Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 14.3s

Precision: binary64

Cost: 19776

Math

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

↓

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

FPCore

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

↓

(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));
}

↓

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

↓

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]

↓

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 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

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

3. Final simplification 0.5

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

Alternatives

Alternative 1
Error	13.8
Cost	19780

\[\begin{array}{l} \mathbf{if}\;\cos th \leq 0.99999993:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \left(\cos th \cdot \sqrt{0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2 + a1 \cdot a1}{\sqrt{2} \cdot \left(0.5 \cdot \left(th \cdot th\right) + 1\right)}\\ \end{array} \]

Alternative 2
Error	21.1
Cost	13644

\[\begin{array}{l} t_1 := \cos th \cdot \sqrt{0.5}\\ t_2 := \left(a2 \cdot a2\right) \cdot t_1\\ t_3 := a1 \cdot \left(a1 \cdot t_1\right)\\ \mathbf{if}\;a1 \leq -2.918360011523314 \cdot 10^{-123}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a1 \leq -3.611730819470189 \cdot 10^{-135}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a1 \leq -3.4881544671587515 \cdot 10^{-156}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	21.1
Cost	13644

\[\begin{array}{l} t_1 := \cos th \cdot \frac{a2 \cdot a2}{\sqrt{2}}\\ t_2 := a1 \cdot \left(a1 \cdot \left(\cos th \cdot \sqrt{0.5}\right)\right)\\ \mathbf{if}\;a1 \leq -2.918360011523314 \cdot 10^{-123}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a1 \leq -3.611730819470189 \cdot 10^{-135}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a1 \leq -3.4881544671587515 \cdot 10^{-156}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	21.1
Cost	13644

\[\begin{array}{l} t_1 := \cos th \cdot \frac{a2 \cdot a2}{\sqrt{2}}\\ t_2 := \cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{if}\;a1 \leq -2.918360011523314 \cdot 10^{-123}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a1 \leq -3.611730819470189 \cdot 10^{-135}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a1 \leq -3.4881544671587515 \cdot 10^{-156}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	21.1
Cost	13644

\[\begin{array}{l} t_1 := \cos th \cdot \frac{a2 \cdot a2}{\sqrt{2}}\\ \mathbf{if}\;a1 \leq -2.918360011523314 \cdot 10^{-123}:\\ \;\;\;\;\cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{elif}\;a1 \leq -3.611730819470189 \cdot 10^{-135}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a1 \leq -3.4881544671587515 \cdot 10^{-156}:\\ \;\;\;\;\frac{a1 \cdot \left(\cos th \cdot a1\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	21.1
Cost	13644

\[\begin{array}{l} \mathbf{if}\;a1 \leq -2.918360011523314 \cdot 10^{-123}:\\ \;\;\;\;\cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{elif}\;a1 \leq -3.611730819470189 \cdot 10^{-135}:\\ \;\;\;\;\cos th \cdot \frac{a2 \cdot a2}{\sqrt{2}}\\ \mathbf{elif}\;a1 \leq -3.4881544671587515 \cdot 10^{-156}:\\ \;\;\;\;\frac{a1 \cdot \left(\cos th \cdot a1\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot \left(\cos th \cdot a2\right)\right)\\ \end{array} \]

Alternative 7
Error	21.1
Cost	13644

\[\begin{array}{l} \mathbf{if}\;a1 \leq -2.918360011523314 \cdot 10^{-123}:\\ \;\;\;\;\cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{elif}\;a1 \leq -3.611730819470189 \cdot 10^{-135}:\\ \;\;\;\;\cos th \cdot \frac{a2 \cdot a2}{\sqrt{2}}\\ \mathbf{elif}\;a1 \leq -3.4881544671587515 \cdot 10^{-156}:\\ \;\;\;\;\frac{a1 \cdot \left(\cos th \cdot a1\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(\cos th \cdot \frac{a2}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 8
Error	0.4
Cost	13504

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

Alternative 9
Error	36.8
Cost	6980

\[\begin{array}{l} \mathbf{if}\;a2 \leq 1.2660112740737315 \cdot 10^{-130}:\\ \;\;\;\;\frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \sqrt{\frac{a2 \cdot a2}{2}}\\ \end{array} \]

Alternative 10
Error	25.8
Cost	6976

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

Alternative 11
Error	36.8
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a2 \leq 1.2660112740737315 \cdot 10^{-130}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 12
Error	36.8
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a2 \leq 1.2660112740737315 \cdot 10^{-130}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]

Alternative 13
Error	36.8
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a2 \leq 1.2660112740737315 \cdot 10^{-130}:\\ \;\;\;\;\frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]

Alternative 14
Error	36.8
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a2 \leq 1.2660112740737315 \cdot 10^{-130}:\\ \;\;\;\;\frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]

Alternative 15
Error	36.8
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a2 \leq 1.2660112740737315 \cdot 10^{-130}:\\ \;\;\;\;\frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \end{array} \]

Alternative 16
Error	41.0
Cost	6720

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

Reproduce

herbie shell --seed 2022217 
(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))))