Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.5%

Time: 5.9s

Alternatives: 13

Speedup: 1.0×

• 44.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% 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]]

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

Alternative 2: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \left(a1 \cdot a1\right)\right) \cdot \cos th + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (+ (* (* 0.5 (* a1 a1)) (cos th)) (* (/ (cos th) (sqrt 2.0)) (* a2 a2))))

C

double code(double a1, double a2, double th) {
	return ((0.5 * (a1 * a1)) * cos(th)) + ((cos(th) / sqrt(2.0)) * (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
    code = ((0.5d0 * (a1 * a1)) * cos(th)) + ((cos(th) / sqrt(2.0d0)) * (a2 * a2))
end function

Java

public static double code(double a1, double a2, double th) {
	return ((0.5 * (a1 * a1)) * Math.cos(th)) + ((Math.cos(th) / Math.sqrt(2.0)) * (a2 * a2));
}

Python

def code(a1, a2, th):
	return ((0.5 * (a1 * a1)) * math.cos(th)) + ((math.cos(th) / math.sqrt(2.0)) * (a2 * a2))

Julia

function code(a1, a2, th)
	return Float64(Float64(Float64(0.5 * Float64(a1 * a1)) * cos(th)) + Float64(Float64(cos(th) / sqrt(2.0)) * Float64(a2 * a2)))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = ((0.5 * (a1 * a1)) * cos(th)) + ((cos(th) / sqrt(2.0)) * (a2 * a2));
end

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(0.5 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 80.1%

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

4. Taylor expanded in a1 around 0

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

6. Applied rewrites 80.1%

   \[\leadsto \color{blue}{\left(0.5 \cdot \left(a1 \cdot a1\right)\right) \cdot \cos th} + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
7. Add Preprocessing

Alternative 3: 74.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (+ (* 0.5 (* a1 a1)) (* (/ (cos th) (sqrt 2.0)) (* a2 a2))))

C

double code(double a1, double a2, double th) {
	return (0.5 * (a1 * a1)) + ((cos(th) / sqrt(2.0)) * (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
    code = (0.5d0 * (a1 * a1)) + ((cos(th) / sqrt(2.0d0)) * (a2 * a2))
end function

Java

public static double code(double a1, double a2, double th) {
	return (0.5 * (a1 * a1)) + ((Math.cos(th) / Math.sqrt(2.0)) * (a2 * a2));
}

Python

def code(a1, a2, th):
	return (0.5 * (a1 * a1)) + ((math.cos(th) / math.sqrt(2.0)) * (a2 * a2))

Julia

function code(a1, a2, th)
	return Float64(Float64(0.5 * Float64(a1 * a1)) + Float64(Float64(cos(th) / sqrt(2.0)) * Float64(a2 * a2)))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = (0.5 * (a1 * a1)) + ((cos(th) / sqrt(2.0)) * (a2 * a2));
end

Wolfram

code[a1_, a2_, th_] := N[(N[(0.5 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 80.1%

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

4. Taylor expanded in th around 0

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

6. Applied rewrites 72.1%

   \[\leadsto \color{blue}{0.5 \cdot \left(a1 \cdot a1\right)} + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
7. Add Preprocessing

Alternative 4: 72.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\log a1}\\ \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \leq 0.5:\\ \;\;\;\;\left(0.5 \cdot \cos th\right) \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, t\_1, a2 \cdot a2\right)}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (exp (log a1))))
   (if (<= (/ (cos th) (sqrt 2.0)) 0.5)
     (* (* 0.5 (cos th)) (fma a1 a1 (* a2 a2)))
     (/ (fma t_1 t_1 (* a2 a2)) (sqrt 2.0)))))

C

double code(double a1, double a2, double th) {
	double t_1 = exp(log(a1));
	double tmp;
	if ((cos(th) / sqrt(2.0)) <= 0.5) {
		tmp = (0.5 * cos(th)) * fma(a1, a1, (a2 * a2));
	} else {
		tmp = fma(t_1, t_1, (a2 * a2)) / sqrt(2.0);
	}
	return tmp;
}

