Average Error: 0.6 → 0.5
Time: 8.5s
Precision: binary64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
↓
\[\frac{\mathsf{fma}\left(1, a2 \cdot a2, a1 \cdot a1\right)}{\frac{\sqrt{2}}{\cos th}} \]
(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 1.0 (* a2 a2) (* a1 a1)) (/ (sqrt 2.0) (cos th))))
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(1.0, (a2 * a2), (a1 * a1)) / (sqrt(2.0) / cos(th));
}

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(fma(1.0, Float64(a2 * a2), Float64(a1 * a1)) / Float64(sqrt(2.0) / cos(th)))
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[(1.0 * N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] / N[Cos[th], $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)
↓
\frac{\mathsf{fma}\left(1, a2 \cdot a2, a1 \cdot a1\right)}{\frac{\sqrt{2}}{\cos th}}

Error

Bits error versus a1

Bits error versus a2

Bits error versus th

Derivation

  1. Initial program 0.6

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

  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. Applied egg-rr0.5

    \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{2}}{\frac{\sqrt{2}}{\cos th}} \cdot 1} \]

  5. Applied egg-rr0.5

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

  6. Final simplification0.5

    \[\leadsto \frac{\mathsf{fma}\left(1, a2 \cdot a2, a1 \cdot a1\right)}{\frac{\sqrt{2}}{\cos th}} \]

Reproduce

herbie shell --seed 2022134 
(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))))