Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 5.0s

Alternatives: 12

Speedup: 1.9×

• 43.4% 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

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathsf{fma}\left(t\_1 \cdot a2, a2, \left(a1\_m \cdot a1\_m\right) \cdot t\_1\right) \end{array} \end{array} \]

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))))
   (fma (* t_1 a2) a2 (* (* a1_m a1_m) t_1))))

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);
	return fma((t_1 * a2), a2, ((a1_m * a1_m) * t_1));
}

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))
	return fma(Float64(t_1 * a2), a2, Float64(Float64(a1_m * a1_m) * t_1))
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]}, N[(N[(t$95$1 * a2), $MachinePrecision] * a2 + N[(N[(a1$95$m * a1$95$m), $MachinePrecision] * t$95$1), $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. 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. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. +-commutative N/A

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

   13. associate-*r* N/A

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

   14. lower-fma.f64 N/A

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

4. Applied rewrites 99.6%

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

Alternative 2: 77.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1\_m \cdot a1\_m\right) + t\_1 \cdot \left(a2 \cdot a2\right) \leq -5 \cdot 10^{-189}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \frac{\mathsf{fma}\left(th \cdot th, -0.5, 1\right)}{\sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2, a2 \cdot \sqrt{2}, \left(a1\_m \cdot a1\_m\right) \cdot \sqrt{2}\right)}{2}\\ \end{array} \end{array} \]

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))) -5e-189)
     (* a2 (* a2 (/ (fma (* th th) -0.5 1.0) (sqrt 2.0))))
     (/ (fma a2 (* a2 (sqrt 2.0)) (* (* a1_m a1_m) (sqrt 2.0))) 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))) <= -5e-189) {
		tmp = a2 * (a2 * (fma((th * th), -0.5, 1.0) / sqrt(2.0)));
	} else {
		tmp = fma(a2, (a2 * sqrt(2.0)), ((a1_m * a1_m) * sqrt(2.0))) / 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))) <= -5e-189)
		tmp = Float64(a2 * Float64(a2 * Float64(fma(Float64(th * th), -0.5, 1.0) / sqrt(2.0))));
	else
		tmp = Float64(fma(a2, Float64(a2 * sqrt(2.0)), Float64(Float64(a1_m * a1_m) * sqrt(2.0))) / 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], -5e-189], N[(a2 * N[(a2 * N[(N[(N[(th * th), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a2 * N[(a2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(a1$95$m * a1$95$m), $MachinePrecision] * N[Sqrt[2.0], $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))) < -4.9999999999999997e-189

   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. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-/.f64 77.0

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

   5. Applied rewrites 77.0%

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

   6. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 47.5

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

   8. Applied rewrites 47.5%

      \[\leadsto \left(a2 \cdot a2\right) \cdot \frac{\mathsf{fma}\left(th \cdot th, -0.5, 1\right)}{\sqrt{\color{blue}{2}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 49.4

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

   10. Applied rewrites 49.4%

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

   if -4.9999999999999997e-189 < (+.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. 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. +-commutative N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sqrt.f64 85.5

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

   5. Applied rewrites 85.5%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. div-add N/A

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

      8. frac-add N/A

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

      9. rem-square-sqrt N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-sqrt.f64 85.5

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

   7. Applied rewrites 85.5%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-*.f64 85.5

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

   9. Applied rewrites 85.5%

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

Alternative 3: 77.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1\_m \cdot a1\_m\right) + t\_1 \cdot \left(a2 \cdot a2\right) \leq -5 \cdot 10^{-189}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \frac{\mathsf{fma}\left(th \cdot th, -0.5, 1\right)}{\sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(a2, a2, a1\_m \cdot a1\_m\right)\right)\\ \end{array} \end{array} \]

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))) -5e-189)
     (* a2 (* a2 (/ (fma (* th th) -0.5 1.0) (sqrt 2.0))))
     (* 0.5 (* (sqrt 2.0) (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) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_1 * (a1_m * a1_m)) + (t_1 * (a2 * a2))) <= -5e-189) {
		tmp = a2 * (a2 * (fma((th * th), -0.5, 1.0) / sqrt(2.0)));
	} else {
		tmp = 0.5 * (sqrt(2.0) * fma(a2, a2, (a1_m * a1_m)));
	}
	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))) <= -5e-189)
		tmp = Float64(a2 * Float64(a2 * Float64(fma(Float64(th * th), -0.5, 1.0) / sqrt(2.0))));
	else
		tmp = Float64(0.5 * Float64(sqrt(2.0) * fma(a2, a2, Float64(a1_m * a1_m))));
	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], -5e-189], N[(a2 * N[(a2 * N[(N[(N[(th * th), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(a2 * a2 + N[(a1$95$m * a1$95$m), $MachinePrecision]), $MachinePrecision]), $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))) < -4.9999999999999997e-189

   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. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-/.f64 77.0

