Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 8.9s

Alternatives: 14

Speedup: 1.9×

• 44.1% 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: 99.5% 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

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \frac{\mathsf{fma}\left(\left(a2 \cdot \cos th\right) \cdot \left(-a2\right), -\sqrt{2}, \sqrt{2} \cdot \left(\left(a1\_m \cdot \cos th\right) \cdot a1\_m\right)\right)}{2} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th)
 :precision binary64
 (/
  (fma
   (* (* a2 (cos th)) (- a2))
   (- (sqrt 2.0))
   (* (sqrt 2.0) (* (* a1_m (cos th)) a1_m)))
  2.0))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	return fma(((a2 * cos(th)) * -a2), -sqrt(2.0), (sqrt(2.0) * ((a1_m * cos(th)) * a1_m))) / 2.0;
}

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	return Float64(fma(Float64(Float64(a2 * cos(th)) * Float64(-a2)), Float64(-sqrt(2.0)), Float64(sqrt(2.0) * Float64(Float64(a1_m * cos(th)) * a1_m))) / 2.0)
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := N[(N[(N[(N[(a2 * N[Cos[th], $MachinePrecision]), $MachinePrecision] * (-a2)), $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision]) + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(a1$95$m * N[Cos[th], $MachinePrecision]), $MachinePrecision] * a1$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 77.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1\_m \cdot a1\_m\right) + t\_1 \cdot \left(a2 \cdot a2\right) \leq -1 \cdot 10^{-73}:\\ \;\;\;\;\left(\left(-0.5 \cdot \left(a2 \cdot \frac{a2}{\sqrt{2}}\right)\right) \cdot th\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2 \cdot a2, \sqrt{2}, \sqrt{2} \cdot \left(a1\_m \cdot a1\_m\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a1_m a1_m)) (* t_1 (* a2 a2))) -1e-73)
     (* (* (* -0.5 (* a2 (/ a2 (sqrt 2.0)))) th) th)
     (/ (fma (* a2 a2) (sqrt 2.0) (* (sqrt 2.0) (* a1_m a1_m))) 2.0))))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_1 * (a1_m * a1_m)) + (t_1 * (a2 * a2))) <= -1e-73) {
		tmp = ((-0.5 * (a2 * (a2 / sqrt(2.0)))) * th) * th;
	} else {
		tmp = fma((a2 * a2), sqrt(2.0), (sqrt(2.0) * (a1_m * a1_m))) / 2.0;
	}
	return tmp;
}

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(t_1 * Float64(a1_m * a1_m)) + Float64(t_1 * Float64(a2 * a2))) <= -1e-73)
		tmp = Float64(Float64(Float64(-0.5 * Float64(a2 * Float64(a2 / sqrt(2.0)))) * th) * th);
	else
		tmp = Float64(fma(Float64(a2 * a2), sqrt(2.0), Float64(sqrt(2.0) * Float64(a1_m * a1_m))) / 2.0);
	end
	return tmp
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, 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$95$m * a1$95$m), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-73], N[(N[(N[(-0.5 * N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision] * th), $MachinePrecision], N[(N[(N[(a2 * a2), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + N[(N[Sqrt[2.0], $MachinePrecision] * N[(a1$95$m * a1$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $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))) < -9.99999999999999997e-74

   1. Initial program 99.7%

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

   3. Taylor expanded in a1 around 0

      \[\leadsto \color{blue}{\frac{{a2}^{2} \cdot \cos th}{\sqrt{2}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 59.4

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

   5. Applied rewrites 59.4%

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

   6. Taylor expanded in th around 0

      \[\leadsto \frac{-1}{2} \cdot \frac{{a2}^{2} \cdot {th}^{2}}{\sqrt{2}} + \color{blue}{\frac{{a2}^{2}}{\sqrt{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 9.7%

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

   9. Taylor expanded in th around inf

      \[\leadsto \frac{-1}{2} \cdot \frac{{a2}^{2} \cdot {th}^{2}}{\color{blue}{\sqrt{2}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 39.8%

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

   if -9.99999999999999997e-74 < (+.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 99.5%

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

   3. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\frac{{a1}^{2}}{\sqrt{2}} + \frac{{a2}^{2}}{\sqrt{2}}} \]
   4. Step-by-step derivation

