Average Error: 0.5 → 0.5
Time: 16.1s
Precision: binary64
Cost: 19840
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
↓
\[\left({2}^{-0.5} \cdot \cos th\right) \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \]
(FPCore (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))))
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));
}

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

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]

\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)

Error

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. Simplified0.5

    \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)} \]
    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)))): 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)))): 1 points increase in error, 2 points decrease in error

  3. Applied egg-rr0.5

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

  4. Final simplification0.5

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

Alternatives

Alternative 1
Error14.3
Cost19780
\[\begin{array}{l} \mathbf{if}\;\cos th \leq 0.99:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \left(\cos th \cdot \sqrt{0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \frac{1}{\sqrt{2}}\\ \end{array} \]
Alternative 2
Error0.5
Cost13504
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]
Alternative 3
Error20.9
Cost13380
\[\begin{array}{l} \mathbf{if}\;a2 \leq 4.788879256446438 \cdot 10^{-143}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \left(\cos th \cdot \sqrt{0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \frac{\cos th}{\sqrt{2}}\right)\\ \end{array} \]
Alternative 4
Error20.9
Cost13380
\[\begin{array}{l} \mathbf{if}\;a2 \leq 4.788879256446438 \cdot 10^{-143}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \left(\cos th \cdot \sqrt{0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\right)\\ \end{array} \]
Alternative 5
Error25.8
Cost7104
\[\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \frac{1}{\sqrt{2}} \]
Alternative 6
Error36.6
Cost6916
\[\begin{array}{l} \mathbf{if}\;a2 \leq 4.788879256446438 \cdot 10^{-143}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left({2}^{-0.5} \cdot a2\right)\\ \end{array} \]
Alternative 7
Error36.6
Cost6852
\[\begin{array}{l} \mathbf{if}\;a2 \leq 4.788879256446438 \cdot 10^{-143}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]
Alternative 8
Error36.6
Cost6852
\[\begin{array}{l} \mathbf{if}\;a2 \leq 4.788879256446438 \cdot 10^{-143}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \end{array} \]
Alternative 9
Error40.3
Cost6720
\[a1 \cdot \left(a1 \cdot \sqrt{0.5}\right) \]

Error

Reproduce

herbie shell --seed 2022321 
(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))))