Average Error: 0.5 → 0.5
Time: 14.3s
Precision: binary64
Cost: 13504
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
↓
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
(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) (sqrt 2.0)) (+ (* a1 a1) (* a2 a2))))
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) / sqrt(2.0)) * ((a1 * a1) + (a2 * a2));
}

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

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) / math.sqrt(2.0)) * ((a1 * a1) + (a2 * a2))

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(cos(th) / sqrt(2.0)) * Float64(Float64(a1 * a1) + Float64(a2 * a2)))
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) / sqrt(2.0)) * ((a1 * a1) + (a2 * a2));
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[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $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{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\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. Simplified0.5

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

    [Start]0.5

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

    distribute-lft-out [=>]0.5

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

  3. Final simplification0.5

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

Alternatives

Alternative 1
Error20.6
Cost13708
\[\begin{array}{l} t_1 := \cos th \cdot \left(a1 \cdot \left(a1 \cdot {2}^{-0.5}\right)\right)\\ \mathbf{if}\;a2 \leq 3.6 \cdot 10^{-115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a2 \leq 2.7 \cdot 10^{-104}:\\ \;\;\;\;a2 \cdot \left(\sqrt{0.5} \cdot \left(\cos th \cdot a2\right)\right)\\ \mathbf{elif}\;a2 \leq 5.8 \cdot 10^{-47}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\\ \end{array} \]
Alternative 2
Error20.6
Cost13645
\[\begin{array}{l} \mathbf{if}\;a2 \leq 4.6 \cdot 10^{-116} \lor \neg \left(a2 \leq 8.5 \cdot 10^{-102}\right) \land a2 \leq 1.8 \cdot 10^{-46}:\\ \;\;\;\;\cos th \cdot \frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(\sqrt{0.5} \cdot \left(\cos th \cdot a2\right)\right)\\ \end{array} \]
Alternative 3
Error20.6
Cost13645
\[\begin{array}{l} \mathbf{if}\;a2 \leq 4.1 \cdot 10^{-116}:\\ \;\;\;\;\cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 2.05 \cdot 10^{-104} \lor \neg \left(a2 \leq 5.2 \cdot 10^{-48}\right):\\ \;\;\;\;a2 \cdot \left(\sqrt{0.5} \cdot \left(\cos th \cdot a2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \frac{a1}{\frac{\sqrt{2}}{a1}}\\ \end{array} \]
Alternative 4
Error20.6
Cost13644
\[\begin{array}{l} t_1 := \cos th \cdot a2\\ \mathbf{if}\;a2 \leq 9.6 \cdot 10^{-117}:\\ \;\;\;\;\cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 3.1 \cdot 10^{-102}:\\ \;\;\;\;a2 \cdot \left(\sqrt{0.5} \cdot t_1\right)\\ \mathbf{elif}\;a2 \leq 3.1 \cdot 10^{-46}:\\ \;\;\;\;\cos th \cdot \frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot t_1\right)\\ \end{array} \]
Alternative 5
Error20.6
Cost13644
\[\begin{array}{l} t_1 := \cos th \cdot a2\\ \mathbf{if}\;a2 \leq 3.4 \cdot 10^{-115}:\\ \;\;\;\;\cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 3.3 \cdot 10^{-102}:\\ \;\;\;\;a2 \cdot \left(\sqrt{0.5} \cdot t_1\right)\\ \mathbf{elif}\;a2 \leq 10^{-43}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\cos th \cdot \left(a1 \cdot a1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot t_1\right)\\ \end{array} \]
Alternative 6
Error20.6
Cost13644
\[\begin{array}{l} t_1 := \cos th \cdot a2\\ t_2 := \sqrt{0.5} \cdot \left(a1 \cdot \left(\cos th \cdot a1\right)\right)\\ \mathbf{if}\;a2 \leq 1.85 \cdot 10^{-115}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a2 \leq 1.45 \cdot 10^{-103}:\\ \;\;\;\;a2 \cdot \left(\sqrt{0.5} \cdot t_1\right)\\ \mathbf{elif}\;a2 \leq 8.2 \cdot 10^{-48}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot t_1\right)\\ \end{array} \]
Alternative 7
Error20.6
Cost13644
\[\begin{array}{l} t_1 := \sqrt{0.5} \cdot \left(a1 \cdot \left(\cos th \cdot a1\right)\right)\\ \mathbf{if}\;a2 \leq 6.2 \cdot 10^{-116}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a2 \leq 3.2 \cdot 10^{-102}:\\ \;\;\;\;a2 \cdot \left(\sqrt{0.5} \cdot \left(\cos th \cdot a2\right)\right)\\ \mathbf{elif}\;a2 \leq 3.5 \cdot 10^{-47}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\\ \end{array} \]
Alternative 8
Error14.6
Cost13513
\[\begin{array}{l} \mathbf{if}\;th \leq -2.55 \lor \neg \left(th \leq 0.24\right):\\ \;\;\;\;a2 \cdot \left(\sqrt{0.5} \cdot \left(\cos th \cdot a2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \left(\sqrt{0.5} \cdot \left(-0.5 \cdot \left(th \cdot th\right) + 1\right)\right)\\ \end{array} \]
Alternative 9
Error0.5
Cost13504
\[\sqrt{0.5} \cdot \left(\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right) \]
Alternative 10
Error26.1
Cost6976
\[\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5} \]
Alternative 11
Error26.1
Cost6976
\[\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}} \]
Alternative 12
Error37.5
Cost6916
\[\begin{array}{l} \mathbf{if}\;a1 \leq -1.9 \cdot 10^{-146}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot {2}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]
Alternative 13
Error37.5
Cost6852
\[\begin{array}{l} \mathbf{if}\;a1 \leq -4.4 \cdot 10^{-148}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]
Alternative 14
Error37.4
Cost6852
\[\begin{array}{l} \mathbf{if}\;a1 \leq -4.2 \cdot 10^{-146}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]
Alternative 15
Error37.4
Cost6852
\[\begin{array}{l} \mathbf{if}\;a1 \leq -4.5 \cdot 10^{-146}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]
Alternative 16
Error37.4
Cost6852
\[\begin{array}{l} \mathbf{if}\;a1 \leq -5.2 \cdot 10^{-146}:\\ \;\;\;\;\frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]
Alternative 17
Error40.8
Cost6720
\[\left(a1 \cdot a1\right) \cdot \sqrt{0.5} \]

Error

Reproduce

herbie shell --seed 2023010 
(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))))