      1. div-add-rev N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 81.4

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

   5. Applied rewrites 81.4%

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

   7. Applied rewrites 81.6%

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

Alternative 3: 77.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1\_m \cdot a1\_m\right) + t\_1 \cdot \left(a2 \cdot a2\right) \leq -1 \cdot 10^{-73}:\\ \;\;\;\;\left(\left(-0.5 \cdot \left(a2 \cdot \frac{a2}{\sqrt{2}}\right)\right) \cdot th\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \mathsf{fma}\left(a1\_m, a1\_m, a2 \cdot a2\right)\right) \cdot \sqrt{2}\\ \end{array} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a1_m a1_m)) (* t_1 (* a2 a2))) -1e-73)
     (* (* (* -0.5 (* a2 (/ a2 (sqrt 2.0)))) th) th)
     (* (* 0.5 (fma a1_m a1_m (* a2 a2))) (sqrt 2.0)))))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_1 * (a1_m * a1_m)) + (t_1 * (a2 * a2))) <= -1e-73) {
		tmp = ((-0.5 * (a2 * (a2 / sqrt(2.0)))) * th) * th;
	} else {
		tmp = (0.5 * fma(a1_m, a1_m, (a2 * a2))) * sqrt(2.0);
	}
	return tmp;
}

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(t_1 * Float64(a1_m * a1_m)) + Float64(t_1 * Float64(a2 * a2))) <= -1e-73)
		tmp = Float64(Float64(Float64(-0.5 * Float64(a2 * Float64(a2 / sqrt(2.0)))) * th) * th);
	else
		tmp = Float64(Float64(0.5 * fma(a1_m, a1_m, Float64(a2 * a2))) * sqrt(2.0));
	end
	return tmp
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, 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$95$m * a1$95$m), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-73], N[(N[(N[(-0.5 * N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision] * th), $MachinePrecision], N[(N[(0.5 * N[(a1$95$m * a1$95$m + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $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))) < -9.99999999999999997e-74

   1. Initial program 99.7%

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

   3. Taylor expanded in a1 around 0

      \[\leadsto \color{blue}{\frac{{a2}^{2} \cdot \cos th}{\sqrt{2}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 59.4

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

   5. Applied rewrites 59.4%

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

   6. Taylor expanded in th around 0

      \[\leadsto \frac{-1}{2} \cdot \frac{{a2}^{2} \cdot {th}^{2}}{\sqrt{2}} + \color{blue}{\frac{{a2}^{2}}{\sqrt{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 9.7%

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

   9. Taylor expanded in th around inf

      \[\leadsto \frac{-1}{2} \cdot \frac{{a2}^{2} \cdot {th}^{2}}{\color{blue}{\sqrt{2}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 39.8%

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

   if -9.99999999999999997e-74 < (+.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 99.5%

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

   4. Applied rewrites 99.7%

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

   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

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

      2. distribute-rgt-out N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in th around 0

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 81.6

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

   10. Applied rewrites 81.6%

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

Alternative 4: 99.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \left(\mathsf{fma}\left(a1\_m, a1\_m, a2 \cdot a2\right) \cdot \cos th\right) \cdot \left(\sqrt{2} \cdot 0.5\right) \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th)
 :precision binary64
 (* (* (fma a1_m a1_m (* a2 a2)) (cos th)) (* (sqrt 2.0) 0.5)))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	return (fma(a1_m, a1_m, (a2 * a2)) * cos(th)) * (sqrt(2.0) * 0.5);
}

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	return Float64(Float64(fma(a1_m, a1_m, Float64(a2 * a2)) * cos(th)) * Float64(sqrt(2.0) * 0.5))
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := N[(N[(N[(a1$95$m * a1$95$m + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 99.7%

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

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

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

   2. distribute-rgt-out N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 99.7

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

7. Applied rewrites 99.7%

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

9. Applied rewrites 99.7%

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

Alternative 5: 99.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \left(\left(0.5 \cdot \sqrt{2}\right) \cdot \cos th\right) \cdot \mathsf{fma}\left(a2, a2, a1\_m \cdot a1\_m\right) \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th)
 :precision binary64
 (* (* (* 0.5 (sqrt 2.0)) (cos th)) (fma a2 a2 (* a1_m a1_m))))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	return ((0.5 * sqrt(2.0)) * cos(th)) * fma(a2, a2, (a1_m * a1_m));
}

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	return Float64(Float64(Float64(0.5 * sqrt(2.0)) * cos(th)) * fma(a2, a2, Float64(a1_m * a1_m)))
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := N[(N[(N[(0.5 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision] * N[(a2 * a2 + N[(a1$95$m * a1$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 99.7%

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

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

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

   2. distribute-rgt-out N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 99.7