Julia

function code(a1, a2, th)
	t_1 = exp(log(a1))
	tmp = 0.0
	if (Float64(cos(th) / sqrt(2.0)) <= 0.5)
		tmp = Float64(Float64(0.5 * cos(th)) * fma(a1, a1, Float64(a2 * a2)));
	else
		tmp = Float64(fma(t_1, t_1, Float64(a2 * a2)) / sqrt(2.0));
	end
	return tmp
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[Exp[N[Log[a1], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], 0.5], N[(N[(0.5 * N[Cos[th], $MachinePrecision]), $MachinePrecision] * N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * t$95$1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) < 0.5

   1. Initial program 99.5%

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

   3. Applied rewrites 60.3%

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

   4. Taylor expanded in th around inf

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

   6. Applied rewrites 60.3%

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

   if 0.5 < (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64)))

   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. Taylor expanded in th around 0

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

   4. Applied rewrites 60.4%

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

   6. Applied rewrites 30.5%

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

   8. Applied rewrites 45.3%

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

Alternative 5: 70.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\log a1}\\ \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \leq 0.5:\\ \;\;\;\;\left(0.5 \cdot \left(a2 \cdot a2\right)\right) \cdot \cos th\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, t\_1, a2 \cdot a2\right)}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (exp (log a1))))
   (if (<= (/ (cos th) (sqrt 2.0)) 0.5)
     (* (* 0.5 (* a2 a2)) (cos th))
     (/ (fma t_1 t_1 (* a2 a2)) (sqrt 2.0)))))

C

double code(double a1, double a2, double th) {
	double t_1 = exp(log(a1));
	double tmp;
	if ((cos(th) / sqrt(2.0)) <= 0.5) {
		tmp = (0.5 * (a2 * a2)) * cos(th);
	} else {
		tmp = fma(t_1, t_1, (a2 * a2)) / sqrt(2.0);
	}
	return tmp;
}

Julia

function code(a1, a2, th)
	t_1 = exp(log(a1))
	tmp = 0.0
	if (Float64(cos(th) / sqrt(2.0)) <= 0.5)
		tmp = Float64(Float64(0.5 * Float64(a2 * a2)) * cos(th));
	else
		tmp = Float64(fma(t_1, t_1, Float64(a2 * a2)) / sqrt(2.0));
	end
	return tmp
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[Exp[N[Log[a1], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], 0.5], N[(N[(0.5 * N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * t$95$1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) < 0.5

   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. Taylor expanded in th around 0

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

   4. Applied rewrites 22.6%

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

   5. Taylor expanded in a1 around 0

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

   7. Applied rewrites 36.4%

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

   if 0.5 < (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64)))

   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. Taylor expanded in th around 0

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

   4. Applied rewrites 60.4%

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

   6. Applied rewrites 30.5%

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

   8. Applied rewrites 45.3%

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

Alternative 6: 65.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{\sqrt{2}}\\ \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \leq 0.5:\\ \;\;\;\;\left(0.5 \cdot \left(a2 \cdot a2\right)\right) \cdot \cos th\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(a1 \cdot a1\right) + t\_1 \cdot \left(a2 \cdot a2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ 1.0 (sqrt 2.0))))
   (if (<= (/ (cos th) (sqrt 2.0)) 0.5)
     (* (* 0.5 (* a2 a2)) (cos th))
     (+ (* t_1 (* a1 a1)) (* t_1 (* a2 a2))))))

C

double code(double a1, double a2, double th) {
	double t_1 = 1.0 / sqrt(2.0);
	double tmp;
	if ((cos(th) / sqrt(2.0)) <= 0.5) {
		tmp = (0.5 * (a2 * a2)) * cos(th);
	} else {
		tmp = (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
	}
	return tmp;
}

