Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 3.8s

Alternatives: 7

Speedup: 1.9×

• 44.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ t\_1 \cdot \left(a1 \cdot a1\right) + t\_1 \cdot \left(a2 \cdot a2\right) \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (+ (* t_1 (* a1 a1)) (* t_1 (* a2 a2)))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a1, a2, th)
use fmin_fmax_functions
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: t_1
    t_1 = cos(th) / sqrt(2.0d0)
    code = (t_1 * (a1 * a1)) + (t_1 * (a2 * a2))
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = Math.cos(th) / Math.sqrt(2.0);
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
}

Python

def code(a1, a2, th):
	t_1 = math.cos(th) / math.sqrt(2.0)
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2))

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	return Float64(Float64(t_1 * Float64(a1 * a1)) + Float64(t_1 * Float64(a2 * a2)))
end

MATLAB

function tmp = code(a1, a2, th)
	t_1 = cos(th) / sqrt(2.0);
	tmp = (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$1 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Initial Program: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ t\_1 \cdot \left(a1 \cdot a1\right) + t\_1 \cdot \left(a2 \cdot a2\right) \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (+ (* t_1 (* a1 a1)) (* t_1 (* a2 a2)))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a1, a2, th)
use fmin_fmax_functions
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: t_1
    t_1 = cos(th) / sqrt(2.0d0)
    code = (t_1 * (a1 * a1)) + (t_1 * (a2 * a2))
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = Math.cos(th) / Math.sqrt(2.0);
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
}

Python

def code(a1, a2, th):
	t_1 = math.cos(th) / math.sqrt(2.0)
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2))

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	return Float64(Float64(t_1 * Float64(a1 * a1)) + Float64(t_1 * Float64(a2 * a2)))
end

MATLAB

function tmp = code(a1, a2, th)
	t_1 = cos(th) / sqrt(2.0);
	tmp = (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$1 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ [a1, a2_m, th] = \mathsf{sort}([a1, a2_m, th])\\ \\ \frac{\mathsf{fma}\left(\left(\cos th \cdot a2\_m\right) \cdot a2\_m, \sqrt{2}, \left(\left(\sqrt{2} \cdot a1\right) \cdot a1\right) \cdot \cos th\right)}{2} \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2_m th)
 :precision binary64
 (/
  (fma
   (* (* (cos th) a2_m) a2_m)
   (sqrt 2.0)
   (* (* (* (sqrt 2.0) a1) a1) (cos th)))
  2.0))

C

a2_m = fabs(a2);
assert(a1 < a2_m && a2_m < th);
double code(double a1, double a2_m, double th) {
	return fma(((cos(th) * a2_m) * a2_m), sqrt(2.0), (((sqrt(2.0) * a1) * a1) * cos(th))) / 2.0;
}

Julia

a2_m = abs(a2)
a1, a2_m, th = sort([a1, a2_m, th])
function code(a1, a2_m, th)
	return Float64(fma(Float64(Float64(cos(th) * a2_m) * a2_m), sqrt(2.0), Float64(Float64(Float64(sqrt(2.0) * a1) * a1) * cos(th))) / 2.0)
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := N[(N[(N[(N[(N[Cos[th], $MachinePrecision] * a2$95$m), $MachinePrecision] * a2$95$m), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * a1), $MachinePrecision] * a1), $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. pow2 N/A

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

   14. *-commutative N/A

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

   15. associate-/l* N/A

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

   16. *-commutative N/A

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

   17. associate-/l* N/A

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

4. Applied rewrites 99.6%

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

Alternative 2: 76.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ [a1, a2_m, th] = \mathsf{sort}([a1, a2_m, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1 \cdot a1\right) + t\_1 \cdot \left(a2\_m \cdot a2\_m\right) \leq -2 \cdot 10^{-150}:\\ \;\;\;\;\left(0.5 \cdot \left(a2\_m \cdot a2\_m\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.5, th \cdot th, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \mathsf{fma}\left(a1, a1, a2\_m \cdot a2\_m\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2_m th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a1 a1)) (* t_1 (* a2_m a2_m))) -2e-150)
     (* (* 0.5 (* a2_m a2_m)) (* (sqrt 2.0) (fma -0.5 (* th th) 1.0)))
     (* (* (sqrt 2.0) (fma a1 a1 (* a2_m a2_m))) 0.5))))

