Graphics.Rasterific.Svg.PathConverter:arcToSegments from rasterific-svg-0.2.3.1

Percentage Accurate: 66.4% → 92.3%

Time: 6.5s

Alternatives: 9

Speedup: 0.8×

• 47.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (/ (* x x) (* y y)) (/ (* z z) (* t t))))

C

double code(double x, double y, double z, double t) {
	return ((x * x) / (y * y)) + ((z * z) / (t * t));
}

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(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x * x) / (y * y)) + ((z * z) / (t * t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x * x) / (y * y)) + ((z * z) / (t * t));
}

Python

def code(x, y, z, t):
	return ((x * x) / (y * y)) + ((z * z) / (t * t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x * x) / Float64(y * y)) + Float64(Float64(z * z) / Float64(t * t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x * x) / (y * y)) + ((z * z) / (t * t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 66.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (/ (* x x) (* y y)) (/ (* z z) (* t t))))

C

double code(double x, double y, double z, double t) {
	return ((x * x) / (y * y)) + ((z * z) / (t * t));
}

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(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x * x) / (y * y)) + ((z * z) / (t * t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x * x) / (y * y)) + ((z * z) / (t * t));
}

Python

def code(x, y, z, t):
	return ((x * x) / (y * y)) + ((z * z) / (t * t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x * x) / Float64(y * y)) + Float64(Float64(z * z) / Float64(t * t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x * x) / (y * y)) + ((z * z) / (t * t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 92.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ z_m = \left|z\right| \\ t_m = \left|t\right| \\ \begin{array}{l} t_1 := \frac{z\_m \cdot z\_m}{t\_m \cdot t\_m}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+228}:\\ \;\;\;\;\frac{x\_m}{y\_m} \cdot \frac{x\_m}{y\_m} + t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z\_m}{t\_m} + \frac{x\_m}{y\_m}\right) \cdot \frac{z\_m}{t\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
z_m = (fabs.f64 z)
t_m = (fabs.f64 t)
(FPCore (x_m y_m z_m t_m)
 :precision binary64
 (let* ((t_1 (/ (* z_m z_m) (* t_m t_m))))
   (if (<= t_1 2e+228)
     (+ (* (/ x_m y_m) (/ x_m y_m)) t_1)
     (* (+ (/ z_m t_m) (/ x_m y_m)) (/ z_m t_m)))))

C

x_m = fabs(x);
y_m = fabs(y);
z_m = fabs(z);
t_m = fabs(t);
double code(double x_m, double y_m, double z_m, double t_m) {
	double t_1 = (z_m * z_m) / (t_m * t_m);
	double tmp;
	if (t_1 <= 2e+228) {
		tmp = ((x_m / y_m) * (x_m / y_m)) + t_1;
	} else {
		tmp = ((z_m / t_m) + (x_m / y_m)) * (z_m / t_m);
	}
	return tmp;
}

Fortran

x_m =     private
y_m =     private
z_m =     private
t_m =     private
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(x_m, y_m, z_m, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z_m * z_m) / (t_m * t_m)
    if (t_1 <= 2d+228) then
        tmp = ((x_m / y_m) * (x_m / y_m)) + t_1
    else
        tmp = ((z_m / t_m) + (x_m / y_m)) * (z_m / t_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
z_m = Math.abs(z);
t_m = Math.abs(t);
public static double code(double x_m, double y_m, double z_m, double t_m) {
	double t_1 = (z_m * z_m) / (t_m * t_m);
	double tmp;
	if (t_1 <= 2e+228) {
		tmp = ((x_m / y_m) * (x_m / y_m)) + t_1;
	} else {
		tmp = ((z_m / t_m) + (x_m / y_m)) * (z_m / t_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
z_m = math.fabs(z)
t_m = math.fabs(t)
def code(x_m, y_m, z_m, t_m):
	t_1 = (z_m * z_m) / (t_m * t_m)
	tmp = 0
	if t_1 <= 2e+228:
		tmp = ((x_m / y_m) * (x_m / y_m)) + t_1
	else:
		tmp = ((z_m / t_m) + (x_m / y_m)) * (z_m / t_m)
	return tmp

