Average Error: 0.5 → 0.4
Time: 7.0s
Precision: binary64
\[ \begin{array}{c}[a1, a2] = \mathsf{sort}([a1, a2])\\ \end{array} \]
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
↓
\[\cos th \cdot \mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \frac{a2}{\frac{\sqrt{2}}{a2}}\right) \]
(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
 (* (cos th) (fma (/ a1 (sqrt 2.0)) a1 (/ a2 (/ (sqrt 2.0) 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 cos(th) * fma((a1 / sqrt(2.0)), a1, (a2 / (sqrt(2.0) / 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(cos(th) * fma(Float64(a1 / sqrt(2.0)), a1, Float64(a2 / Float64(sqrt(2.0) / 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[Cos[th], $MachinePrecision] * N[(N[(a1 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * a1 + N[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / a2), $MachinePrecision]), $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)
↓
\cos th \cdot \mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \frac{a2}{\frac{\sqrt{2}}{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

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

  3. Taylor expanded in a1 around 0 0.5

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

  4. Simplified0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2022192 
(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))))