Average Error: 0.5 → 0.4
Time: 16.7s
Precision: binary64
Cost: 26048
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
↓
\[\frac{{\left(\mathsf{hypot}\left(a2, a1\right)\right)}^{2}}{\frac{\sqrt{2}}{\cos th}} \]
(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 (hypot a2 a1) 2.0) (/ (sqrt 2.0) (cos th))))
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(hypot(a2, a1), 2.0) / (sqrt(2.0) / cos(th));
}

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(Math.hypot(a2, a1), 2.0) / (Math.sqrt(2.0) / Math.cos(th));
}

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(math.hypot(a2, a1), 2.0) / (math.sqrt(2.0) / math.cos(th))

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((hypot(a2, a1) ^ 2.0) / Float64(sqrt(2.0) / cos(th)))
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 = (hypot(a2, a1) ^ 2.0) / (sqrt(2.0) / cos(th));
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[Power[N[Sqrt[a2 ^ 2 + a1 ^ 2], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] / N[Cos[th], $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{{\left(\mathsf{hypot}\left(a2, a1\right)\right)}^{2}}{\frac{\sqrt{2}}{\cos th}}

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-rr0.4

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

  3. Final simplification0.4

    \[\leadsto \frac{{\left(\mathsf{hypot}\left(a2, a1\right)\right)}^{2}}{\frac{\sqrt{2}}{\cos th}} \]

Alternatives

Alternative 1
Error14.6
Cost19780
\[\begin{array}{l} \mathbf{if}\;\cos th \leq 0.99:\\ \;\;\;\;\left(a2 \cdot \cos th\right) \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \end{array} \]
Alternative 2
Error0.5
Cost19776
\[\frac{\cos th}{\sqrt{2}} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \]
Alternative 3
Error21.0
Cost13644
\[\begin{array}{l} \mathbf{if}\;a1 \leq -6.006095108509966 \cdot 10^{-24}:\\ \;\;\;\;\cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{elif}\;a1 \leq -6.052530258782011 \cdot 10^{-112}:\\ \;\;\;\;\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{elif}\;a1 \leq -3.2040251015145385 \cdot 10^{-132}:\\ \;\;\;\;\frac{a1 \cdot a1}{\frac{\sqrt{2}}{\cos th}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(\cos th \cdot \frac{a2}{\sqrt{2}}\right)\\ \end{array} \]
Alternative 4
Error21.0
Cost13644
\[\begin{array}{l} \mathbf{if}\;a1 \leq -3.7205729887072595 \cdot 10^{-21}:\\ \;\;\;\;\cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{elif}\;a1 \leq -6.052530258782011 \cdot 10^{-112}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt{2}}\\ \mathbf{elif}\;a1 \leq -3.2040251015145385 \cdot 10^{-132}:\\ \;\;\;\;\frac{a1 \cdot a1}{\frac{\sqrt{2}}{\cos th}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(\cos th \cdot \frac{a2}{\sqrt{2}}\right)\\ \end{array} \]
Alternative 5
Error0.5
Cost13504
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]
Alternative 6
Error21.1
Cost13380
\[\begin{array}{l} \mathbf{if}\;a2 \leq 2.8786861742659004 \cdot 10^{-140}:\\ \;\;\;\;\cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot \cos th\right) \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \]
Alternative 7
Error21.1
Cost13380
\[\begin{array}{l} \mathbf{if}\;a2 \leq 2.8786861742659004 \cdot 10^{-140}:\\ \;\;\;\;a1 \cdot \left(\cos th \cdot \left(a1 \cdot \sqrt{0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot \cos th\right) \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \]
Alternative 8
Error21.1
Cost13380
\[\begin{array}{l} \mathbf{if}\;a2 \leq 2.8786861742659004 \cdot 10^{-140}:\\ \;\;\;\;a1 \cdot \left(\cos th \cdot \left(a1 \cdot \sqrt{0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \frac{a2 \cdot a2}{\sqrt{2}}\\ \end{array} \]
Alternative 9
Error21.1
Cost13380
\[\begin{array}{l} \mathbf{if}\;a2 \leq 2.8786861742659004 \cdot 10^{-140}:\\ \;\;\;\;a1 \cdot \left(\cos th \cdot \left(a1 \cdot \sqrt{0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot \left(a2 \cdot \cos th\right)}{\sqrt{2}}\\ \end{array} \]
Alternative 10
Error25.7
Cost13248
\[\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \sqrt{0.5} \]
Alternative 11
Error25.8
Cost7104
\[\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \frac{1}{\sqrt{2}} \]
Alternative 12
Error36.6
Cost6980
\[\begin{array}{l} \mathbf{if}\;a2 \leq 3.162261452390262 \cdot 10^{-78}:\\ \;\;\;\;\frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \sqrt{\frac{a2 \cdot a2}{2}}\\ \end{array} \]
Alternative 13
Error36.6
Cost6852
\[\begin{array}{l} \mathbf{if}\;a2 \leq 3.162261452390262 \cdot 10^{-78}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]
Alternative 14
Error36.6
Cost6852
\[\begin{array}{l} \mathbf{if}\;a2 \leq 3.162261452390262 \cdot 10^{-78}:\\ \;\;\;\;\frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]
Alternative 15
Error36.6
Cost6852
\[\begin{array}{l} \mathbf{if}\;a2 \leq 3.162261452390262 \cdot 10^{-78}:\\ \;\;\;\;\frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \end{array} \]
Alternative 16
Error40.6
Cost6720
\[a1 \cdot \frac{a1}{\sqrt{2}} \]

Error

Reproduce

herbie shell --seed 2022300 
(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))))