Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 16.4s

Precision: binary64

Cost: 19776

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• 43.6% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} [a1, a2] = \mathsf{sort}([a1, a2])\\ \end{array} \cos th \cdot \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (+
  (* (/ (cos th) (sqrt 2.0)) (* a1 a1))
  (* (/ (cos th) (sqrt 2.0)) (* a2 a2))))

↓

;; Ensure these are sorted, for example in Racket, do
(match-define (list a1 a2) (sort (a1 a2) <))

(FPCore (a1 a2 th)
 :precision binary64
 (* (cos th) (/ (fma a1 a1 (* a2 a2)) (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));
}

↓

// Ensure these are sorted
assert(a1 < a2);

double code(double a1, double a2, double th) {
	return cos(th) * (fma(a1, a1, (a2 * a2)) / 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

↓

a1, a2 = sort([a1, a2])

function code(a1, a2, th)
	return Float64(cos(th) * Float64(fma(a1, a1, Float64(a2 * a2)) / 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]

↓

NOTE: a1 and a2 should be sorted in increasing order before calling this function.

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

2. Simplified 99.7%

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

   [Start] 99.6	\[ \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   distribute-lft-out [=>] 99.6	\[ \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]
   associate-*l/ [=>] 99.7	\[ \color{blue}{\frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\sqrt{2}}} \]
   associate-*r/ [<=] 99.7	\[ \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]
   fma-def [=>] 99.7	\[ \cos th \cdot \frac{\color{blue}{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}{\sqrt{2}} \]

3. Final simplification 99.7%

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

Alternatives

Alternative 1
Accuracy	79.1%
Cost	19780

\[\begin{array}{l} \mathbf{if}\;\cos th \leq 0.6:\\ \;\;\;\;\cos th \cdot \left(\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2 + a1 \cdot a1}{\sqrt{2}}\\ \end{array} \]

Alternative 2
Accuracy	87.6%
Cost	13512

\[\begin{array}{l} \mathbf{if}\;a2 \leq 1.95 \cdot 10^{-154}:\\ \;\;\;\;\cos th \cdot \left(\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;a2 \leq 1.2 \cdot 10^{-81}:\\ \;\;\;\;\frac{a2 \cdot a2 + a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]

Alternative 3
Accuracy	87.6%
Cost	13512

\[\begin{array}{l} \mathbf{if}\;a2 \leq 3.7 \cdot 10^{-153}:\\ \;\;\;\;\cos th \cdot \left(\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;a2 \leq 1.2 \cdot 10^{-81}:\\ \;\;\;\;\frac{a2 \cdot a2 + a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot \left(\cos th \cdot a2\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	87.6%
Cost	13512

\[\begin{array}{l} \mathbf{if}\;a2 \leq 1.5 \cdot 10^{-152}:\\ \;\;\;\;\cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 1.2 \cdot 10^{-81}:\\ \;\;\;\;\frac{a2 \cdot a2 + a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot \left(\cos th \cdot a2\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	99.5%
Cost	13504

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

Alternative 6
Accuracy	66.1%
Cost	7241

\[\begin{array}{l} \mathbf{if}\;th \leq -6.8 \cdot 10^{+104} \lor \neg \left(th \leq -1.32 \cdot 10^{+51}\right):\\ \;\;\;\;\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \sqrt{\frac{a1 \cdot a1}{2}}\\ \end{array} \]

Alternative 7
Accuracy	66.1%
Cost	7240

\[\begin{array}{l} t_1 := a2 \cdot a2 + a1 \cdot a1\\ \mathbf{if}\;th \leq -6.8 \cdot 10^{+104}:\\ \;\;\;\;t_1 \cdot \sqrt{0.5}\\ \mathbf{elif}\;th \leq -1.32 \cdot 10^{+51}:\\ \;\;\;\;a1 \cdot \sqrt{\frac{a1 \cdot a1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\sqrt{2}}\\ \end{array} \]

Alternative 8
Accuracy	58.6%
Cost	7117

\[\begin{array}{l} \mathbf{if}\;a2 \leq 9.5 \cdot 10^{-78} \lor \neg \left(a2 \leq 1.26 \cdot 10^{-37}\right) \land a2 \leq 1.1 \cdot 10^{-8}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \]

Alternative 9
Accuracy	58.6%
Cost	7116

\[\begin{array}{l} t_1 := \left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{if}\;a2 \leq 5 \cdot 10^{-76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a2 \leq 9.5 \cdot 10^{-37}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;a2 \leq 1.15 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 10
Accuracy	58.7%
Cost	7116

\[\begin{array}{l} \mathbf{if}\;a2 \leq 4.3 \cdot 10^{-76}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 6.4 \cdot 10^{-37}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;a2 \leq 8.8 \cdot 10^{-10}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 11
Accuracy	58.6%
Cost	7116

\[\begin{array}{l} \mathbf{if}\;a2 \leq 1.3 \cdot 10^{-79}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 1.78 \cdot 10^{-37}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;a2 \leq 2.4 \cdot 10^{-10}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]

Alternative 12
Accuracy	58.6%
Cost	7116

\[\begin{array}{l} \mathbf{if}\;a2 \leq 1.12 \cdot 10^{-76}:\\ \;\;\;\;\frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 2 \cdot 10^{-37}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;a2 \leq 5.3 \cdot 10^{-8}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]

Alternative 13
Accuracy	58.6%
Cost	7116

\[\begin{array}{l} \mathbf{if}\;a2 \leq 8 \cdot 10^{-79}:\\ \;\;\;\;\frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 1.6 \cdot 10^{-37}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 1.05 \cdot 10^{-7}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]

Alternative 14
Accuracy	53.8%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a1 \leq -9.5 \cdot 10^{+133}:\\ \;\;\;\;a1 \cdot a1\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \]

Alternative 15
Accuracy	30.0%
Cost	320

\[a1 \cdot \left(a1 \cdot 0.5\right) \]

Alternative 16
Accuracy	29.9%
Cost	192

\[a1 \cdot a1 \]

Reproduce

herbie shell --seed 2023160 
(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))))