Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 12.7s

Precision: binary64

Cost: 19776

Math

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

↓

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

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

   \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]
   Proof
   (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 2)) (+.f64 (*.f64 a1 a1) (*.f64 a2 a2))): 0 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, 0 points decrease in error

3. Applied egg-rr 0.8

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

4. Taylor expanded in th around inf 0.5

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

5. Simplified 0.5

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt{2}} \cdot \cos th} \]
   Proof
   (*.f64 (/.f64 (fma.f64 a2 a2 (*.f64 a1 a1)) (sqrt.f64 2)) (cos.f64 th)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (Rewrite=> fma-udef_binary64 (+.f64 (*.f64 a2 a2) (*.f64 a1 a1))) (sqrt.f64 2)) (cos.f64 th)): 1 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 a2 2)) (*.f64 a1 a1)) (sqrt.f64 2)) (cos.f64 th)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (+.f64 (pow.f64 a2 2) (Rewrite<= unpow2_binary64 (pow.f64 a1 2))) (sqrt.f64 2)) (cos.f64 th)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/r/_binary64 (/.f64 (+.f64 (pow.f64 a2 2) (pow.f64 a1 2)) (/.f64 (sqrt.f64 2) (cos.f64 th)))): 23 points increase in error, 20 points decrease in error
   (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (+.f64 (pow.f64 a2 2) (pow.f64 a1 2)) (cos.f64 th)) (sqrt.f64 2))): 17 points increase in error, 20 points decrease in error

6. Final simplification 0.5

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

Alternatives

Alternative 1
Error	14.8
Cost	19780

\[\begin{array}{l} \mathbf{if}\;\cos th \leq 0.97:\\ \;\;\;\;a1 \cdot \left(\cos th \cdot \left(a1 \cdot \sqrt{0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 2
Error	21.1
Cost	13644

\[\begin{array}{l} t_1 := a2 \cdot \frac{\cos th}{\frac{\sqrt{2}}{a2}}\\ t_2 := a1 \cdot \left(\cos th \cdot \left(a1 \cdot \sqrt{0.5}\right)\right)\\ \mathbf{if}\;a1 \leq -1.95 \cdot 10^{-81}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a1 \leq -6.9 \cdot 10^{-127}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a1 \leq -3.1 \cdot 10^{-152}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	21.0
Cost	13644

\[\begin{array}{l} t_1 := a2 \cdot \frac{\cos th}{\frac{\sqrt{2}}{a2}}\\ \mathbf{if}\;a1 \leq -1.4 \cdot 10^{-81}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\left(a1 \cdot a1\right) \cdot \cos th\right)\\ \mathbf{elif}\;a1 \leq -1.55 \cdot 10^{-128}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a1 \leq -2.45 \cdot 10^{-152}:\\ \;\;\;\;a1 \cdot \left(\cos th \cdot \left(a1 \cdot \sqrt{0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	21.0
Cost	13644

\[\begin{array}{l} t_1 := \sqrt{0.5} \cdot \left(\cos th \cdot \left(a2 \cdot a2\right)\right)\\ \mathbf{if}\;a1 \leq -1.4 \cdot 10^{-69}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\left(a1 \cdot a1\right) \cdot \cos th\right)\\ \mathbf{elif}\;a1 \leq -7.5 \cdot 10^{-128}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a1 \leq -1.86 \cdot 10^{-152}:\\ \;\;\;\;a1 \cdot \left(\cos th \cdot \left(a1 \cdot \sqrt{0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	21.0
Cost	13644

\[\begin{array}{l} t_1 := \sqrt{0.5} \cdot \left(\cos th \cdot \left(a2 \cdot a2\right)\right)\\ \mathbf{if}\;a1 \leq -1.25 \cdot 10^{-69}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\left(a1 \cdot a1\right) \cdot \cos th\right)\\ \mathbf{elif}\;a1 \leq -5.5 \cdot 10^{-128}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a1 \leq -9.5 \cdot 10^{-153}:\\ \;\;\;\;\cos th \cdot \frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	21.0
Cost	13644

\[\begin{array}{l} t_1 := \cos th \cdot \left(a2 \cdot a2\right)\\ \mathbf{if}\;a1 \leq -1.4 \cdot 10^{-69}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\left(a1 \cdot a1\right) \cdot \cos th\right)\\ \mathbf{elif}\;a1 \leq -2.75 \cdot 10^{-129}:\\ \;\;\;\;\sqrt{0.5} \cdot t_1\\ \mathbf{elif}\;a1 \leq -3.1 \cdot 10^{-152}:\\ \;\;\;\;\cos th \cdot \frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\sqrt{2}}\\ \end{array} \]

Alternative 7
Error	0.5
Cost	13504

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

Alternative 8
Error	0.5
Cost	13504

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

Alternative 9
Error	25.9
Cost	6976

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

Alternative 10
Error	36.7
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a2 \leq 4.8 \cdot 10^{-145}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 11
Error	36.7
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a2 \leq 7 \cdot 10^{-145}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 12
Error	36.7
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a2 \leq 7 \cdot 10^{-145}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]

Alternative 13
Error	40.1
Cost	6720

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

Reproduce

herbie shell --seed 2022338 
(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))))