Average Error: 0.5 → 0.5
Time: 8.7s
Precision: binary64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
↓
\[\frac{a1 \cdot a1}{\frac{\sqrt{2}}{\cos th}} + \cos th \cdot \left(a2 \cdot \frac{a2}{\sqrt{2}}\right) \]
(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
 (+
  (/ (* a1 a1) (/ (sqrt 2.0) (cos th)))
  (* (cos th) (* a2 (/ a2 (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 ((a1 * a1) / (sqrt(2.0) / cos(th))) + (cos(th) * (a2 * (a2 / sqrt(2.0))));
}

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 = ((a1 * a1) / (sqrt(2.0d0) / cos(th))) + (cos(th) * (a2 * (a2 / sqrt(2.0d0))))
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 ((a1 * a1) / (Math.sqrt(2.0) / Math.cos(th))) + (Math.cos(th) * (a2 * (a2 / Math.sqrt(2.0))));
}

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

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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

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

  3. Simplified0.5

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

  4. Applied egg-rr0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2022203 
(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))))