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

7. Applied rewrites 99.7%

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

9. Applied rewrites 99.7%

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

Alternative 6: 99.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \left(\left(0.5 \cdot \sqrt{2}\right) \cdot \cos th\right) \cdot \mathsf{fma}\left(a1\_m, a1\_m, a2 \cdot a2\right) \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th)
 :precision binary64
 (* (* (* 0.5 (sqrt 2.0)) (cos th)) (fma a1_m a1_m (* a2 a2))))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	return ((0.5 * sqrt(2.0)) * cos(th)) * fma(a1_m, a1_m, (a2 * a2));
}

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	return Float64(Float64(Float64(0.5 * sqrt(2.0)) * cos(th)) * fma(a1_m, a1_m, Float64(a2 * a2)))
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := N[(N[(N[(0.5 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision] * N[(a1$95$m * a1$95$m + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 99.7%

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

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

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

   2. distribute-rgt-out N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 99.7

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

7. Applied rewrites 99.7%

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

Alternative 7: 78.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \left(\cos th \cdot a2\right) \cdot \sqrt{\frac{a2 \cdot a2}{2}} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th)
 :precision binary64
 (* (* (cos th) a2) (sqrt (/ (* a2 a2) 2.0))))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	return (cos(th) * a2) * sqrt(((a2 * a2) / 2.0));
}

Fortran

a1_m =     private
NOTE: a1_m, a2, 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_m, a2, th)
use fmin_fmax_functions
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = (cos(th) * a2) * sqrt(((a2 * a2) / 2.0d0))
end function

Java

a1_m = Math.abs(a1);
assert a1_m < a2 && a2 < th;
public static double code(double a1_m, double a2, double th) {
	return (Math.cos(th) * a2) * Math.sqrt(((a2 * a2) / 2.0));
}

Python

a1_m = math.fabs(a1)
[a1_m, a2, th] = sort([a1_m, a2, th])
def code(a1_m, a2, th):
	return (math.cos(th) * a2) * math.sqrt(((a2 * a2) / 2.0))

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	return Float64(Float64(cos(th) * a2) * sqrt(Float64(Float64(a2 * a2) / 2.0)))
end

MATLAB

a1_m = abs(a1);
a1_m, a2, th = num2cell(sort([a1_m, a2, th])){:}
function tmp = code(a1_m, a2, th)
	tmp = (cos(th) * a2) * sqrt(((a2 * a2) / 2.0));
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := N[(N[(N[Cos[th], $MachinePrecision] * a2), $MachinePrecision] * N[Sqrt[N[(N[(a2 * a2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in a1 around 0

   \[\leadsto \color{blue}{\frac{{a2}^{2} \cdot \cos th}{\sqrt{2}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-sqrt.f64 56.5

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

5. Applied rewrites 56.5%

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

7. Applied rewrites 33.4%

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

Alternative 8: 78.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \left(\left(a2 \cdot a2\right) \cdot \cos th\right) \cdot \left(\sqrt{2} \cdot 0.5\right) \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th)
 :precision binary64
 (* (* (* a2 a2) (cos th)) (* (sqrt 2.0) 0.5)))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	return ((a2 * a2) * cos(th)) * (sqrt(2.0) * 0.5);
}

Fortran

a1_m =     private
NOTE: a1_m, a2, 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_m, a2, th)
use fmin_fmax_functions
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = ((a2 * a2) * cos(th)) * (sqrt(2.0d0) * 0.5d0)
end function

Java

a1_m = Math.abs(a1);
assert a1_m < a2 && a2 < th;
public static double code(double a1_m, double a2, double th) {
	return ((a2 * a2) * Math.cos(th)) * (Math.sqrt(2.0) * 0.5);
}

Python

a1_m = math.fabs(a1)
[a1_m, a2, th] = sort([a1_m, a2, th])
def code(a1_m, a2, th):
	return ((a2 * a2) * math.cos(th)) * (math.sqrt(2.0) * 0.5)

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	return Float64(Float64(Float64(a2 * a2) * cos(th)) * Float64(sqrt(2.0) * 0.5))
end

MATLAB

a1_m = abs(a1);
a1_m, a2, th = num2cell(sort([a1_m, a2, th])){:}
function tmp = code(a1_m, a2, th)
	tmp = ((a2 * a2) * cos(th)) * (sqrt(2.0) * 0.5);
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := N[(N[(N[(a2 * a2), $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 99.7%

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

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

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

   2. distribute-rgt-out N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 99.7

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

7. Applied rewrites 99.7%

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

9. Applied rewrites 99.7%

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

10. Taylor expanded in a1 around 0

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

12. Applied rewrites 56.6%

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

Alternative 9: 78.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \left(\left(\cos th \cdot a2\right) \cdot a2\right) \cdot \left(\sqrt{2} \cdot 0.5\right) \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th)
 :precision binary64
 (* (* (* (cos th) a2) a2) (* (sqrt 2.0) 0.5)))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	return ((cos(th) * a2) * a2) * (sqrt(2.0) * 0.5);
}