Julia

x_m = abs(x)
y_m = abs(y)
z_m = abs(z)
t_m = abs(t)
function code(x_m, y_m, z_m, t_m)
	t_1 = Float64(Float64(z_m * z_m) / Float64(t_m * t_m))
	tmp = 0.0
	if (t_1 <= 2e+228)
		tmp = Float64(Float64(Float64(x_m / y_m) * Float64(x_m / y_m)) + t_1);
	else
		tmp = Float64(Float64(Float64(z_m / t_m) + Float64(x_m / y_m)) * Float64(z_m / t_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
y_m = abs(y);
z_m = abs(z);
t_m = abs(t);
function tmp_2 = code(x_m, y_m, z_m, t_m)
	t_1 = (z_m * z_m) / (t_m * t_m);
	tmp = 0.0;
	if (t_1 <= 2e+228)
		tmp = ((x_m / y_m) * (x_m / y_m)) + t_1;
	else
		tmp = ((z_m / t_m) + (x_m / y_m)) * (z_m / t_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[x$95$m_, y$95$m_, z$95$m_, t$95$m_] := Block[{t$95$1 = N[(N[(z$95$m * z$95$m), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+228], N[(N[(N[(x$95$m / y$95$m), $MachinePrecision] * N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(N[(z$95$m / t$95$m), $MachinePrecision] + N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision] * N[(z$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 1.9999999999999998e228

   1. Initial program 77.5%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + \frac{z \cdot z}{t \cdot t} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + \frac{z \cdot z}{t \cdot t} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y}} \cdot \frac{x}{y} + \frac{z \cdot z}{t \cdot t} \]

      7. lower-/.f64 96.6

         \[\leadsto \frac{x}{y} \cdot \color{blue}{\frac{x}{y}} + \frac{z \cdot z}{t \cdot t} \]

   4. Applied rewrites 96.6%

      \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + \frac{z \cdot z}{t \cdot t} \]

   if 1.9999999999999998e228 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 55.8%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing

   4. Applied rewrites 81.5%

      \[\leadsto \color{blue}{\left(\frac{z}{t} + \frac{x}{y}\right) \cdot \left(\frac{z}{t} - \frac{x}{y}\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(\frac{z}{t} + \frac{x}{y}\right) \cdot \color{blue}{\frac{z}{t}} \]
   6. Step-by-step derivation

      1. lower-/.f64 85.9

         \[\leadsto \left(\frac{z}{t} + \frac{x}{y}\right) \cdot \frac{z}{\color{blue}{t}} \]

   7. Applied rewrites 85.9%

      \[\leadsto \left(\frac{z}{t} + \frac{x}{y}\right) \cdot \color{blue}{\frac{z}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 89.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ z_m = \left|z\right| \\ t_m = \left|t\right| \\ \begin{array}{l} t_1 := \frac{z\_m \cdot z\_m}{t\_m \cdot t\_m}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-315}:\\ \;\;\;\;\frac{x\_m}{y\_m} \cdot \frac{x\_m}{y\_m}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+228}:\\ \;\;\;\;\frac{x\_m \cdot x\_m}{y\_m \cdot y\_m} + z\_m \cdot \frac{z\_m}{t\_m \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z\_m}{t\_m} + \frac{x\_m}{y\_m}\right) \cdot \frac{z\_m}{t\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
z_m = (fabs.f64 z)
t_m = (fabs.f64 t)
(FPCore (x_m y_m z_m t_m)
 :precision binary64
 (let* ((t_1 (/ (* z_m z_m) (* t_m t_m))))
   (if (<= t_1 2e-315)
     (* (/ x_m y_m) (/ x_m y_m))
     (if (<= t_1 2e+228)
       (+ (/ (* x_m x_m) (* y_m y_m)) (* z_m (/ z_m (* t_m t_m))))
       (* (+ (/ z_m t_m) (/ x_m y_m)) (/ z_m t_m))))))

