Migdal et al, Equation (64)

Average Accuracy: 99.2% → 99.3%

Time: 15.1s

Precision: binary64

Cost: 19776

\[ \begin{array}{c}[a1, a2] = \mathsf{sort}([a1, a2])\\ \end{array} \]

Math

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

↓

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

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)) (fma 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)) * fma(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)) * fma(a2, a2, Float64(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[(a2 * a2 + N[(a1 * a1), $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%

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

   [Start] 99.2	\[ \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   div-inv [=>] 99.1	\[ \color{blue}{\left(\cos th \cdot \frac{1}{\sqrt{2}}\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   *-commutative [=>] 99.1	\[ \color{blue}{\left(\frac{1}{\sqrt{2}} \cdot \cos th\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   pow1/2 [=>] 99.1	\[ \left(\frac{1}{\color{blue}{{2}^{0.5}}} \cdot \cos th\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   pow-flip [=>] 99.3	\[ \left(\color{blue}{{2}^{\left(-0.5\right)}} \cdot \cos th\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   metadata-eval [=>] 99.3	\[ \left({2}^{\color{blue}{-0.5}} \cdot \cos th\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

4. Taylor expanded in th around inf 99.3%

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

5. Taylor expanded in a1 around 0 99.3%

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

6. Simplified 99.3%

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

   [Start] 99.3	\[ \left(\sqrt{0.5} \cdot \cos th\right) \cdot \left({a2}^{2} + {a1}^{2}\right) \]
   unpow2 [=>] 99.3	\[ \left(\sqrt{0.5} \cdot \cos th\right) \cdot \left(\color{blue}{a2 \cdot a2} + {a1}^{2}\right) \]
   unpow2 [=>] 99.3	\[ \left(\sqrt{0.5} \cdot \cos th\right) \cdot \left(a2 \cdot a2 + \color{blue}{a1 \cdot a1}\right) \]
   fma-udef [<=] 99.3	\[ \left(\sqrt{0.5} \cdot \cos th\right) \cdot \color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)} \]

7. Final simplification 99.3%

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

Alternatives

Alternative 1
Accuracy	77.0%
Cost	13513

\[\begin{array}{l} \mathbf{if}\;th \leq -2.25 \cdot 10^{+21} \lor \neg \left(th \leq 1250000\right):\\ \;\;\;\;a2 \cdot \left(a2 \cdot \frac{\cos th}{\sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \left(\sqrt{0.5} \cdot \left(-0.5 \cdot \left(th \cdot th\right) + 1\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	77.0%
Cost	13512

\[\begin{array}{l} \mathbf{if}\;th \leq -2.25 \cdot 10^{+21}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \frac{\cos th}{\sqrt{2}}\right)\\ \mathbf{elif}\;th \leq 1250000:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \left(\sqrt{0.5} \cdot \left(-0.5 \cdot \left(th \cdot th\right) + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]

Alternative 3
Accuracy	99.3%
Cost	13504

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

Alternative 4
Accuracy	88.3%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;a1 \leq -3.25 \cdot 10^{-143}:\\ \;\;\;\;\cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \frac{\cos th}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 5
Accuracy	88.4%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;a1 \leq -5.4 \cdot 10^{-142}:\\ \;\;\;\;\cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\cos th \cdot \left(a2 \cdot a2\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	88.4%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;a1 \leq -5.4 \cdot 10^{-142}:\\ \;\;\;\;\frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\cos th \cdot \left(a2 \cdot a2\right)\right)\\ \end{array} \]

Alternative 7
Accuracy	54.0%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;a1 \leq -5.7 \cdot 10^{-145}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{2}}{a1 \cdot a1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot a2\right)\\ \end{array} \]

Alternative 8
Accuracy	59.9%
Cost	6976

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

Alternative 9
Accuracy	54.0%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a1 \leq -1.2 \cdot 10^{-145}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]

Alternative 10
Accuracy	54.0%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a1 \leq -5.7 \cdot 10^{-145}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot a2\right)\\ \end{array} \]

Alternative 11
Accuracy	54.0%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a1 \leq -5.7 \cdot 10^{-145}:\\ \;\;\;\;\frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot a2\right)\\ \end{array} \]

Alternative 12
Accuracy	37.3%
Cost	6720

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

Reproduce

herbie shell --seed 2023141 
(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))))