Average Error: 0.5 → 0.5
Time: 15.1s
Precision: binary64
Cost: 13568
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
↓
\[\left({2}^{-0.5} \cdot \cos th\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
(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
 (* (* (pow 2.0 -0.5) (cos th)) (+ (* 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 (pow(2.0, -0.5) * cos(th)) * ((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 = ((2.0d0 ** (-0.5d0)) * cos(th)) * ((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.pow(2.0, -0.5) * Math.cos(th)) * ((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.pow(2.0, -0.5) * math.cos(th)) * ((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((2.0 ^ -0.5) * cos(th)) * 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 = ((2.0 ^ -0.5) * cos(th)) * ((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[Power[2.0, -0.5], $MachinePrecision] * N[Cos[th], $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)
↓
\left({2}^{-0.5} \cdot \cos th\right) \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
    (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 2)) (+.f64 (*.f64 a1 a1) (*.f64 a2 a2))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 2)) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 2)) (*.f64 a2 a2)))): 0 points increase in error, 1 points decrease in error

  3. Applied egg-rr0.5

    \[\leadsto \color{blue}{\left({2}^{-0.5} \cdot \cos th\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

  4. Final simplification0.5

    \[\leadsto \left({2}^{-0.5} \cdot \cos th\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

Alternatives

Alternative 1
Error20.4
Cost13576
\[\begin{array}{l} \mathbf{if}\;a2 \leq 1.46 \cdot 10^{-133}:\\ \;\;\;\;\cos th \cdot \left(a1 \cdot \frac{a1}{\sqrt{2}}\right)\\ \mathbf{elif}\;a2 \leq 5.6 \cdot 10^{-42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(\cos th \cdot \left({2}^{-0.5} \cdot a2\right)\right)\\ \end{array} \]
Alternative 2
Error14.4
Cost13512
\[\begin{array}{l} t_1 := a1 \cdot \frac{a1}{\frac{\sqrt{2}}{\cos th}}\\ \mathbf{if}\;th \leq -7.6 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;th \leq 33000000:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 3
Error20.4
Cost13512
\[\begin{array}{l} \mathbf{if}\;a2 \leq 1.3 \cdot 10^{-132}:\\ \;\;\;\;a1 \cdot \frac{a1}{\frac{\sqrt{2}}{\cos th}}\\ \mathbf{elif}\;a2 \leq 4.4 \cdot 10^{-43}:\\ \;\;\;\;\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \frac{\cos th}{\sqrt{2}}\right)\\ \end{array} \]
Alternative 4
Error20.3
Cost13512
\[\begin{array}{l} \mathbf{if}\;a2 \leq 1.8 \cdot 10^{-132}:\\ \;\;\;\;a1 \cdot \frac{a1}{\frac{\sqrt{2}}{\cos th}}\\ \mathbf{elif}\;a2 \leq 3.2 \cdot 10^{-42}:\\ \;\;\;\;\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(\cos th \cdot \frac{a2}{\sqrt{2}}\right)\\ \end{array} \]
Alternative 5
Error20.3
Cost13512
\[\begin{array}{l} \mathbf{if}\;a2 \leq 5.6 \cdot 10^{-133}:\\ \;\;\;\;a1 \cdot \frac{a1}{\frac{\sqrt{2}}{\cos th}}\\ \mathbf{elif}\;a2 \leq 1.6 \cdot 10^{-42}:\\ \;\;\;\;\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{\cos th \cdot a2}{\sqrt{2}}\\ \end{array} \]
Alternative 6
Error20.3
Cost13512
\[\begin{array}{l} \mathbf{if}\;a2 \leq 6 \cdot 10^{-133}:\\ \;\;\;\;a1 \cdot \frac{a1}{\frac{\sqrt{2}}{\cos th}}\\ \mathbf{elif}\;a2 \leq 5.2 \cdot 10^{-44}:\\ \;\;\;\;\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]
Alternative 7
Error20.4
Cost13512
\[\begin{array}{l} \mathbf{if}\;a2 \leq 7 \cdot 10^{-133}:\\ \;\;\;\;\cos th \cdot \left(a1 \cdot \frac{a1}{\sqrt{2}}\right)\\ \mathbf{elif}\;a2 \leq 1.36 \cdot 10^{-42}:\\ \;\;\;\;\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]
Alternative 8
Error20.3
Cost13512
\[\begin{array}{l} \mathbf{if}\;a2 \leq 7 \cdot 10^{-133}:\\ \;\;\;\;\cos th \cdot \left(a1 \cdot \frac{a1}{\sqrt{2}}\right)\\ \mathbf{elif}\;a2 \leq 8.5 \cdot 10^{-44}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]
Alternative 9
Error0.5
Cost13504
\[\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \]
Alternative 10
Error0.5
Cost13504
\[\frac{a1 \cdot a1 + a2 \cdot a2}{\frac{\sqrt{2}}{\cos th}} \]
Alternative 11
Error36.7
Cost7180
\[\begin{array}{l} \mathbf{if}\;a2 \leq 3.3 \cdot 10^{-146}:\\ \;\;\;\;\frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 3.7 \cdot 10^{-119}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 5.2 \cdot 10^{-58}:\\ \;\;\;\;a1 \cdot \left({2}^{-0.5} \cdot a1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]
Alternative 12
Error36.7
Cost7180
\[\begin{array}{l} t_1 := {2}^{-0.5} \cdot \left(a2 \cdot a2\right)\\ \mathbf{if}\;a2 \leq 6.2 \cdot 10^{-145}:\\ \;\;\;\;\frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 8.2 \cdot 10^{-120}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a2 \leq 1.2 \cdot 10^{-56}:\\ \;\;\;\;a1 \cdot \left({2}^{-0.5} \cdot a1\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 13
Error36.7
Cost7116
\[\begin{array}{l} t_1 := a2 \cdot \frac{a2}{\sqrt{2}}\\ \mathbf{if}\;a2 \leq 2.3 \cdot 10^{-144}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 8.8 \cdot 10^{-119}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a2 \leq 1.12 \cdot 10^{-57}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 14
Error36.7
Cost7116
\[\begin{array}{l} t_1 := a2 \cdot \frac{a2}{\sqrt{2}}\\ \mathbf{if}\;a2 \leq 3.2 \cdot 10^{-148}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 1.3 \cdot 10^{-119}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a2 \leq 4.8 \cdot 10^{-60}:\\ \;\;\;\;\frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 15
Error36.7
Cost7116
\[\begin{array}{l} \mathbf{if}\;a2 \leq 8.5 \cdot 10^{-145}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 8 \cdot 10^{-119}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 2.6 \cdot 10^{-59}:\\ \;\;\;\;\frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]
Alternative 16
Error36.9
Cost7116
\[\begin{array}{l} t_1 := \frac{a1 \cdot a1}{\sqrt{2}}\\ t_2 := \frac{a2}{\frac{\sqrt{2}}{a2}}\\ \mathbf{if}\;a2 \leq 2.6 \cdot 10^{-148}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a2 \leq 1.25 \cdot 10^{-91}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a2 \leq 2.8 \cdot 10^{-58}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 17
Error25.9
Cost6976
\[\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5} \]
Alternative 18
Error40.6
Cost6720
\[\left(a1 \cdot a1\right) \cdot \sqrt{0.5} \]
Alternative 19
Error40.6
Cost6720
\[a1 \cdot \frac{a1}{\sqrt{2}} \]

Error

Reproduce

herbie shell --seed 2022325 
(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))))