Average Error: 0.5 → 0.7
Time: 19.1s
Precision: binary64
Cost: 26048
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
↓
\[\frac{\cos th}{{\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{-2} \cdot \sqrt{2}} \]
(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) (* (pow (hypot a1 a2) -2.0) (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) / (pow(hypot(a1, a2), -2.0) * sqrt(2.0));
}

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.pow(Math.hypot(a1, a2), -2.0) * 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) / (math.pow(math.hypot(a1, a2), -2.0) * 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((hypot(a1, a2) ^ -2.0) * 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) / ((hypot(a1, a2) ^ -2.0) * 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[Power[N[Sqrt[a1 ^ 2 + a2 ^ 2], $MachinePrecision], -2.0], $MachinePrecision] * N[Sqrt[2.0], $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}{{\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{-2} \cdot \sqrt{2}}

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

    \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)} \]
    Proof
    (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 2)) (fma.f64 a1 a1 (*.f64 a2 a2))): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 2)) (Rewrite<= fma-def_binary64 (+.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-rr0.7

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

  4. Applied egg-rr0.7

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

  5. Final simplification0.7

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

Alternatives

Alternative 1
Error14.0
Cost19780
\[\begin{array}{l} \mathbf{if}\;\cos th \leq 0.996:\\ \;\;\;\;\cos th \cdot \left(\sqrt{0.5} \cdot \left(a2 \cdot a2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right)\\ \end{array} \]
Alternative 2
Error0.5
Cost13504
\[\left(\cos th \cdot \sqrt{0.5}\right) \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]
Alternative 3
Error20.9
Cost13444
\[\begin{array}{l} \mathbf{if}\;a1 \leq -2.7561877992818123 \cdot 10^{-146}:\\ \;\;\;\;{2}^{-0.5} \cdot \left(a1 \cdot \left(\cos th \cdot a1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos th \cdot a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]
Alternative 4
Error20.9
Cost13380
\[\begin{array}{l} \mathbf{if}\;a1 \leq -2.7561877992818123 \cdot 10^{-146}:\\ \;\;\;\;\cos th \cdot \left(\sqrt{0.5} \cdot \left(a1 \cdot a1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \left(\sqrt{0.5} \cdot \left(a2 \cdot a2\right)\right)\\ \end{array} \]
Alternative 5
Error20.9
Cost13380
\[\begin{array}{l} \mathbf{if}\;a1 \leq -2.7561877992818123 \cdot 10^{-146}:\\ \;\;\;\;\cos th \cdot \left(\sqrt{0.5} \cdot \left(a1 \cdot a1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos th \cdot a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]
Alternative 6
Error20.9
Cost13380
\[\begin{array}{l} \mathbf{if}\;a1 \leq -2.7561877992818123 \cdot 10^{-146}:\\ \;\;\;\;\frac{a1 \cdot \left(\cos th \cdot a1\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos th \cdot a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]
Alternative 7
Error20.9
Cost13380
\[\begin{array}{l} \mathbf{if}\;a1 \leq -2.7561877992818123 \cdot 10^{-146}:\\ \;\;\;\;a1 \cdot \frac{\cos th}{\frac{\sqrt{2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos th \cdot a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]
Alternative 8
Error36.9
Cost7244
\[\begin{array}{l} t_1 := \sqrt{0.5} \cdot \left(a1 \cdot a1\right)\\ t_2 := a2 \cdot \sqrt{\frac{a2 \cdot a2}{2}}\\ \mathbf{if}\;a2 \leq 1.9566498730172507 \cdot 10^{-159}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a2 \leq 7.783531299346684 \cdot 10^{-70}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a2 \leq 1.4614409330228602 \cdot 10^{-45}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 9
Error37.0
Cost7116
\[\begin{array}{l} t_1 := \sqrt{0.5} \cdot \left(a1 \cdot a1\right)\\ \mathbf{if}\;a2 \leq 1.9566498730172507 \cdot 10^{-159}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a2 \leq 7.783531299346684 \cdot 10^{-70}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \mathbf{elif}\;a2 \leq 1.4614409330228602 \cdot 10^{-45}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \end{array} \]
Alternative 10
Error37.0
Cost7116
\[\begin{array}{l} t_1 := \sqrt{0.5} \cdot \left(a1 \cdot a1\right)\\ \mathbf{if}\;a2 \leq 1.9566498730172507 \cdot 10^{-159}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a2 \leq 7.783531299346684 \cdot 10^{-70}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 1.4614409330228602 \cdot 10^{-45}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \end{array} \]
Alternative 11
Error25.5
Cost6976
\[\sqrt{0.5} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]
Alternative 12
Error40.2
Cost6720
\[\frac{a2}{\frac{\sqrt{2}}{a2}} \]
Alternative 13
Error40.2
Cost6720
\[\frac{a2 \cdot a2}{\sqrt{2}} \]
Alternative 14
Error55.4
Cost192
\[a1 \cdot 0 \]

Error

Reproduce

herbie shell --seed 2022298 
(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))))