Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 16.4s

Precision: binary64

Cost: 26304

Math

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

↓

\[\frac{\frac{\cos th}{\frac{1}{\mathsf{hypot}\left(a1, a2\right)}}}{\frac{\sqrt{2}}{\mathsf{hypot}\left(a1, a2\right)}} \]

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) (/ 1.0 (hypot a1 a2))) (/ (sqrt 2.0) (hypot a1 a2))))

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) / (1.0 / hypot(a1, a2))) / (sqrt(2.0) / hypot(a1, a2));
}

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) / (1.0 / Math.hypot(a1, a2))) / (Math.sqrt(2.0) / Math.hypot(a1, a2));
}

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) / (1.0 / math.hypot(a1, a2))) / (math.sqrt(2.0) / math.hypot(a1, a2))

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(cos(th) / Float64(1.0 / hypot(a1, a2))) / Float64(sqrt(2.0) / hypot(a1, a2)))
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) / (1.0 / hypot(a1, a2))) / (sqrt(2.0) / hypot(a1, a2));
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[Cos[th], $MachinePrecision] / N[(1.0 / N[Sqrt[a1 ^ 2 + a2 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[a1 ^ 2 + a2 ^ 2], $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

   \[\leadsto \color{blue}{\cos th \cdot \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}}} \]
   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	\[ \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]
   associate-*l/ [=>] 0.5	\[ \color{blue}{\frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\sqrt{2}}} \]
   associate-*r/ [<=] 0.5	\[ \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]
   fma-def [=>] 0.5	\[ \cos th \cdot \frac{\color{blue}{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}{\sqrt{2}} \]

3. Applied egg-rr 0.5

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

4. Final simplification 0.5

   \[\leadsto \frac{\frac{\cos th}{\frac{1}{\mathsf{hypot}\left(a1, a2\right)}}}{\frac{\sqrt{2}}{\mathsf{hypot}\left(a1, a2\right)}} \]

Alternatives

Alternative 1
Error	0.5
Cost	26240

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

Alternative 2
Error	0.5
Cost	20160

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

Alternative 3
Error	20.7
Cost	13645

\[\begin{array}{l} \mathbf{if}\;a2 \leq 2.6 \cdot 10^{-148} \lor \neg \left(a2 \leq 3.8 \cdot 10^{-135}\right) \land a2 \leq 3.7 \cdot 10^{-109}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \left(\cos th \cdot \sqrt{0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\right)\\ \end{array} \]

Alternative 4
Error	14.7
Cost	13513

\[\begin{array}{l} \mathbf{if}\;th \leq -1400000000 \lor \neg \left(th \leq 0.0023\right):\\ \;\;\;\;a1 \cdot \left(a1 \cdot \left(\cos th \cdot \sqrt{0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right)\\ \end{array} \]

Alternative 5
Error	0.5
Cost	13504

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

Alternative 6
Error	0.5
Cost	13504

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

Alternative 7
Error	26.3
Cost	6976

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

Alternative 8
Error	37.0
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a2 \leq 2.45 \cdot 10^{-147}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \]

Alternative 9
Error	41.1
Cost	6720

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

Reproduce

herbie shell --seed 2023083 
(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))))