Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 7.9s

Precision: binary64

Math

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

↓

\[\cos th \cdot \left(\mathsf{hypot}\left(a1, a2\right) \cdot \left(\mathsf{hypot}\left(a1, a2\right) \cdot {2}^{-0.5}\right)\right) \]

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) (* (hypot a1 a2) (* (hypot a1 a2) (pow 2.0 -0.5)))))

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) * (hypot(a1, a2) * (hypot(a1, a2) * pow(2.0, -0.5)));
}

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) * (Math.hypot(a1, a2) * (Math.hypot(a1, a2) * Math.pow(2.0, -0.5)));
}

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) * (math.hypot(a1, a2) * (math.hypot(a1, a2) * math.pow(2.0, -0.5)))

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(hypot(a1, a2) * Float64(hypot(a1, a2) * (2.0 ^ -0.5))))
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) * (hypot(a1, a2) * (hypot(a1, a2) * (2.0 ^ -0.5)));
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[Sqrt[a1 ^ 2 + a2 ^ 2], $MachinePrecision] * N[(N[Sqrt[a1 ^ 2 + a2 ^ 2], $MachinePrecision] * N[Power[2.0, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Error

Bits error versus a1

Bits error versus a2

Bits error versus th

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

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

3. Applied egg-rr 0.5

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

4. Final simplification 0.5

   \[\leadsto \cos th \cdot \left(\mathsf{hypot}\left(a1, a2\right) \cdot \left(\mathsf{hypot}\left(a1, a2\right) \cdot {2}^{-0.5}\right)\right) \]

Reproduce

herbie shell --seed 2022169 
(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))))