C

a2_m = fabs(a2);
assert(a1 < a2_m && a2_m < th);
double code(double a1, double a2_m, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_1 * (a1 * a1)) + (t_1 * (a2_m * a2_m))) <= -2e-150) {
		tmp = (0.5 * (a2_m * a2_m)) * (sqrt(2.0) * fma(-0.5, (th * th), 1.0));
	} else {
		tmp = (sqrt(2.0) * fma(a1, a1, (a2_m * a2_m))) * 0.5;
	}
	return tmp;
}

Julia

a2_m = abs(a2)
a1, a2_m, th = sort([a1, a2_m, th])
function code(a1, a2_m, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(t_1 * Float64(a1 * a1)) + Float64(t_1 * Float64(a2_m * a2_m))) <= -2e-150)
		tmp = Float64(Float64(0.5 * Float64(a2_m * a2_m)) * Float64(sqrt(2.0) * fma(-0.5, Float64(th * th), 1.0)));
	else
		tmp = Float64(Float64(sqrt(2.0) * fma(a1, a1, Float64(a2_m * a2_m))) * 0.5);
	end
	return tmp
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-150], N[(N[(0.5 * N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.5 * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(a1 * a1 + N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < -2.00000000000000001e-150

   1. Initial program 99.6%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in a1 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a2}^{2} \cdot \left(\cos th \cdot \sqrt{2}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-cos.f64 76.6

          \[\leadsto \left(0.5 \cdot \left(a2 \cdot a2\right)\right) \cdot \left(\sqrt{2} \cdot \cos th\right) \]

   7. Applied rewrites 76.6%

      \[\leadsto \color{blue}{\left(0.5 \cdot \left(a2 \cdot a2\right)\right) \cdot \left(\sqrt{2} \cdot \cos th\right)} \]

   8. Taylor expanded in th around 0

      \[\leadsto \left(\frac{1}{2} \cdot \left(a2 \cdot a2\right)\right) \cdot \left(\sqrt{2} \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {th}^{2}}\right)\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(a2 \cdot a2\right)\right) \cdot \left(\sqrt{2} \cdot \left(\frac{-1}{2} \cdot {th}^{2} + 1\right)\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(a2 \cdot a2\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{2}, {th}^{\color{blue}{2}}, 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(a2 \cdot a2\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{2}, th \cdot th, 1\right)\right) \]

      4. lower-*.f64 49.1

         \[\leadsto \left(0.5 \cdot \left(a2 \cdot a2\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.5, th \cdot th, 1\right)\right) \]

   10. Applied rewrites 49.1%

       \[\leadsto \left(0.5 \cdot \left(a2 \cdot a2\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.5, \color{blue}{th \cdot th}, 1\right)\right) \]

   if -2.00000000000000001e-150 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.6%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a1}^{2} \cdot \sqrt{2} + {a2}^{2} \cdot \sqrt{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lift-fma.f64 N/A

         \[\leadsto \left(\sqrt{2} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right) \cdot \frac{1}{2} \]

      9. lift-*.f64 84.1

         \[\leadsto \left(\sqrt{2} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right) \cdot 0.5 \]

   7. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right) \cdot 0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ [a1, a2_m, th] = \mathsf{sort}([a1, a2_m, th])\\ \\ \left(\left(\mathsf{fma}\left(a1, a1, a2\_m \cdot a2\_m\right) \cdot \sqrt{2}\right) \cdot \cos th\right) \cdot 0.5 \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2_m th)
 :precision binary64
 (* (* (* (fma a1 a1 (* a2_m a2_m)) (sqrt 2.0)) (cos th)) 0.5))

C

a2_m = fabs(a2);
assert(a1 < a2_m && a2_m < th);
double code(double a1, double a2_m, double th) {
	return ((fma(a1, a1, (a2_m * a2_m)) * sqrt(2.0)) * cos(th)) * 0.5;
}

Julia

a2_m = abs(a2)
a1, a2_m, th = sort([a1, a2_m, th])
function code(a1, a2_m, th)
	return Float64(Float64(Float64(fma(a1, a1, Float64(a2_m * a2_m)) * sqrt(2.0)) * cos(th)) * 0.5)
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := N[(N[(N[(N[(a1 * a1 + N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. pow2 N/A

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

   14. *-commutative N/A

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

   15. associate-/l* N/A

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

   16. *-commutative N/A

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

   17. associate-/l* N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in a1 around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a1}^{2} \cdot \left(\cos th \cdot \sqrt{2}\right)\right) + \frac{1}{2} \cdot \left({a2}^{2} \cdot \left(\cos th \cdot \sqrt{2}\right)\right)} \]
6. Step-by-step derivation

