?

Average Error: 0.5 → 0.5
Time: 25.8s
Precision: binary64
Cost: 20032

?

\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
↓
\[\cos th \cdot \left(a1 \cdot \frac{a1}{\sqrt{2}} + a2 \cdot \frac{a2}{\sqrt{2}}\right) \]
(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
 (* (cos th) (+ (* a1 (/ a1 (sqrt 2.0))) (* 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 cos(th) * ((a1 * (a1 / sqrt(2.0))) + (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 = cos(th) * ((a1 * (a1 / sqrt(2.0d0))) + (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 Math.cos(th) * ((a1 * (a1 / Math.sqrt(2.0))) + (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 math.cos(th) * ((a1 * (a1 / math.sqrt(2.0))) + (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(cos(th) * Float64(Float64(a1 * Float64(a1 / sqrt(2.0))) + 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 = cos(th) * ((a1 * (a1 / sqrt(2.0))) + (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[Cos[th], $MachinePrecision] * N[(N[(a1 * N[(a1 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $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)
↓
\cos th \cdot \left(a1 \cdot \frac{a1}{\sqrt{2}} + 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. Applied egg-rr0.5

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

  3. Simplified0.5

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

    [Start]0.5

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

    rational.json-simplify-1 [=>]0.5

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

    rational.json-simplify-49 [=>]0.5

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

    rational.json-simplify-49 [=>]0.5

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

  4. Final simplification0.5

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

Alternatives

Alternative 1
Error0.5
Cost13504
\[\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}} \]
Alternative 2
Error25.7
Cost7104
\[\frac{\sqrt{2} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right)}{2} \]
Alternative 3
Error25.7
Cost6976
\[\frac{a2 \cdot a2 + a1 \cdot a1}{\sqrt{2}} \]

Error

Reproduce?

herbie shell --seed 2023073 
(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))))