Average Error: 0.5 → 0.4
Time: 15.3s
Precision: binary64
Cost: 26176
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
↓
\[\frac{\mathsf{hypot}\left(a1, a2\right)}{\frac{\frac{\sqrt{2}}{\cos th}}{\mathsf{hypot}\left(a1, a2\right)}} \]
(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
 (/ (hypot a1 a2) (/ (/ (sqrt 2.0) (cos th)) (hypot a1 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 hypot(a1, a2) / ((sqrt(2.0) / cos(th)) / hypot(a1, a2));
}

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.hypot(a1, a2) / ((Math.sqrt(2.0) / Math.cos(th)) / Math.hypot(a1, 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.hypot(a1, a2) / ((math.sqrt(2.0) / math.cos(th)) / math.hypot(a1, 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(hypot(a1, a2) / Float64(Float64(sqrt(2.0) / cos(th)) / hypot(a1, 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 = hypot(a1, a2) / ((sqrt(2.0) / cos(th)) / hypot(a1, 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[Sqrt[a1 ^ 2 + a2 ^ 2], $MachinePrecision] / N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[Cos[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[a1 ^ 2 + a2 ^ 2], $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{\mathsf{hypot}\left(a1, a2\right)}{\frac{\frac{\sqrt{2}}{\cos th}}{\mathsf{hypot}\left(a1, 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. Applied egg-rr0.4

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

  4. Final simplification0.4

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

Alternatives

Alternative 1
Error0.5
Cost19776
\[\cos th \cdot \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}} \]
Alternative 2
Error0.5
Cost19776
\[\frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\frac{\sqrt{2}}{\cos th}} \]
Alternative 3
Error14.4
Cost13513
\[\begin{array}{l} \mathbf{if}\;th \leq -490000000 \lor \neg \left(th \leq 21\right):\\ \;\;\;\;\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \left(\sqrt{0.5} \cdot \left(-0.5 \cdot \left(th \cdot th\right) + 1\right)\right)\\ \end{array} \]
Alternative 4
Error14.4
Cost13512
\[\begin{array}{l} \mathbf{if}\;th \leq -490000000:\\ \;\;\;\;\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;th \leq 21:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \left(\sqrt{0.5} \cdot \left(-0.5 \cdot \left(th \cdot th\right) + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]
Alternative 5
Error0.5
Cost13504
\[\sqrt{0.5} \cdot \left(\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right) \]
Alternative 6
Error0.5
Cost13504
\[\left(\cos th \cdot \sqrt{0.5}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
Alternative 7
Error0.5
Cost13504
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
Alternative 8
Error20.6
Cost13380
\[\begin{array}{l} \mathbf{if}\;a2 \leq 3.2 \cdot 10^{-143}:\\ \;\;\;\;\cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\right)\\ \end{array} \]
Alternative 9
Error20.6
Cost13380
\[\begin{array}{l} \mathbf{if}\;a2 \leq 3 \cdot 10^{-143}:\\ \;\;\;\;\cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\sqrt{2}} \cdot \left(a2 \cdot \cos th\right)\\ \end{array} \]
Alternative 10
Error20.6
Cost13380
\[\begin{array}{l} \mathbf{if}\;a2 \leq 3.2 \cdot 10^{-143}:\\ \;\;\;\;\frac{a1 \cdot a1}{\frac{\sqrt{2}}{\cos th}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\sqrt{2}} \cdot \left(a2 \cdot \cos th\right)\\ \end{array} \]
Alternative 11
Error25.5
Cost6976
\[\sqrt{0.5} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
Alternative 12
Error25.5
Cost6976
\[\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}} \]
Alternative 13
Error36.1
Cost6852
\[\begin{array}{l} \mathbf{if}\;a2 \leq 9 \cdot 10^{-146}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]
Alternative 14
Error36.1
Cost6852
\[\begin{array}{l} \mathbf{if}\;a2 \leq 1.25 \cdot 10^{-146}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]
Alternative 15
Error36.1
Cost6852
\[\begin{array}{l} \mathbf{if}\;a2 \leq 3.8 \cdot 10^{-145}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]
Alternative 16
Error36.1
Cost6852
\[\begin{array}{l} \mathbf{if}\;a2 \leq 4 \cdot 10^{-147}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]
Alternative 17
Error36.0
Cost6852
\[\begin{array}{l} \mathbf{if}\;a2 \leq 3.9 \cdot 10^{-145}:\\ \;\;\;\;\frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]
Alternative 18
Error36.0
Cost6852
\[\begin{array}{l} \mathbf{if}\;a2 \leq 3.9 \cdot 10^{-145}:\\ \;\;\;\;\frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \end{array} \]
Alternative 19
Error40.0
Cost6720
\[\left(a1 \cdot a1\right) \cdot \sqrt{0.5} \]

Error

Reproduce

herbie shell --seed 2023012 
(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))))