   1. distribute-lft-in N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. distribute-rgt-out N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. pow2 N/A

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

   11. pow2 N/A

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

   12. lift-fma.f64 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \cos th\right) \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right) \cdot \frac{1}{2} \]

   13. lift-*.f64 99.6

       \[\leadsto \left(\left(\sqrt{2} \cdot \cos th\right) \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right) \cdot 0.5 \]

7. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \cos th\right) \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right) \cdot 0.5} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\left(\sqrt{2} \cdot \cos th\right) \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right) \cdot \frac{1}{2} \]

   2. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(\sqrt{2} \cdot \cos th\right) \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right) \cdot \frac{1}{2} \]

   3. lift-cos.f64 N/A

      \[\leadsto \left(\left(\sqrt{2} \cdot \cos th\right) \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right) \cdot \frac{1}{2} \]

   4. *-commutative N/A

      \[\leadsto \left(\left(\cos th \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right) \cdot \frac{1}{2} \]

   5. lift-*.f64 N/A

      \[\leadsto \left(\left(\cos th \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right) \cdot \frac{1}{2} \]

   6. lift-fma.f64 N/A

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

   7. pow2 N/A

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

   8. pow2 N/A

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

   9. lower-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. pow2 N/A

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

   16. pow2 N/A

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

   17. lift-fma.f64 N/A

       \[\leadsto \left(\left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \sqrt{2}\right) \cdot \cos th\right) \cdot \frac{1}{2} \]

   18. lift-*.f64 N/A

       \[\leadsto \left(\left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \sqrt{2}\right) \cdot \cos th\right) \cdot \frac{1}{2} \]

   19. lift-sqrt.f64 N/A

       \[\leadsto \left(\left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \sqrt{2}\right) \cdot \cos th\right) \cdot \frac{1}{2} \]

   20. lift-cos.f64 99.6

       \[\leadsto \left(\left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \sqrt{2}\right) \cdot \cos th\right) \cdot 0.5 \]

9. Applied rewrites 99.6%

   \[\leadsto \left(\left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \sqrt{2}\right) \cdot \cos th\right) \cdot 0.5 \]
10. Add Preprocessing

Alternative 4: 99.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ [a1, a2_m, th] = \mathsf{sort}([a1, a2_m, th])\\ \\ \left(\left(\sqrt{2} \cdot \cos th\right) \cdot \mathsf{fma}\left(a1, a1, a2\_m \cdot a2\_m\right)\right) \cdot 0.5 \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2_m th)
 :precision binary64
 (* (* (* (sqrt 2.0) (cos th)) (fma a1 a1 (* a2_m a2_m))) 0.5))

C

a2_m = fabs(a2);
assert(a1 < a2_m && a2_m < th);
double code(double a1, double a2_m, double th) {
	return ((sqrt(2.0) * cos(th)) * fma(a1, a1, (a2_m * a2_m))) * 0.5;
}

Julia

a2_m = abs(a2)
a1, a2_m, th = sort([a1, a2_m, th])
function code(a1, a2_m, th)
	return Float64(Float64(Float64(sqrt(2.0) * cos(th)) * fma(a1, a1, Float64(a2_m * a2_m))) * 0.5)
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision] * N[(a1 * a1 + N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. pow2 N/A

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

   14. *-commutative N/A

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

   15. associate-/l* N/A

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

   16. *-commutative N/A

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

   17. associate-/l* N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in a1 around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a1}^{2} \cdot \left(\cos th \cdot \sqrt{2}\right)\right) + \frac{1}{2} \cdot \left({a2}^{2} \cdot \left(\cos th \cdot \sqrt{2}\right)\right)} \]
6. Step-by-step derivation

   1. distribute-lft-in N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. distribute-rgt-out N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. pow2 N/A

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

   11. pow2 N/A

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

   12. lift-fma.f64 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \cos th\right) \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right) \cdot \frac{1}{2} \]

   13. lift-*.f64 99.6

       \[\leadsto \left(\left(\sqrt{2} \cdot \cos th\right) \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right) \cdot 0.5 \]

7. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \cos th\right) \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right) \cdot 0.5} \]
8. Add Preprocessing