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

   5. Applied rewrites 77.0%

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

   6. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 47.5

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

   8. Applied rewrites 47.5%

      \[\leadsto \left(a2 \cdot a2\right) \cdot \frac{\mathsf{fma}\left(th \cdot th, -0.5, 1\right)}{\sqrt{\color{blue}{2}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 49.4

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

   10. Applied rewrites 49.4%

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

   if -4.9999999999999997e-189 < (+.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. 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. +-commutative N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sqrt.f64 85.5

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

   5. Applied rewrites 85.5%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. div-add N/A

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

      8. frac-add N/A

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

      9. rem-square-sqrt N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-sqrt.f64 85.5

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

   7. Applied rewrites 85.5%

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

   8. Taylor expanded in a1 around 0

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

   10. Applied rewrites 85.5%

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

Alternative 4: 77.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1\_m \cdot a1\_m\right) + t\_1 \cdot \left(a2 \cdot a2\right) \leq -5 \cdot 10^{-189}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \frac{\left(th \cdot th\right) \cdot -0.5}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(a2, a2, a1\_m \cdot a1\_m\right)\right)\\ \end{array} \end{array} \]

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))) -5e-189)
     (* (* a2 a2) (/ (* (* th th) -0.5) (sqrt 2.0)))
     (* 0.5 (* (sqrt 2.0) (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) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_1 * (a1_m * a1_m)) + (t_1 * (a2 * a2))) <= -5e-189) {
		tmp = (a2 * a2) * (((th * th) * -0.5) / sqrt(2.0));
	} else {
		tmp = 0.5 * (sqrt(2.0) * fma(a2, a2, (a1_m * a1_m)));
	}
	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))) <= -5e-189)
		tmp = Float64(Float64(a2 * a2) * Float64(Float64(Float64(th * th) * -0.5) / sqrt(2.0)));
	else
		tmp = Float64(0.5 * Float64(sqrt(2.0) * fma(a2, a2, Float64(a1_m * a1_m))));
	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], -5e-189], N[(N[(a2 * a2), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * -0.5), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(a2 * a2 + N[(a1$95$m * a1$95$m), $MachinePrecision]), $MachinePrecision]), $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))) < -4.9999999999999997e-189

   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. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-/.f64 77.0

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

   5. Applied rewrites 77.0%

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

   6. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 47.5

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

   8. Applied rewrites 47.5%

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

   9. Taylor expanded in th around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 47.5

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

   11. Applied rewrites 47.5%

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

   if -4.9999999999999997e-189 < (+.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. 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. +-commutative N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sqrt.f64 85.5

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

   5. Applied rewrites 85.5%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. div-add N/A

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

      8. frac-add N/A

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

      9. rem-square-sqrt N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-sqrt.f64 85.5

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

   7. Applied rewrites 85.5%

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

   8. Taylor expanded in a1 around 0

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

   10. Applied rewrites 85.5%

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

Alternative 5: 89.7% accurate, 1.7× speedup

Math

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

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 (* (/ (cos th) (sqrt 2.0)) a2) a2 (/ (* 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 fma(((cos(th) / sqrt(2.0)) * a2), a2, ((a1_m * a1_m) / sqrt(2.0)));
}

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	return fma(Float64(Float64(cos(th) / sqrt(2.0)) * a2), a2, Float64(Float64(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[(N[(N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * a2), $MachinePrecision] * a2 + N[(N[(a1$95$m * a1$95$m), $MachinePrecision] / 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. 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. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. +-commutative N/A

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

   13. associate-*r* N/A

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

   14. lower-fma.f64 N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in th around 0

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

   1. pow2 N/A

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

   2. *-commutative N/A

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

   3. pow2 N/A

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

   4. lower-/.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-sqrt.f64 89.7

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

7. Applied rewrites 89.7%

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

Alternative 6: 99.5% accurate, 1.9× speedup

Math

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

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
 (* (cos th) (/ (* (sqrt 2.0) (fma a2 a2 (* 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) {
	return cos(th) * ((sqrt(2.0) * fma(a2, a2, (a1_m * 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(cos(th) * Float64(Float64(sqrt(2.0) * fma(a2, a2, Float64(a1_m * 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[Cos[th], $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(a2 * a2 + N[(a1$95$m * a1$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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. 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%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. associate-/l* N/A

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

   8. pow2 N/A

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

   9. pow2 N/A

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

   10. div-add-rev N/A

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

   11. lower-*.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. div-add-rev N/A

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

   14. lower-/.f64 N/A

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

   15. pow2 N/A

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

   16. pow2 N/A

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

   17. lift-fma.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. lift-sqrt.f64 99.6

