Migdal et al, Equation (64)

Average Error: 0.6 → 0.5

Time: 12.4s

Precision: binary64

Cost: 26624

\[ \begin{array}{c}[a1, a2] = \mathsf{sort}([a1, a2])\\ \end{array} \]

Math

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

↓

\[\frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) - a1 \cdot \left(-\frac{a1 \cdot \cos th}{\sqrt{2}}\right) \]

FPCore

(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) (sqrt 2.0)) (* a2 a2))
  (* a1 (- (/ (* a1 (cos th)) (sqrt 2.0))))))

C

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) / sqrt(2.0)) * (a2 * a2)) - (a1 * -((a1 * cos(th)) / sqrt(2.0)));
}

Fortran

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 = ((cos(th) / sqrt(2.0d0)) * (a2 * a2)) - (a1 * -((a1 * cos(th)) / sqrt(2.0d0)))
end function

Java

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.sqrt(2.0)) * (a2 * a2)) - (a1 * -((a1 * Math.cos(th)) / Math.sqrt(2.0)));
}

Python

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.sqrt(2.0)) * (a2 * a2)) - (a1 * -((a1 * math.cos(th)) / math.sqrt(2.0)))

Julia

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(Float64(cos(th) / sqrt(2.0)) * Float64(a2 * a2)) - Float64(a1 * Float64(-Float64(Float64(a1 * cos(th)) / sqrt(2.0)))))
end

MATLAB

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) / sqrt(2.0)) * (a2 * a2)) - (a1 * -((a1 * cos(th)) / sqrt(2.0)));
end

Wolfram

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[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(a2 * a2), $MachinePrecision]), $MachinePrecision] - N[(a1 * (-N[(N[(a1 * N[Cos[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.6

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

2. Applied egg-rr 0.6

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

3. Simplified 0.5

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

   [Start] 0.6	\[ \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) - \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot \left(-a1\right)\right) \]
   rational_best_oopsla_all_46_json_45_simplify-7 [=>] 0.5	\[ \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) - \color{blue}{a1 \cdot \left(\frac{\cos th}{\sqrt{2}} \cdot \left(-a1\right)\right)} \]

4. Taylor expanded in th around inf 0.5

   \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) - a1 \cdot \color{blue}{\left(-1 \cdot \frac{a1 \cdot \cos th}{\sqrt{2}}\right)} \]

5. Simplified 0.5

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

   [Start] 0.5	\[ \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) - a1 \cdot \left(-1 \cdot \frac{a1 \cdot \cos th}{\sqrt{2}}\right) \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.5	\[ \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) - a1 \cdot \color{blue}{\left(\frac{a1 \cdot \cos th}{\sqrt{2}} \cdot -1\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-92 [=>] 0.5	\[ \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) - a1 \cdot \color{blue}{\left(-\frac{a1 \cdot \cos th}{\sqrt{2}}\right)} \]

6. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.5
Cost	26176

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

Alternative 2
Error	0.6
Cost	13504

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

Alternative 3
Error	25.8
Cost	7104

\[\frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

Reproduce

herbie shell --seed 2023090 
(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))))