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
    real(8) :: tmp
    t_1 = 1.0d0 / sqrt(2.0d0)
    if ((cos(th) / sqrt(2.0d0)) <= 0.5d0) then
        tmp = (0.5d0 * (a2 * a2)) * cos(th)
    else
        tmp = (t_1 * (a1 * a1)) + (t_1 * (a2 * a2))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = 1.0 / Math.sqrt(2.0);
	double tmp;
	if ((Math.cos(th) / Math.sqrt(2.0)) <= 0.5) {
		tmp = (0.5 * (a2 * a2)) * Math.cos(th);
	} else {
		tmp = (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = 1.0 / math.sqrt(2.0)
	tmp = 0
	if (math.cos(th) / math.sqrt(2.0)) <= 0.5:
		tmp = (0.5 * (a2 * a2)) * math.cos(th)
	else:
		tmp = (t_1 * (a1 * a1)) + (t_1 * (a2 * a2))
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(1.0 / sqrt(2.0))
	tmp = 0.0
	if (Float64(cos(th) / sqrt(2.0)) <= 0.5)
		tmp = Float64(Float64(0.5 * Float64(a2 * a2)) * cos(th));
	else
		tmp = Float64(Float64(t_1 * Float64(a1 * a1)) + Float64(t_1 * Float64(a2 * a2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = 1.0 / sqrt(2.0);
	tmp = 0.0;
	if ((cos(th) / sqrt(2.0)) <= 0.5)
		tmp = (0.5 * (a2 * a2)) * cos(th);
	else
		tmp = (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], 0.5], N[(N[(0.5 * N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) < 0.5

   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. Taylor expanded in th around 0

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

   4. Applied rewrites 22.6%

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

   5. Taylor expanded in a1 around 0

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

   7. Applied rewrites 36.4%

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

   if 0.5 < (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64)))

   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. Taylor expanded in th around 0

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

   4. Applied rewrites 95.8%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 91.8%

      \[\leadsto \frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\color{blue}{1}}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 55.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ t_2 := \frac{1}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1 \cdot a1\right) + t\_1 \cdot \left(a2 \cdot a2\right) \leq -5 \cdot 10^{-157}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \mathsf{fma}\left(-0.25, th \cdot th, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(a1 \cdot a1\right) + t\_2 \cdot \left(a2 \cdot a2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))) (t_2 (/ 1.0 (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a1 a1)) (* t_1 (* a2 a2))) -5e-157)
     (* (* a2 a2) (fma -0.25 (* th th) 0.5))
     (+ (* t_2 (* a1 a1)) (* t_2 (* a2 a2))))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double t_2 = 1.0 / sqrt(2.0);
	double tmp;
	if (((t_1 * (a1 * a1)) + (t_1 * (a2 * a2))) <= -5e-157) {
		tmp = (a2 * a2) * fma(-0.25, (th * th), 0.5);
	} else {
		tmp = (t_2 * (a1 * a1)) + (t_2 * (a2 * a2));
	}
	return tmp;
}

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	t_2 = Float64(1.0 / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(t_1 * Float64(a1 * a1)) + Float64(t_1 * Float64(a2 * a2))) <= -5e-157)
		tmp = Float64(Float64(a2 * a2) * fma(-0.25, Float64(th * th), 0.5));
	else
		tmp = Float64(Float64(t_2 * Float64(a1 * a1)) + Float64(t_2 * Float64(a2 * a2)));
	end
	return tmp
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-157], N[(N[(a2 * a2), $MachinePrecision] * N[(-0.25 * N[(th * th), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * N[(a2 * a2), $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))) < -5.0000000000000002e-157

   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. Taylor expanded in th around 0

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

   4. Applied rewrites 2.5%

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

   5. Taylor expanded in a2 around inf

      \[\leadsto {a2}^{2} \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {th}^{2}\right)} \]

   7. Applied rewrites 41.7%

      \[\leadsto \left(a2 \cdot a2\right) \cdot \color{blue}{\mathsf{fma}\left(-0.25, th \cdot th, 0.5\right)} \]

   if -5.0000000000000002e-157 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.5%

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

   2. Taylor expanded in th around 0

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

   4. Applied rewrites 92.1%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 84.1%

      \[\leadsto \frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\color{blue}{1}}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 55.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1 \cdot a1\right) + t\_1 \cdot \left(a2 \cdot a2\right) \leq -5 \cdot 10^{-157}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \mathsf{fma}\left(-0.25, th \cdot th, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \left(a1 \cdot a1\right)\right) \cdot 1 + \frac{1}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a1 a1)) (* t_1 (* a2 a2))) -5e-157)
     (* (* a2 a2) (fma -0.25 (* th th) 0.5))
     (+ (* (* 0.5 (* a1 a1)) 1.0) (* (/ 1.0 (sqrt 2.0)) (* a2 a2))))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_1 * (a1 * a1)) + (t_1 * (a2 * a2))) <= -5e-157) {
		tmp = (a2 * a2) * fma(-0.25, (th * th), 0.5);
	} else {
		tmp = ((0.5 * (a1 * a1)) * 1.0) + ((1.0 / sqrt(2.0)) * (a2 * a2));
	}
	return tmp;
}