Fortran

a1_m =     private
NOTE: a1_m, a2, 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_m, a2, th)
use fmin_fmax_functions
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = ((cos(th) * a2) * a2) * (sqrt(2.0d0) * 0.5d0)
end function

Java

a1_m = Math.abs(a1);
assert a1_m < a2 && a2 < th;
public static double code(double a1_m, double a2, double th) {
	return ((Math.cos(th) * a2) * a2) * (Math.sqrt(2.0) * 0.5);
}

Python

a1_m = math.fabs(a1)
[a1_m, a2, th] = sort([a1_m, a2, th])
def code(a1_m, a2, th):
	return ((math.cos(th) * a2) * a2) * (math.sqrt(2.0) * 0.5)

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	return Float64(Float64(Float64(cos(th) * a2) * a2) * Float64(sqrt(2.0) * 0.5))
end

MATLAB

a1_m = abs(a1);
a1_m, a2, th = num2cell(sort([a1_m, a2, th])){:}
function tmp = code(a1_m, a2, th)
	tmp = ((cos(th) * a2) * a2) * (sqrt(2.0) * 0.5);
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := N[(N[(N[(N[Cos[th], $MachinePrecision] * a2), $MachinePrecision] * a2), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 99.7%

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

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

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

   2. distribute-rgt-out N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 99.7

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

7. Applied rewrites 99.7%

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

9. Applied rewrites 99.7%

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

10. Taylor expanded in a1 around 0

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

12. Applied rewrites 56.6%

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

Alternative 10: 67.4% accurate, 8.3× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \left(0.5 \cdot \mathsf{fma}\left(a1\_m, a1\_m, a2 \cdot a2\right)\right) \cdot \sqrt{2} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th)
 :precision binary64
 (* (* 0.5 (fma a1_m a1_m (* a2 a2))) (sqrt 2.0)))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	return (0.5 * fma(a1_m, a1_m, (a2 * a2))) * sqrt(2.0);
}

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	return Float64(Float64(0.5 * fma(a1_m, a1_m, Float64(a2 * a2))) * sqrt(2.0))
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := N[(N[(0.5 * N[(a1$95$m * a1$95$m + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 99.7%

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

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

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

   2. distribute-rgt-out N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in th around 0

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

   1. distribute-rgt-in N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 64.5

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

10. Applied rewrites 64.5%

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

Alternative 11: 67.4% accurate, 8.3× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \left(0.5 \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(a1\_m, a1\_m, a2 \cdot a2\right) \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th)
 :precision binary64
 (* (* 0.5 (sqrt 2.0)) (fma a1_m a1_m (* a2 a2))))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	return (0.5 * sqrt(2.0)) * fma(a1_m, a1_m, (a2 * a2));
}

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	return Float64(Float64(0.5 * sqrt(2.0)) * fma(a1_m, a1_m, Float64(a2 * a2)))
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := N[(N[(0.5 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(a1$95$m * a1$95$m + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 99.7%

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

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. distribute-rgt-out N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 64.5

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

7. Applied rewrites 64.5%

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

Alternative 12: 53.5% accurate, 9.9× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \frac{a2 \cdot a2}{\sqrt{2}} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th) :precision binary64 (/ (* a2 a2) (sqrt 2.0)))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	return (a2 * a2) / sqrt(2.0);
}

Fortran

a1_m =     private
NOTE: a1_m, a2, 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_m, a2, th)
use fmin_fmax_functions
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = (a2 * a2) / sqrt(2.0d0)
end function

Java

a1_m = Math.abs(a1);
assert a1_m < a2 && a2 < th;
public static double code(double a1_m, double a2, double th) {
	return (a2 * a2) / Math.sqrt(2.0);
}

Python

a1_m = math.fabs(a1)
[a1_m, a2, th] = sort([a1_m, a2, th])
def code(a1_m, a2, th):
	return (a2 * a2) / math.sqrt(2.0)

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	return Float64(Float64(a2 * a2) / sqrt(2.0))
end

MATLAB

a1_m = abs(a1);
a1_m, a2, th = num2cell(sort([a1_m, a2, th])){:}
function tmp = code(a1_m, a2, th)
	tmp = (a2 * a2) / sqrt(2.0);
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := N[(N[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in th around 0

   \[\leadsto \color{blue}{\frac{{a1}^{2}}{\sqrt{2}} + \frac{{a2}^{2}}{\sqrt{2}}} \]
4. Step-by-step derivation

