Average Error: 0.5 → 0.5
Time: 10.8s
Precision: binary64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
↓
\[\mathsf{fma}\left(\cos th \cdot \frac{{\left(\mathsf{hypot}\left(a2, a1\right)\right)}^{2}}{\sqrt{2}}, 1, 0\right) \]
(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
 (fma (* (cos th) (/ (pow (hypot a2 a1) 2.0) (sqrt 2.0))) 1.0 0.0))
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 fma((cos(th) * (pow(hypot(a2, a1), 2.0) / sqrt(2.0))), 1.0, 0.0);
}

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 fma(Float64(cos(th) * Float64((hypot(a2, a1) ^ 2.0) / sqrt(2.0))), 1.0, 0.0)
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[(N[Power[N[Sqrt[a2 ^ 2 + a1 ^ 2], $MachinePrecision], 2.0], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0 + 0.0), $MachinePrecision]

\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\mathsf{fma}\left(\cos th \cdot \frac{{\left(\mathsf{hypot}\left(a2, a1\right)\right)}^{2}}{\sqrt{2}}, 1, 0\right)

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. Taylor expanded in th around inf 0.5

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

  3. Simplified0.5

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

  4. Applied egg-rr0.5

    \[\leadsto \color{blue}{\mathsf{fma}\left(\cos th \cdot \frac{{\left(\mathsf{hypot}\left(a2, a1\right)\right)}^{2}}{\sqrt{2}}, 1, 0\right)} \]

  5. Final simplification0.5

    \[\leadsto \mathsf{fma}\left(\cos th \cdot \frac{{\left(\mathsf{hypot}\left(a2, a1\right)\right)}^{2}}{\sqrt{2}}, 1, 0\right) \]

Reproduce

herbie shell --seed 2022194 
(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))))