C

x_m = fabs(x);
y_m = fabs(y);
z_m = fabs(z);
t_m = fabs(t);
double code(double x_m, double y_m, double z_m, double t_m) {
	double t_1 = (z_m * z_m) / (t_m * t_m);
	double tmp;
	if (t_1 <= 2e-315) {
		tmp = (x_m / y_m) * (x_m / y_m);
	} else if (t_1 <= 2e+228) {
		tmp = ((x_m * x_m) / (y_m * y_m)) + (z_m * (z_m / (t_m * t_m)));
	} else {
		tmp = ((z_m / t_m) + (x_m / y_m)) * (z_m / t_m);
	}
	return tmp;
}

Fortran

x_m =     private
y_m =     private
z_m =     private
t_m =     private
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(x_m, y_m, z_m, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z_m * z_m) / (t_m * t_m)
    if (t_1 <= 2d-315) then
        tmp = (x_m / y_m) * (x_m / y_m)
    else if (t_1 <= 2d+228) then
        tmp = ((x_m * x_m) / (y_m * y_m)) + (z_m * (z_m / (t_m * t_m)))
    else
        tmp = ((z_m / t_m) + (x_m / y_m)) * (z_m / t_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
z_m = Math.abs(z);
t_m = Math.abs(t);
public static double code(double x_m, double y_m, double z_m, double t_m) {
	double t_1 = (z_m * z_m) / (t_m * t_m);
	double tmp;
	if (t_1 <= 2e-315) {
		tmp = (x_m / y_m) * (x_m / y_m);
	} else if (t_1 <= 2e+228) {
		tmp = ((x_m * x_m) / (y_m * y_m)) + (z_m * (z_m / (t_m * t_m)));
	} else {
		tmp = ((z_m / t_m) + (x_m / y_m)) * (z_m / t_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
z_m = math.fabs(z)
t_m = math.fabs(t)
def code(x_m, y_m, z_m, t_m):
	t_1 = (z_m * z_m) / (t_m * t_m)
	tmp = 0
	if t_1 <= 2e-315:
		tmp = (x_m / y_m) * (x_m / y_m)
	elif t_1 <= 2e+228:
		tmp = ((x_m * x_m) / (y_m * y_m)) + (z_m * (z_m / (t_m * t_m)))
	else:
		tmp = ((z_m / t_m) + (x_m / y_m)) * (z_m / t_m)
	return tmp

Julia

x_m = abs(x)
y_m = abs(y)
z_m = abs(z)
t_m = abs(t)
function code(x_m, y_m, z_m, t_m)
	t_1 = Float64(Float64(z_m * z_m) / Float64(t_m * t_m))
	tmp = 0.0
	if (t_1 <= 2e-315)
		tmp = Float64(Float64(x_m / y_m) * Float64(x_m / y_m));
	elseif (t_1 <= 2e+228)
		tmp = Float64(Float64(Float64(x_m * x_m) / Float64(y_m * y_m)) + Float64(z_m * Float64(z_m / Float64(t_m * t_m))));
	else
		tmp = Float64(Float64(Float64(z_m / t_m) + Float64(x_m / y_m)) * Float64(z_m / t_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
y_m = abs(y);
z_m = abs(z);
t_m = abs(t);
function tmp_2 = code(x_m, y_m, z_m, t_m)
	t_1 = (z_m * z_m) / (t_m * t_m);
	tmp = 0.0;
	if (t_1 <= 2e-315)
		tmp = (x_m / y_m) * (x_m / y_m);
	elseif (t_1 <= 2e+228)
		tmp = ((x_m * x_m) / (y_m * y_m)) + (z_m * (z_m / (t_m * t_m)));
	else
		tmp = ((z_m / t_m) + (x_m / y_m)) * (z_m / t_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[x$95$m_, y$95$m_, z$95$m_, t$95$m_] := Block[{t$95$1 = N[(N[(z$95$m * z$95$m), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-315], N[(N[(x$95$m / y$95$m), $MachinePrecision] * N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+228], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] + N[(z$95$m * N[(z$95$m / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z$95$m / t$95$m), $MachinePrecision] + N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision] * N[(z$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 2.0000000019e-315

   1. Initial program 70.3%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z \cdot z}{t \cdot t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{\color{blue}{z \cdot z}}{t \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{\color{blue}{t \cdot t}} \]

      4. times-frac N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{t}} \cdot \frac{z}{t} \]

      7. lower-/.f64 74.6

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{z}{t} \cdot \color{blue}{\frac{z}{t}} \]