Alternative 5: 78.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ [a1, a2_m, th] = \mathsf{sort}([a1, a2_m, th])\\ \\ \left(\left(\left(a2\_m \cdot \cos th\right) \cdot a2\_m\right) \cdot \sqrt{2}\right) \cdot 0.5 \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2_m th)
 :precision binary64
 (* (* (* (* a2_m (cos th)) a2_m) (sqrt 2.0)) 0.5))

C

a2_m = fabs(a2);
assert(a1 < a2_m && a2_m < th);
double code(double a1, double a2_m, double th) {
	return (((a2_m * cos(th)) * a2_m) * sqrt(2.0)) * 0.5;
}

Fortran

a2_m =     private
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a1, a2_m, th)
use fmin_fmax_functions
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    code = (((a2_m * cos(th)) * a2_m) * sqrt(2.0d0)) * 0.5d0
end function

Java

a2_m = Math.abs(a2);
assert a1 < a2_m && a2_m < th;
public static double code(double a1, double a2_m, double th) {
	return (((a2_m * Math.cos(th)) * a2_m) * Math.sqrt(2.0)) * 0.5;
}

Python

a2_m = math.fabs(a2)
[a1, a2_m, th] = sort([a1, a2_m, th])
def code(a1, a2_m, th):
	return (((a2_m * math.cos(th)) * a2_m) * math.sqrt(2.0)) * 0.5

Julia

a2_m = abs(a2)
a1, a2_m, th = sort([a1, a2_m, th])
function code(a1, a2_m, th)
	return Float64(Float64(Float64(Float64(a2_m * cos(th)) * a2_m) * sqrt(2.0)) * 0.5)
end

MATLAB

a2_m = abs(a2);
a1, a2_m, th = num2cell(sort([a1, a2_m, th])){:}
function tmp = code(a1, a2_m, th)
	tmp = (((a2_m * cos(th)) * a2_m) * sqrt(2.0)) * 0.5;
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := N[(N[(N[(N[(a2$95$m * N[Cos[th], $MachinePrecision]), $MachinePrecision] * a2$95$m), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. pow2 N/A

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

   14. *-commutative N/A

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

   15. associate-/l* N/A

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

   16. *-commutative N/A

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

   17. associate-/l* N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in a1 around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a1}^{2} \cdot \left(\cos th \cdot \sqrt{2}\right)\right) + \frac{1}{2} \cdot \left({a2}^{2} \cdot \left(\cos th \cdot \sqrt{2}\right)\right)} \]
6. Step-by-step derivation

   1. distribute-lft-in N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. distribute-rgt-out N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. pow2 N/A

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

   11. pow2 N/A

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

   12. lift-fma.f64 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \cos th\right) \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right) \cdot \frac{1}{2} \]

   13. lift-*.f64 99.6

       \[\leadsto \left(\left(\sqrt{2} \cdot \cos th\right) \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right) \cdot 0.5 \]

7. Applied rewrites 99.6%

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

8. Taylor expanded in a1 around 0

   \[\leadsto \left({a2}^{2} \cdot \left(\cos th \cdot \sqrt{2}\right)\right) \cdot \frac{1}{2} \]
9. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\left({a2}^{2} \cdot \cos th\right) \cdot \sqrt{2}\right) \cdot \frac{1}{2} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\left({a2}^{2} \cdot \cos th\right) \cdot \sqrt{2}\right) \cdot \frac{1}{2} \]

   3. *-commutative N/A

      \[\leadsto \left(\left(\cos th \cdot {a2}^{2}\right) \cdot \sqrt{2}\right) \cdot \frac{1}{2} \]

   4. pow2 N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lift-sqrt.f64 78.0

       \[\leadsto \left(\left(\left(a2 \cdot \cos th\right) \cdot a2\right) \cdot \sqrt{2}\right) \cdot 0.5 \]

10. Applied rewrites 78.0%

    \[\leadsto \left(\left(\left(a2 \cdot \cos th\right) \cdot a2\right) \cdot \sqrt{2}\right) \cdot 0.5 \]
11. Add Preprocessing

Alternative 6: 65.8% accurate, 8.3× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ [a1, a2_m, th] = \mathsf{sort}([a1, a2_m, th])\\ \\ \left(\sqrt{2} \cdot \mathsf{fma}\left(a1, a1, a2\_m \cdot a2\_m\right)\right) \cdot 0.5 \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2_m th)
 :precision binary64
 (* (* (sqrt 2.0) (fma a1 a1 (* a2_m a2_m))) 0.5))

