Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 14.4s

Precision: binary64

Cost: 19776

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

Math

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

↓

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

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 (* a1 a1)) (/ (sqrt 2.0) (cos th))))

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, (a1 * a1)) / (sqrt(2.0) / cos(th));
}

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(fma(a2, a2, Float64(a1 * a1)) / Float64(sqrt(2.0) / cos(th)))
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[(a2 * a2 + N[(a1 * a1), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] / N[Cos[th], $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. Taylor expanded in th around inf 0.5

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

4. Simplified 0.5

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

   [Start] 0.5	\[ \frac{\left({a2}^{2} + {a1}^{2}\right) \cdot \cos th}{\sqrt{2}} \]
   associate-/l* [=>] 0.5	\[ \color{blue}{\frac{{a2}^{2} + {a1}^{2}}{\frac{\sqrt{2}}{\cos th}}} \]
   unpow2 [=>] 0.5	\[ \frac{\color{blue}{a2 \cdot a2} + {a1}^{2}}{\frac{\sqrt{2}}{\cos th}} \]
   unpow2 [=>] 0.5	\[ \frac{a2 \cdot a2 + \color{blue}{a1 \cdot a1}}{\frac{\sqrt{2}}{\cos th}} \]
   fma-def [=>] 0.5	\[ \frac{\color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}}{\frac{\sqrt{2}}{\cos th}} \]

5. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.5
Cost	19776

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

Alternative 2
Error	8.1
Cost	13644

\[\begin{array}{l} t_1 := \cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{if}\;a1 \leq -4.8 \cdot 10^{-59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a1 \leq -2.8 \cdot 10^{-96}:\\ \;\;\;\;\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\\ \mathbf{elif}\;a1 \leq -1.8 \cdot 10^{-155}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]

Alternative 3
Error	8.1
Cost	13644

\[\begin{array}{l} \mathbf{if}\;a1 \leq -2.35 \cdot 10^{-58}:\\ \;\;\;\;\cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{elif}\;a1 \leq -4 \cdot 10^{-96}:\\ \;\;\;\;\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\\ \mathbf{elif}\;a1 \leq -1.8 \cdot 10^{-155}:\\ \;\;\;\;\frac{a1}{\frac{\frac{\sqrt{2}}{\cos th}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]

Alternative 4
Error	8.0
Cost	13644

\[\begin{array}{l} t_1 := \frac{\sqrt{2}}{\cos th}\\ \mathbf{if}\;a1 \leq -7.5 \cdot 10^{-60}:\\ \;\;\;\;\frac{a1 \cdot a1}{t_1}\\ \mathbf{elif}\;a1 \leq -3.6 \cdot 10^{-96}:\\ \;\;\;\;\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\\ \mathbf{elif}\;a1 \leq -1.8 \cdot 10^{-155}:\\ \;\;\;\;\frac{a1}{\frac{t_1}{a1}}\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]

Alternative 5
Error	14.6
Cost	13513

\[\begin{array}{l} \mathbf{if}\;th \leq -0.00044 \lor \neg \left(th \leq 3.6 \cdot 10^{+18}\right):\\ \;\;\;\;\cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\\ \end{array} \]

Alternative 6
Error	0.5
Cost	13504

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

Alternative 7
Error	25.7
Cost	6976

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

Alternative 8
Error	25.7
Cost	6976

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

Alternative 9
Error	30.8
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a1 \leq -3.9 \cdot 10^{-50}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \]

Alternative 10
Error	30.8
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a1 \leq -5.2 \cdot 10^{-51}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \]

Alternative 11
Error	30.8
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a1 \leq -3 \cdot 10^{-51}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]

Alternative 12
Error	40.8
Cost	6720

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

Reproduce

herbie shell --seed 2023025 
(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))))