Average Error: 0.6 → 0.5
Time: 8.2s
Precision: binary64
\[ \begin{array}{c}[a1, a2] = \mathsf{sort}([a1, a2])\\ \end{array} \]
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
↓
\[\cos th \cdot \left(\frac{a2 \cdot a2}{\sqrt{2}} + \frac{a1}{\frac{\sqrt{2}}{a1}}\right) \]
(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) (+ (/ (* a2 a2) (sqrt 2.0)) (/ a1 (/ (sqrt 2.0) a1)))))
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) * (((a2 * a2) / sqrt(2.0)) + (a1 / (sqrt(2.0) / a1)));
}

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = ((cos(th) / sqrt(2.0d0)) * (a1 * a1)) + ((cos(th) / sqrt(2.0d0)) * (a2 * a2))
end function
↓
real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = cos(th) * (((a2 * a2) / sqrt(2.0d0)) + (a1 / (sqrt(2.0d0) / a1)))
end function

public static double code(double a1, double a2, double th) {
	return ((Math.cos(th) / Math.sqrt(2.0)) * (a1 * a1)) + ((Math.cos(th) / Math.sqrt(2.0)) * (a2 * a2));
}
↓
public static double code(double a1, double a2, double th) {
	return Math.cos(th) * (((a2 * a2) / Math.sqrt(2.0)) + (a1 / (Math.sqrt(2.0) / a1)));
}

def code(a1, a2, th):
	return ((math.cos(th) / math.sqrt(2.0)) * (a1 * a1)) + ((math.cos(th) / math.sqrt(2.0)) * (a2 * a2))
↓
def code(a1, a2, th):
	return math.cos(th) * (((a2 * a2) / math.sqrt(2.0)) + (a1 / (math.sqrt(2.0) / a1)))

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) * Float64(Float64(Float64(a2 * a2) / sqrt(2.0)) + Float64(a1 / Float64(sqrt(2.0) / a1))))
end

function tmp = code(a1, a2, th)
	tmp = ((cos(th) / sqrt(2.0)) * (a1 * a1)) + ((cos(th) / sqrt(2.0)) * (a2 * a2));
end
↓
function tmp = code(a1, a2, th)
	tmp = cos(th) * (((a2 * a2) / sqrt(2.0)) + (a1 / (sqrt(2.0) / a1)));
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[(N[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + N[(a1 / N[(N[Sqrt[2.0], $MachinePrecision] / a1), $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 \left(\frac{a2 \cdot a2}{\sqrt{2}} + \frac{a1}{\frac{\sqrt{2}}{a1}}\right)

Error

Bits error versus a1

Bits error versus a2

Bits error versus th

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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 a1 around 0 0.5

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

  4. Simplified0.5

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

  5. Applied egg-rr0.5

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

  6. Final simplification0.5

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

Reproduce

herbie shell --seed 2022150 
(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))))