Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 13.6s

Precision: binary64

Cost: 19840

Math

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

↓

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

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
 (* (* (pow 2.0 -0.5) (cos th)) (fma a1 a1 (* a2 a2))))

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 (pow(2.0, -0.5) * cos(th)) * fma(a1, a1, (a2 * a2));
}

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((2.0 ^ -0.5) * cos(th)) * fma(a1, a1, Float64(a2 * a2)))
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[Power[2.0, -0.5], $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision] * N[(a1 * a1 + N[(a2 * a2), $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

   \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)} \]
   Proof
   (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 2)) (fma.f64 a1 a1 (*.f64 a2 a2))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 2)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 a1 a1) (*.f64 a2 a2)))): 1 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 2)) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 2)) (*.f64 a2 a2)))): 0 points increase in error, 1 points decrease in error

3. Applied egg-rr 0.5

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

4. Final simplification 0.5

   \[\leadsto \left({2}^{-0.5} \cdot \cos th\right) \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \]

Alternatives

Alternative 1
Error	14.8
Cost	19780

\[\begin{array}{l} \mathbf{if}\;\cos th \leq 0.9995:\\ \;\;\;\;\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right)\\ \end{array} \]

Alternative 2
Error	14.8
Cost	19780

\[\begin{array}{l} \mathbf{if}\;\cos th \leq 0.9995:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right)\\ \end{array} \]

Alternative 3
Error	20.3
Cost	13512

\[\begin{array}{l} \mathbf{if}\;a2 \leq 2.2729402104295545 \cdot 10^{-134}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 6.195574097221892 \cdot 10^{-46}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\cos th \cdot \left(a2 \cdot a2\right)\right)\\ \end{array} \]

Alternative 4
Error	20.3
Cost	13512

\[\begin{array}{l} \mathbf{if}\;a2 \leq 2.2729402104295545 \cdot 10^{-134}:\\ \;\;\;\;\frac{a1 \cdot a1}{\frac{\sqrt{2}}{\cos th}}\\ \mathbf{elif}\;a2 \leq 6.195574097221892 \cdot 10^{-46}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\cos th \cdot \left(a2 \cdot a2\right)\right)\\ \end{array} \]

Alternative 5
Error	20.3
Cost	13512

\[\begin{array}{l} \mathbf{if}\;a2 \leq 2.2729402104295545 \cdot 10^{-134}:\\ \;\;\;\;\cos th \cdot \left(a1 \cdot \frac{a1}{\sqrt{2}}\right)\\ \mathbf{elif}\;a2 \leq 6.195574097221892 \cdot 10^{-46}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\cos th \cdot \left(a2 \cdot a2\right)\right)\\ \end{array} \]

Alternative 6
Error	20.3
Cost	13512

\[\begin{array}{l} \mathbf{if}\;a2 \leq 2.2729402104295545 \cdot 10^{-134}:\\ \;\;\;\;\cos th \cdot \left(a1 \cdot \frac{a1}{\sqrt{2}}\right)\\ \mathbf{elif}\;a2 \leq 6.195574097221892 \cdot 10^{-46}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \left(\cos th \cdot \sqrt{0.5}\right)\\ \end{array} \]

Alternative 7
Error	20.3
Cost	13512

\[\begin{array}{l} \mathbf{if}\;a2 \leq 2.2729402104295545 \cdot 10^{-134}:\\ \;\;\;\;\cos th \cdot \left(a1 \cdot \frac{a1}{\sqrt{2}}\right)\\ \mathbf{elif}\;a2 \leq 6.195574097221892 \cdot 10^{-46}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \left(\cos th \cdot \sqrt{0.5}\right)\\ \end{array} \]

Alternative 8
Error	0.5
Cost	13504

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

Alternative 9
Error	36.9
Cost	7116

\[\begin{array}{l} t_1 := \frac{a1 \cdot a1}{\sqrt{2}}\\ t_2 := a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \mathbf{if}\;a2 \leq 7.4663191542435486 \cdot 10^{-152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a2 \leq 1.6237979464953404 \cdot 10^{-93}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a2 \leq 4.204125763592496 \cdot 10^{-60}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Error	25.9
Cost	6976

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

Alternative 11
Error	40.5
Cost	6720

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

Alternative 12
Error	40.5
Cost	6720

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

Reproduce

herbie shell --seed 2022325 
(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))))