Migdal et al, Equation (64)

Average Accuracy: 99.2% → 99.3%

Time: 14.8s

Precision: binary64

Cost: 32896

Math

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

↓

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

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
 (fma
  (* (* a2 a2) (cos th))
  (pow 2.0 -0.5)
  (/ (* (cos th) (* a1 a1)) (sqrt 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 fma(((a2 * a2) * cos(th)), pow(2.0, -0.5), ((cos(th) * (a1 * a1)) / sqrt(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 fma(Float64(Float64(a2 * a2) * cos(th)), (2.0 ^ -0.5), Float64(Float64(cos(th) * Float64(a1 * a1)) / sqrt(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[(a2 * a2), $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision] * N[Power[2.0, -0.5], $MachinePrecision] + N[(N[(N[Cos[th], $MachinePrecision] * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

2. Simplified 99.2%

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

   [Start] 99.2	\[ \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   distribute-lft-out [=>] 99.2	\[ \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]

3. Applied egg-rr 99.3%

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

   [Start] 99.2	\[ \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   +-commutative [=>] 99.2	\[ \frac{\cos th}{\sqrt{2}} \cdot \color{blue}{\left(a2 \cdot a2 + a1 \cdot a1\right)} \]
   distribute-lft-in [=>] 99.2	\[ \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right)} \]
   *-commutative [<=] 99.2	\[ \color{blue}{\left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}}} + \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) \]
   div-inv [=>] 99.1	\[ \left(a2 \cdot a2\right) \cdot \color{blue}{\left(\cos th \cdot \frac{1}{\sqrt{2}}\right)} + \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) \]
   associate-*r* [=>] 99.1	\[ \color{blue}{\left(\left(a2 \cdot a2\right) \cdot \cos th\right) \cdot \frac{1}{\sqrt{2}}} + \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) \]
   fma-def [=>] 99.2	\[ \color{blue}{\mathsf{fma}\left(\left(a2 \cdot a2\right) \cdot \cos th, \frac{1}{\sqrt{2}}, \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right)\right)} \]
   pow1/2 [=>] 99.2	\[ \mathsf{fma}\left(\left(a2 \cdot a2\right) \cdot \cos th, \frac{1}{\color{blue}{{2}^{0.5}}}, \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right)\right) \]
   pow-flip [=>] 99.2	\[ \mathsf{fma}\left(\left(a2 \cdot a2\right) \cdot \cos th, \color{blue}{{2}^{\left(-0.5\right)}}, \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right)\right) \]
   metadata-eval [=>] 99.2	\[ \mathsf{fma}\left(\left(a2 \cdot a2\right) \cdot \cos th, {2}^{\color{blue}{-0.5}}, \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right)\right) \]
   associate-*l/ [=>] 99.3	\[ \mathsf{fma}\left(\left(a2 \cdot a2\right) \cdot \cos th, {2}^{-0.5}, \color{blue}{\frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{2}}}\right) \]

4. Final simplification 99.3%

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

Alternatives

Alternative 1
Accuracy	99.2%
Cost	19776

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

Alternative 2
Accuracy	77.0%
Cost	13513

\[\begin{array}{l} \mathbf{if}\;th \leq -0.0005 \lor \neg \left(th \leq 0.8\right):\\ \;\;\;\;a2 \cdot \left(\sqrt{0.5} \cdot \left(a2 \cdot \cos th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2 + a1 \cdot a1}{\sqrt{2}}\\ \end{array} \]

Alternative 3
Accuracy	77.0%
Cost	13512

\[\begin{array}{l} \mathbf{if}\;th \leq -0.007:\\ \;\;\;\;a2 \cdot \frac{a2}{\frac{\sqrt{2}}{\cos th}}\\ \mathbf{elif}\;th \leq 0.8:\\ \;\;\;\;\frac{a2 \cdot a2 + a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(\sqrt{0.5} \cdot \left(a2 \cdot \cos th\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	99.3%
Cost	13504

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

Alternative 5
Accuracy	68.0%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;a2 \leq 2.6 \cdot 10^{-132}:\\ \;\;\;\;\left(\cos th \cdot \left(a1 \cdot a1\right)\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\frac{\sqrt{2}}{\cos th}}\\ \end{array} \]

Alternative 6
Accuracy	68.0%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;a2 \leq 2.6 \cdot 10^{-132}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \left(\cos th \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\frac{\sqrt{2}}{\cos th}}\\ \end{array} \]

Alternative 7
Accuracy	68.1%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;a2 \leq 2.6 \cdot 10^{-132}:\\ \;\;\;\;\frac{a1}{\frac{\sqrt{2}}{\cos th \cdot a1}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\frac{\sqrt{2}}{\cos th}}\\ \end{array} \]

Alternative 8
Accuracy	59.3%
Cost	6976

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

Alternative 9
Accuracy	59.3%
Cost	6976

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

Alternative 10
Accuracy	42.7%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a2 \leq 2.6 \cdot 10^{-133}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \]

Alternative 11
Accuracy	42.7%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a2 \leq 1.8 \cdot 10^{-133}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 12
Accuracy	42.7%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a2 \leq 2.5 \cdot 10^{-133}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 13
Accuracy	42.7%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a2 \leq 2.5 \cdot 10^{-133}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]

Alternative 14
Accuracy	42.7%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a2 \leq 2.6 \cdot 10^{-133}:\\ \;\;\;\;\frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]

Alternative 15
Accuracy	36.7%
Cost	6720

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

Alternative 16
Accuracy	14.1%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023136 
(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))))