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

6. Applied rewrites 99.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. pow2 N/A

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

   5. pow2 N/A

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

   6. +-commutative N/A

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

   7. lift-sqrt.f64 N/A

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

   8. div-add N/A

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

   9. frac-add N/A

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

   10. *-commutative N/A

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

   11. pow2 N/A

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

   12. pow2 N/A

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

   13. rem-square-sqrt N/A

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

   14. lower-/.f64 N/A

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

8. Applied rewrites 99.5%

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

Alternative 7: 99.6% accurate, 1.9× speedup

Math

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

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
 (* (cos th) (/ (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 cos(th) * (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(cos(th) * Float64(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[Cos[th], $MachinePrecision] * N[(N[(a1$95$m * a1$95$m + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] / 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. 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%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. associate-/l* N/A

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

   8. pow2 N/A

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

   9. pow2 N/A

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

   10. div-add-rev N/A

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

   11. lower-*.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. div-add-rev N/A

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

   14. lower-/.f64 N/A

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

   15. pow2 N/A

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

   16. pow2 N/A

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

   17. lift-fma.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. lift-sqrt.f64 99.6

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

6. Applied rewrites 99.6%

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

Alternative 8: 78.0% accurate, 2.0× speedup

Math

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

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 2.0)) a2))

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(2.0)) * a2;
}

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(2.0d0)) * a2
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(2.0)) * a2;
}

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(2.0)) * 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(cos(th) * a2) / sqrt(2.0)) * a2)
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(2.0)) * 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[(N[Cos[th], $MachinePrecision] * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * a2), $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. associate-/l* N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-/.f64 78.0

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

5. Applied rewrites 78.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. associate-/l* N/A

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

   12. lower-/.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lift-cos.f64 N/A

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

   16. lift-sqrt.f64 78.0

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

7. Applied rewrites 78.0%

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

Alternative 9: 78.0% accurate, 2.0× speedup

Math

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

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) (/ (cos th) (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) * (cos(th) / 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) * (cos(th) / 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.cos(th) / 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.cos(th) / 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) * Float64(cos(th) / 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) * (cos(th) / 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[(N[Cos[th], $MachinePrecision] / 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 a1 around 0

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

   1. associate-/l* N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-/.f64 78.0

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

5. Applied rewrites 78.0%

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

Alternative 10: 66.8% accurate, 8.3× speedup

Math

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

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 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) * 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(0.5 * Float64(sqrt(2.0) * 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[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(a2 * a2 + N[(a1$95$m * a1$95$m), $MachinePrecision]), $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. +-commutative N/A

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

   2. div-add-rev N/A

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

   3. lower-/.f64 N/A

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

   4. pow2 N/A

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

   5. lower-fma.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-sqrt.f64 66.8

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

5. Applied rewrites 66.8%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. div-add N/A

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

   8. frac-add N/A

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

   9. rem-square-sqrt N/A

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

   10. lower-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-sqrt.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. pow2 N/A

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

   18. lift-*.f64 N/A

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

   19. lift-sqrt.f64 66.8

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

7. Applied rewrites 66.8%

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

8. Taylor expanded in a1 around 0

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

10. Applied rewrites 66.8%

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

Alternative 11: 53.2% accurate, 10.2× speedup

Math

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

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 (* 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 * (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 = (0.5d0 * (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 (0.5 * (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 (0.5 * (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(0.5 * 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 = (0.5 * (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[(a2 * a2), $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

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. +-commutative N/A

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

   2. div-add-rev N/A

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

   3. lower-/.f64 N/A

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

   4. pow2 N/A

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

   5. lower-fma.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-sqrt.f64 66.8

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

5. Applied rewrites 66.8%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. div-add N/A

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

   8. frac-add N/A

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

   9. rem-square-sqrt N/A

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

   10. lower-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-sqrt.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. pow2 N/A

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

   18. lift-*.f64 N/A

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

   19. lift-sqrt.f64 66.8

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

7. Applied rewrites 66.8%

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

8. Taylor expanded in a1 around 0

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

10. Applied rewrites 53.2%

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

Alternative 12: 27.1% accurate, 10.2× speedup

Math

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

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 (* 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 (0.5 * (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 = (0.5d0 * (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 (0.5 * (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 (0.5 * (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(Float64(0.5 * Float64(a1_m * 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 = (0.5 * (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[(N[(0.5 * N[(a1$95$m * a1$95$m), $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

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. +-commutative N/A

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

   2. div-add-rev N/A

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

   3. lower-/.f64 N/A

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

   4. pow2 N/A

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

   5. lower-fma.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-sqrt.f64 66.8

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

5. Applied rewrites 66.8%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. div-add N/A

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

   8. frac-add N/A

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

   9. rem-square-sqrt N/A

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

   10. lower-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-sqrt.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. pow2 N/A

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

   18. lift-*.f64 N/A

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

   19. lift-sqrt.f64 66.8

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

7. Applied rewrites 66.8%

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

8. Taylor expanded in a1 around inf

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

10. Applied rewrites 27.1%

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

Reproduce

herbie shell --seed 2025086 
(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))))