   1. div-add-rev N/A

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

   2. lower-/.f64 N/A

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

   3. +-commutative N/A

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

   4. unpow2 N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sqrt.f64 64.3

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

5. Applied rewrites 64.3%

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

6. Taylor expanded in a1 around 0

   \[\leadsto \frac{{a2}^{2}}{\sqrt{\color{blue}{2}}} \]
7. Step-by-step derivation

8. Applied rewrites 39.0%

   \[\leadsto \frac{a2 \cdot a2}{\sqrt{\color{blue}{2}}} \]

9. Final simplification 39.0%

   \[\leadsto \frac{a2 \cdot a2}{\sqrt{2}} \]
10. Add Preprocessing

Alternative 13: 53.5% accurate, 9.9× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ a2 \cdot \frac{a2}{\sqrt{2}} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th) :precision binary64 (* a2 (/ a2 (sqrt 2.0))))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	return a2 * (a2 / sqrt(2.0));
}

Fortran

a1_m =     private
NOTE: a1_m, a2, 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_m, a2, th)
use fmin_fmax_functions
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = a2 * (a2 / sqrt(2.0d0))
end function

Java

a1_m = Math.abs(a1);
assert a1_m < a2 && a2 < th;
public static double code(double a1_m, double a2, double th) {
	return a2 * (a2 / Math.sqrt(2.0));
}

Python

a1_m = math.fabs(a1)
[a1_m, a2, th] = sort([a1_m, a2, th])
def code(a1_m, a2, th):
	return a2 * (a2 / math.sqrt(2.0))

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	return Float64(a2 * Float64(a2 / sqrt(2.0)))
end

MATLAB

a1_m = abs(a1);
a1_m, a2, th = num2cell(sort([a1_m, a2, th])){:}
function tmp = code(a1_m, a2, th)
	tmp = a2 * (a2 / sqrt(2.0));
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in th around 0

   \[\leadsto \color{blue}{\frac{{a1}^{2}}{\sqrt{2}} + \frac{{a2}^{2}}{\sqrt{2}}} \]
4. Step-by-step derivation

   1. div-add-rev N/A

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

   2. lower-/.f64 N/A

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

   3. +-commutative N/A

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

   4. unpow2 N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sqrt.f64 64.3

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

5. Applied rewrites 64.3%

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

6. Taylor expanded in a1 around 0

   \[\leadsto \frac{{a2}^{2}}{\color{blue}{\sqrt{2}}} \]
7. Step-by-step derivation

8. Applied rewrites 39.0%

   \[\leadsto a2 \cdot \color{blue}{\frac{a2}{\sqrt{2}}} \]
9. Add Preprocessing

Alternative 14: 27.4% accurate, 9.9× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ a1\_m \cdot \frac{a1\_m}{\sqrt{2}} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th) :precision binary64 (* a1_m (/ a1_m (sqrt 2.0))))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	return a1_m * (a1_m / sqrt(2.0));
}

Fortran

a1_m =     private
NOTE: a1_m, a2, 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_m, a2, th)
use fmin_fmax_functions
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = a1_m * (a1_m / sqrt(2.0d0))
end function

Java

a1_m = Math.abs(a1);
assert a1_m < a2 && a2 < th;
public static double code(double a1_m, double a2, double th) {
	return a1_m * (a1_m / Math.sqrt(2.0));
}

Python

a1_m = math.fabs(a1)
[a1_m, a2, th] = sort([a1_m, a2, th])
def code(a1_m, a2, th):
	return a1_m * (a1_m / math.sqrt(2.0))

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	return Float64(a1_m * Float64(a1_m / sqrt(2.0)))
end

MATLAB

a1_m = abs(a1);
a1_m, a2, th = num2cell(sort([a1_m, a2, th])){:}
function tmp = code(a1_m, a2, th)
	tmp = a1_m * (a1_m / sqrt(2.0));
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := N[(a1$95$m * N[(a1$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in th around 0

   \[\leadsto \color{blue}{\frac{{a1}^{2}}{\sqrt{2}} + \frac{{a2}^{2}}{\sqrt{2}}} \]
4. Step-by-step derivation

   1. div-add-rev N/A

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

   2. lower-/.f64 N/A

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

   3. +-commutative N/A

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

   4. unpow2 N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sqrt.f64 64.3

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

5. Applied rewrites 64.3%

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

6. Taylor expanded in a1 around inf

   \[\leadsto \frac{{a1}^{2}}{\color{blue}{\sqrt{2}}} \]
7. Step-by-step derivation

8. Applied rewrites 40.6%

   \[\leadsto a1 \cdot \color{blue}{\frac{a1}{\sqrt{2}}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024361 
(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))))