Average Error: 0.5 → 0.5
Time: 10.9s
Precision: binary64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
↓
\[\sqrt{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)} \cdot \left(\cos th \cdot \frac{\mathsf{hypot}\left(a2, a1\right)}{\sqrt{2}}\right) \]
(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
 (* (sqrt (fma a1 a1 (* a2 a2))) (* (cos th) (/ (hypot a2 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 sqrt(fma(a1, a1, (a2 * a2))) * (cos(th) * (hypot(a2, 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(sqrt(fma(a1, a1, Float64(a2 * a2))) * Float64(cos(th) * Float64(hypot(a2, 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[Sqrt[N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Cos[th], $MachinePrecision] * N[(N[Sqrt[a2 ^ 2 + a1 ^ 2], $MachinePrecision] / N[Sqrt[2.0], $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)
↓
\sqrt{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)} \cdot \left(\cos th \cdot \frac{\mathsf{hypot}\left(a2, a1\right)}{\sqrt{2}}\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. Simplified0.5

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

  3. Applied *-commutative_binary640.5

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

  4. Applied *-un-lft-identity_binary640.5

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

  5. Applied add-sqr-sqrt_binary640.5

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

  6. Applied times-frac_binary640.5

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

  7. Applied associate-*l*_binary640.5

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

  8. Simplified0.5

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

  9. Final simplification0.5

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

Reproduce

herbie shell --seed 2022131 
(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))))