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(t_1 * Float64(a1 * a1)) + Float64(t_1 * Float64(a2 * a2))) <= -5e-157)
		tmp = Float64(Float64(a2 * a2) * fma(-0.25, Float64(th * th), 0.5));
	else
		tmp = Float64(Float64(Float64(0.5 * Float64(a1 * a1)) * 1.0) + Float64(Float64(1.0 / sqrt(2.0)) * Float64(a2 * a2)));
	end
	return tmp
end

Wolfram

code[a1_, 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 * a1), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-157], N[(N[(a2 * a2), $MachinePrecision] * N[(-0.25 * N[(th * th), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] + N[(N[(1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(a2 * a2), $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))) < -5.0000000000000002e-157

   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. Taylor expanded in th around 0

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

   4. Applied rewrites 2.5%

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

   5. Taylor expanded in a2 around inf

      \[\leadsto {a2}^{2} \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {th}^{2}\right)} \]

   7. Applied rewrites 41.7%

      \[\leadsto \left(a2 \cdot a2\right) \cdot \color{blue}{\mathsf{fma}\left(-0.25, th \cdot th, 0.5\right)} \]

   if -5.0000000000000002e-157 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.5%

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

   3. Applied rewrites 79.9%

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

   4. Taylor expanded in a1 around 0

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

   6. Applied rewrites 79.9%

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

   7. Taylor expanded in th around 0

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

   9. Applied rewrites 79.8%

      \[\leadsto \left(0.5 \cdot \left(a1 \cdot a1\right)\right) \cdot 1 + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]

   10. Taylor expanded in th around 0

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

   12. Applied rewrites 71.8%

       \[\leadsto \left(0.5 \cdot \left(a1 \cdot a1\right)\right) \cdot 1 + \frac{\color{blue}{1}}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 50.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1 \cdot a1\right) + t\_1 \cdot \left(a2 \cdot a2\right) \leq -5 \cdot 10^{-157}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \mathsf{fma}\left(-0.25, th \cdot th, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a1 a1)) (* t_1 (* a2 a2))) -5e-157)
     (* (* a2 a2) (fma -0.25 (* th th) 0.5))
     (* 0.5 (fma a1 a1 (* a2 a2))))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_1 * (a1 * a1)) + (t_1 * (a2 * a2))) <= -5e-157) {
		tmp = (a2 * a2) * fma(-0.25, (th * th), 0.5);
	} else {
		tmp = 0.5 * fma(a1, a1, (a2 * a2));
	}
	return tmp;
}

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(t_1 * Float64(a1 * a1)) + Float64(t_1 * Float64(a2 * a2))) <= -5e-157)
		tmp = Float64(Float64(a2 * a2) * fma(-0.25, Float64(th * th), 0.5));
	else
		tmp = Float64(0.5 * fma(a1, a1, Float64(a2 * a2)));
	end
	return tmp
end

Wolfram

code[a1_, 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 * a1), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-157], N[(N[(a2 * a2), $MachinePrecision] * N[(-0.25 * N[(th * th), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(a1 * a1 + N[(a2 * a2), $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))) < -5.0000000000000002e-157

   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. Taylor expanded in th around 0

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

   4. Applied rewrites 2.5%

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

   5. Taylor expanded in a2 around inf

      \[\leadsto {a2}^{2} \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {th}^{2}\right)} \]

   7. Applied rewrites 41.7%

      \[\leadsto \left(a2 \cdot a2\right) \cdot \color{blue}{\mathsf{fma}\left(-0.25, th \cdot th, 0.5\right)} \]

   if -5.0000000000000002e-157 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.5%

