Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 12.0s

Precision: binary64

Cost: 19904

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

• 43.3% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

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

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 2.0) (* (fma a2 a2 (* a1 a1)) (cos th))) 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 (sqrt(2.0) * (fma(a2, a2, (a1 * a1)) * cos(th))) / 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 Float64(Float64(sqrt(2.0) * Float64(fma(a2, a2, Float64(a1 * a1)) * cos(th))) / 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[Sqrt[2.0], $MachinePrecision] * N[(N[(a2 * a2 + N[(a1 * a1), $MachinePrecision]), $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $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 99.6%

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

3. Applied egg-rr 99.7%

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

   [Start] 99.6%	\[ \cos th \cdot \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}} \]
   fma-def [<=] 99.6%	\[ \cos th \cdot \frac{\color{blue}{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{2}} \]
   associate-*r/ [=>] 99.6%	\[ \color{blue}{\frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\sqrt{2}}} \]
   associate-*l/ [<=] 99.6%	\[ \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]
   distribute-lft-in [=>] 99.6%	\[ \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)} \]
   associate-*l/ [=>] 99.6%	\[ \color{blue}{\frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{2}}} + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   associate-*l/ [=>] 99.6%	\[ \frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{2}} + \color{blue}{\frac{\cos th \cdot \left(a2 \cdot a2\right)}{\sqrt{2}}} \]
   frac-add [=>] 99.3%	\[ \color{blue}{\frac{\left(\cos th \cdot \left(a1 \cdot a1\right)\right) \cdot \sqrt{2} + \sqrt{2} \cdot \left(\cos th \cdot \left(a2 \cdot a2\right)\right)}{\sqrt{2} \cdot \sqrt{2}}} \]
   fma-def [=>] 99.3%	\[ \frac{\color{blue}{\mathsf{fma}\left(\cos th \cdot \left(a1 \cdot a1\right), \sqrt{2}, \sqrt{2} \cdot \left(\cos th \cdot \left(a2 \cdot a2\right)\right)\right)}}{\sqrt{2} \cdot \sqrt{2}} \]
   *-commutative [=>] 99.3%	\[ \frac{\mathsf{fma}\left(\cos th \cdot \left(a1 \cdot a1\right), \sqrt{2}, \sqrt{2} \cdot \color{blue}{\left(\left(a2 \cdot a2\right) \cdot \cos th\right)}\right)}{\sqrt{2} \cdot \sqrt{2}} \]
   add-sqr-sqrt [<=] 99.7%	\[ \frac{\mathsf{fma}\left(\cos th \cdot \left(a1 \cdot a1\right), \sqrt{2}, \sqrt{2} \cdot \left(\left(a2 \cdot a2\right) \cdot \cos th\right)\right)}{\color{blue}{2}} \]

