Average Error: 0.5 → 0.5
Time: 8.0s
Precision: binary64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
↓
\[\frac{\cos th \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt{2}} \]
(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 a2 a2 (* a1 a1))) (sqrt 2.0)))
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(a2, a2, (a1 * a1))) / sqrt(2.0);
}

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

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

Error

Bits error versus a1

Bits error versus a2

Bits error versus th

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 th around inf 0.5

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

  4. Taylor expanded in a1 around 0 0.5

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

  5. Simplified0.5

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

  6. Final simplification0.5

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

Reproduce

herbie shell --seed 2022141 
(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))))