in error
   (+.f64 (/.f64 (*.f64 (pow.f64 a2 2) (cos.f64 th)) (sqrt.f64 2)) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 a1 2) (cos.f64 th))) (sqrt.f64 2))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (*.f64 (pow.f64 a1 2) (cos.f64 th)) (sqrt.f64 2)) (/.f64 (*.f64 (pow.f64 a2 2) (cos.f64 th)) (sqrt.f64 2)))): 0 points increase in error, 0 points decrease in error

4. Applied egg-rr 0.5

   \[\leadsto \color{blue}{\left({2}^{-0.5} \cdot \cos th\right)} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

5. Taylor expanded in th around inf 0.5

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

6. Final simplification 0.5

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

Alternatives

Alternative 1
Error	14.3
Cost	19780

\[\begin{array}{l} \mathbf{if}\;\cos th \leq 0.99995:\\ \;\;\;\;\left(\cos th \cdot a1\right) \cdot \left(\sqrt{0.5} \cdot a1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \left(\sqrt{0.5} \cdot \left(\left(th \cdot th\right) \cdot \left(-0.5 + \left(th \cdot th\right) \cdot 0.041666666666666664\right) + 1\right)\right)\\ \end{array} \]

Alternative 2
Error	20.7
Cost	13380

\[\begin{array}{l} \mathbf{if}\;a1 \leq -1.5205107275450004 \cdot 10^{-141}:\\ \;\;\;\;\left(\cos th \cdot a1\right) \cdot \left(\sqrt{0.5} \cdot a1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot \left(\cos th \cdot a2\right)\right)\\ \end{array} \]

Alternative 3
Error	20.7
Cost	13380

\[\begin{array}{l} \mathbf{if}\;a1 \leq -1.5205107275450004 \cdot 10^{-141}:\\ \;\;\;\;\cos th \cdot \left(\sqrt{0.5} \cdot \left(a1 \cdot a1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot \left(\cos th \cdot a2\right)\right)\\ \end{array} \]

Alternative 4
Error	25.9
Cost	6976

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

Alternative 5
Error	36.8
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a2 \leq 2.2056188796933025 \cdot 10^{-150}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot a1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]

Alternative 6
Error	36.8
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a2 \leq 2.2056188796933025 \cdot 10^{-150}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot a1\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]

Alternative 7
Error	40.3
Cost	6720

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

Alternative 8
Error	40.3
Cost	6720

\[\frac{a2}{\frac{\sqrt{2}}{a2}} \]

Reproduce

herbie shell --seed 2022295 
(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))))