4. Simplified 99.6%

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

   [Start] 99.7%	\[ \frac{\mathsf{fma}\left(\cos th \cdot \left(a1 \cdot a1\right), \sqrt{2}, \sqrt{2} \cdot \left(\left(a2 \cdot a2\right) \cdot \cos th\right)\right)}{2} \]
   fma-udef [=>] 99.7%	\[ \frac{\color{blue}{\left(\cos th \cdot \left(a1 \cdot a1\right)\right) \cdot \sqrt{2} + \sqrt{2} \cdot \left(\left(a2 \cdot a2\right) \cdot \cos th\right)}}{2} \]
   *-commutative [=>] 99.7%	\[ \frac{\left(\cos th \cdot \left(a1 \cdot a1\right)\right) \cdot \sqrt{2} + \color{blue}{\left(\left(a2 \cdot a2\right) \cdot \cos th\right) \cdot \sqrt{2}}}{2} \]
   distribute-rgt-out [=>] 99.6%	\[ \frac{\color{blue}{\sqrt{2} \cdot \left(\cos th \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot \cos th\right)}}{2} \]
   *-commutative [<=] 99.6%	\[ \frac{\sqrt{2} \cdot \left(\cos th \cdot \left(a1 \cdot a1\right) + \color{blue}{\cos th \cdot \left(a2 \cdot a2\right)}\right)}{2} \]
   distribute-lft-in [<=] 99.6%	\[ \frac{\sqrt{2} \cdot \color{blue}{\left(\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right)}}{2} \]
   +-commutative [=>] 99.6%	\[ \frac{\sqrt{2} \cdot \left(\cos th \cdot \color{blue}{\left(a2 \cdot a2 + a1 \cdot a1\right)}\right)}{2} \]
   unpow2 [<=] 99.7%	\[ \frac{\sqrt{2} \cdot \left(\cos th \cdot \left(\color{blue}{{a2}^{2}} + a1 \cdot a1\right)\right)}{2} \]
   unpow2 [<=] 99.7%	\[ \frac{\sqrt{2} \cdot \left(\cos th \cdot \left({a2}^{2} + \color{blue}{{a1}^{2}}\right)\right)}{2} \]
   *-commutative [<=] 99.7%	\[ \frac{\sqrt{2} \cdot \color{blue}{\left(\left({a2}^{2} + {a1}^{2}\right) \cdot \cos th\right)}}{2} \]
   unpow2 [=>] 99.6%	\[ \frac{\sqrt{2} \cdot \left(\left(\color{blue}{a2 \cdot a2} + {a1}^{2}\right) \cdot \cos th\right)}{2} \]
   unpow2 [=>] 99.6%	\[ \frac{\sqrt{2} \cdot \left(\left(a2 \cdot a2 + \color{blue}{a1 \cdot a1}\right) \cdot \cos th\right)}{2} \]
   fma-def [=>] 99.6%	\[ \frac{\sqrt{2} \cdot \left(\color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)} \cdot \cos th\right)}{2} \]

5. Final simplification 99.6%

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

Alternatives

Alternative 1
Accuracy	99.6%
Cost	19904

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

Alternative 2
Accuracy	78.4%
Cost	19780

\[\begin{array}{l} \mathbf{if}\;\cos th \leq -0.005:\\ \;\;\;\;a1 \cdot \frac{a1 \cdot \cos th}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 3
Accuracy	99.6%
Cost	13568

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

Alternative 4
Accuracy	68.7%
Cost	13512

\[\begin{array}{l} \mathbf{if}\;a1 \leq -1.9 \cdot 10^{+46}:\\ \;\;\;\;a1 \cdot \frac{a1 \cdot \cos th}{\sqrt{2}}\\ \mathbf{elif}\;a1 \leq -1.75 \cdot 10^{-129}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\right)\\ \end{array} \]

Alternative 5
Accuracy	68.7%
Cost	13512

\[\begin{array}{l} \mathbf{if}\;a1 \leq -1.25 \cdot 10^{+46}:\\ \;\;\;\;a1 \cdot \frac{a1 \cdot \cos th}{\sqrt{2}}\\ \mathbf{elif}\;a1 \leq -5.3 \cdot 10^{-129}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\sqrt{2}} \cdot \left(a2 \cdot \cos th\right)\\ \end{array} \]

Alternative 6
Accuracy	68.7%
Cost	13512

\[\begin{array}{l} \mathbf{if}\;a1 \leq -2 \cdot 10^{+46}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \left(\cos th \cdot {2}^{-0.5}\right)\right)\\ \mathbf{elif}\;a1 \leq -5.2 \cdot 10^{-129}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\sqrt{2}} \cdot \left(a2 \cdot \cos th\right)\\ \end{array} \]

Alternative 7
Accuracy	99.6%
Cost	13504

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

Alternative 8
Accuracy	35.4%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;a1 \leq -1.6 \cdot 10^{+70}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \sqrt{\left(a2 \cdot a2\right) \cdot 0.5}\\ \end{array} \]

Alternative 9
Accuracy	65.5%
Cost	6976

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

Alternative 10
Accuracy	47.0%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a1 \leq -2.65 \cdot 10^{+54}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 11
Accuracy	38.8%
Cost	6720

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

Alternative 12
Accuracy	38.8%
Cost	6720

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

Reproduce

herbie shell --seed 2023272 
(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))))