Average Error: 0.5 → 0.5
Time: 6.5s
Precision: binary64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
↓
\[\left({\left(\mathsf{hypot}\left(a2, a1\right)\right)}^{2} \cdot \cos th\right) \cdot {2}^{-0.5} \]
(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) (cos th)) (pow 2.0 -0.5)))
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) * cos(th)) * pow(2.0, -0.5);
}

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.cos(th)) * Math.pow(2.0, -0.5);
}

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.cos(th)) * math.pow(2.0, -0.5)

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((hypot(a2, a1) ^ 2.0) * cos(th)) * (2.0 ^ -0.5))
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) * cos(th)) * (2.0 ^ -0.5);
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[N[Sqrt[a2 ^ 2 + a1 ^ 2], $MachinePrecision], 2.0], $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision] * N[Power[2.0, -0.5], $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({\left(\mathsf{hypot}\left(a2, a1\right)\right)}^{2} \cdot \cos th\right) \cdot {2}^{-0.5}

Error

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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

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

  3. Taylor expanded in th around inf 0.5

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

  4. Applied egg-rr0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2022206 
(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))))