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

   3. Applied rewrites 59.5%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 59.1%

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

Alternative 10: 46.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1 \cdot a1\right) + t\_1 \cdot \left(a2 \cdot a2\right) \leq -5 \cdot 10^{-157}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \mathsf{fma}\left(-0.25, th \cdot th, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a1 a1)) (* t_1 (* a2 a2))) -5e-157)
     (* (* a1 a1) (fma -0.25 (* th th) 0.5))
     (* 0.5 (fma a1 a1 (* a2 a2))))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_1 * (a1 * a1)) + (t_1 * (a2 * a2))) <= -5e-157) {
		tmp = (a1 * a1) * fma(-0.25, (th * th), 0.5);
	} else {
		tmp = 0.5 * fma(a1, a1, (a2 * a2));
	}
	return tmp;
}

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(t_1 * Float64(a1 * a1)) + Float64(t_1 * Float64(a2 * a2))) <= -5e-157)
		tmp = Float64(Float64(a1 * a1) * fma(-0.25, Float64(th * th), 0.5));
	else
		tmp = Float64(0.5 * fma(a1, a1, Float64(a2 * a2)));
	end
	return tmp
end

Wolfram

code[a1_, 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 * a1), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-157], N[(N[(a1 * a1), $MachinePrecision] * N[(-0.25 * N[(th * th), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(a1 * a1 + N[(a2 * a2), $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))) < -5.0000000000000002e-157

   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. Taylor expanded in th around 0

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

   4. Applied rewrites 2.5%

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

   5. Taylor expanded in a1 around inf

      \[\leadsto {a1}^{2} \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {th}^{2}\right)} \]

   7. Applied rewrites 44.0%

      \[\leadsto \left(a1 \cdot a1\right) \cdot \color{blue}{\mathsf{fma}\left(-0.25, th \cdot th, 0.5\right)} \]

   if -5.0000000000000002e-157 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.5%

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

   3. Applied rewrites 59.5%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 59.1%

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

Alternative 11: 41.9% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* 0.5 (fma a1 a1 (* a2 a2))))

C

double code(double a1, double a2, double th) {
	return 0.5 * fma(a1, a1, (a2 * a2));
}

Julia

function code(a1, a2, th)
	return Float64(0.5 * fma(a1, a1, Float64(a2 * a2)))
end

Wolfram

code[a1_, a2_, th_] := N[(0.5 * N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 60.3%

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

4. Taylor expanded in th around 0

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

6. Applied rewrites 46.2%

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

Alternative 12: 30.1% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(a2 \cdot a2\right) \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* 0.5 (* a2 a2)))

C

double code(double a1, double a2, double th) {
	return 0.5 * (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
    code = 0.5d0 * (a2 * a2)
end function

Java

public static double code(double a1, double a2, double th) {
	return 0.5 * (a2 * a2);
}

Python

def code(a1, a2, th):
	return 0.5 * (a2 * a2)

Julia

function code(a1, a2, th)
	return Float64(0.5 * Float64(a2 * a2))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = 0.5 * (a2 * a2);
end

Wolfram

code[a1_, a2_, th_] := N[(0.5 * N[(a2 * a2), $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. Taylor expanded in th around 0

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

4. Applied rewrites 46.2%

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

5. Taylor expanded in a1 around 0

   \[\leadsto \frac{1}{2} \cdot \color{blue}{{a2}^{2}} \]

7. Applied rewrites 29.8%

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

Alternative 13: 29.8% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(a1 \cdot a1\right) \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* 0.5 (* a1 a1)))

C

double code(double a1, double a2, double th) {
	return 0.5 * (a1 * a1);
}

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
    code = 0.5d0 * (a1 * a1)
end function

Java

public static double code(double a1, double a2, double th) {
	return 0.5 * (a1 * a1);
}

Python

def code(a1, a2, th):
	return 0.5 * (a1 * a1)

Julia

function code(a1, a2, th)
	return Float64(0.5 * Float64(a1 * a1))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = 0.5 * (a1 * a1);
end

Wolfram

code[a1_, a2_, th_] := N[(0.5 * N[(a1 * a1), $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. Taylor expanded in th around 0

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

4. Applied rewrites 46.2%

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

5. Taylor expanded in a1 around inf

   \[\leadsto \frac{1}{2} \cdot \color{blue}{{a1}^{2}} \]

7. Applied rewrites 30.1%

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

Reproduce

herbie shell --seed 2025132 
(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))))