C

a2_m = fabs(a2);
assert(a1 < a2_m && a2_m < th);
double code(double a1, double a2_m, double th) {
	return (sqrt(2.0) * fma(a1, a1, (a2_m * a2_m))) * 0.5;
}

Julia

a2_m = abs(a2)
a1, a2_m, th = sort([a1, a2_m, th])
function code(a1, a2_m, th)
	return Float64(Float64(sqrt(2.0) * fma(a1, a1, Float64(a2_m * a2_m))) * 0.5)
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(a1 * a1 + N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. pow2 N/A

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

   14. *-commutative N/A

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

   15. associate-/l* N/A

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

   16. *-commutative N/A

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

   17. associate-/l* N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in th around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a1}^{2} \cdot \sqrt{2} + {a2}^{2} \cdot \sqrt{2}\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. distribute-rgt-out N/A

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

   4. lower-*.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. pow2 N/A

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

   7. pow2 N/A

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

   8. lift-fma.f64 N/A

      \[\leadsto \left(\sqrt{2} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right) \cdot \frac{1}{2} \]

   9. lift-*.f64 65.8

      \[\leadsto \left(\sqrt{2} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right) \cdot 0.5 \]

7. Applied rewrites 65.8%

   \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right) \cdot 0.5} \]
8. Add Preprocessing

Alternative 7: 52.4% accurate, 10.2× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ [a1, a2_m, th] = \mathsf{sort}([a1, a2_m, th])\\ \\ \left(0.5 \cdot \left(a2\_m \cdot a2\_m\right)\right) \cdot \sqrt{2} \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2_m th) :precision binary64 (* (* 0.5 (* a2_m a2_m)) (sqrt 2.0)))

C

a2_m = fabs(a2);
assert(a1 < a2_m && a2_m < th);
double code(double a1, double a2_m, double th) {
	return (0.5 * (a2_m * a2_m)) * sqrt(2.0);
}

Fortran

a2_m =     private
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a1, a2_m, th)
use fmin_fmax_functions
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    code = (0.5d0 * (a2_m * a2_m)) * sqrt(2.0d0)
end function

Java

a2_m = Math.abs(a2);
assert a1 < a2_m && a2_m < th;
public static double code(double a1, double a2_m, double th) {
	return (0.5 * (a2_m * a2_m)) * Math.sqrt(2.0);
}

Python

a2_m = math.fabs(a2)
[a1, a2_m, th] = sort([a1, a2_m, th])
def code(a1, a2_m, th):
	return (0.5 * (a2_m * a2_m)) * math.sqrt(2.0)

Julia

a2_m = abs(a2)
a1, a2_m, th = sort([a1, a2_m, th])
function code(a1, a2_m, th)
	return Float64(Float64(0.5 * Float64(a2_m * a2_m)) * sqrt(2.0))
end

MATLAB

a2_m = abs(a2);
a1, a2_m, th = num2cell(sort([a1, a2_m, th])){:}
function tmp = code(a1, a2_m, th)
	tmp = (0.5 * (a2_m * a2_m)) * sqrt(2.0);
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := N[(N[(0.5 * N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. pow2 N/A

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

   14. *-commutative N/A

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

   15. associate-/l* N/A

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

   16. *-commutative N/A

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

   17. associate-/l* N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in a1 around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a2}^{2} \cdot \left(\cos th \cdot \sqrt{2}\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

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

   2. metadata-eval N/A

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

   3. lower-*.f64 N/A

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

   4. metadata-eval N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. lift-cos.f64 78.0

       \[\leadsto \left(0.5 \cdot \left(a2 \cdot a2\right)\right) \cdot \left(\sqrt{2} \cdot \cos th\right) \]

7. Applied rewrites 78.0%

   \[\leadsto \color{blue}{\left(0.5 \cdot \left(a2 \cdot a2\right)\right) \cdot \left(\sqrt{2} \cdot \cos th\right)} \]

8. Taylor expanded in th around 0

   \[\leadsto \left(\frac{1}{2} \cdot \left(a2 \cdot a2\right)\right) \cdot \sqrt{2} \]
9. Step-by-step derivation

   1. lift-sqrt.f64 52.4

      \[\leadsto \left(0.5 \cdot \left(a2 \cdot a2\right)\right) \cdot \sqrt{2} \]

10. Applied rewrites 52.4%

    \[\leadsto \left(0.5 \cdot \left(a2 \cdot a2\right)\right) \cdot \sqrt{2} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025091 
(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))))