   4. Applied rewrites 74.6%

      \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y} + \frac{z}{t} \cdot \frac{z}{t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y}} + \frac{z}{t} \cdot \frac{z}{t} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{y \cdot y}} + \frac{z}{t} \cdot \frac{z}{t} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x \cdot x}{y}}{y}} + \frac{z}{t} \cdot \frac{z}{t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot x}}{y}}{y} + \frac{z}{t} \cdot \frac{z}{t} \]

      7. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x}}{y} + \frac{z}{t} \cdot \frac{z}{t} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y}} \cdot x}{y} + \frac{z}{t} \cdot \frac{z}{t} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x}}{y} + \frac{z}{t} \cdot \frac{z}{t} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{\frac{x}{y} \cdot x}{y} + \frac{z}{t} \cdot \color{blue}{\frac{z}{t}} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{\frac{x}{y} \cdot x}{y} + \color{blue}{\frac{\frac{z}{t} \cdot z}{t}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\frac{x}{y} \cdot x}{y} + \frac{\color{blue}{\frac{z}{t} \cdot z}}{t} \]

      13. frac-add N/A

          \[\leadsto \color{blue}{\frac{\left(\frac{x}{y} \cdot x\right) \cdot t + y \cdot \left(\frac{z}{t} \cdot z\right)}{y \cdot t}} \]

      14. *-commutative N/A

          \[\leadsto \frac{\color{blue}{t \cdot \left(\frac{x}{y} \cdot x\right)} + y \cdot \left(\frac{z}{t} \cdot z\right)}{y \cdot t} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{t \cdot \left(\frac{x}{y} \cdot x\right)} + y \cdot \left(\frac{z}{t} \cdot z\right)}{y \cdot t} \]

      16. *-commutative N/A

          \[\leadsto \frac{t \cdot \left(\frac{x}{y} \cdot x\right) + \color{blue}{\left(\frac{z}{t} \cdot z\right) \cdot y}}{y \cdot t} \]

      17. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{z}{t} \cdot z\right) \cdot y + t \cdot \left(\frac{x}{y} \cdot x\right)}}{y \cdot t} \]

      18. lift-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t} \cdot z, y, t \cdot \left(\frac{x}{y} \cdot x\right)\right)}}{y \cdot t} \]

   6. Applied rewrites 86.4%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\frac{z}{t} \cdot z, y, \left(\frac{x}{y} \cdot x\right) \cdot t\right)}{y}}{t}} \]

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{{x}^{2}}{{y}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{x \cdot x}{{\color{blue}{y}}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot \color{blue}{y}} \]

      3. times-frac N/A

         \[\leadsto \frac{x}{y} \cdot \color{blue}{\frac{x}{y}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{x}{y} \cdot \color{blue}{\frac{x}{y}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{x}{y} \cdot \frac{\color{blue}{x}}{y} \]

      6. lower-/.f64 95.2

         \[\leadsto \frac{x}{y} \cdot \frac{x}{\color{blue}{y}} \]

   9. Applied rewrites 95.2%

      \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} \]

   if 2.0000000019e-315 < (/.f64 (*.f64 z z) (*.f64 t t)) < 1.9999999999999998e228

   1. Initial program 92.7%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z \cdot z}{t \cdot t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{\color{blue}{z \cdot z}}{t \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{\color{blue}{t \cdot t}} \]

      4. times-frac N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{t}} \cdot \frac{z}{t} \]

      7. lower-/.f64 92.8

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{z}{t} \cdot \color{blue}{\frac{z}{t}} \]

   4. Applied rewrites 92.8%

      \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{t}} \cdot \frac{z}{t} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{z}{t} \cdot \color{blue}{\frac{z}{t}} \]

      4. frac-times N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z \cdot z}{t \cdot t}} \]

      5. sqr-neg-rev N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{\color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \left(\mathsf{neg}\left(z\right)\right)}}{t \cdot t} \]

      6. associate-/l* N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \frac{\mathsf{neg}\left(z\right)}{t \cdot t}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \frac{\mathsf{neg}\left(z\right)}{t \cdot t}} \]

      8. lower-neg.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\left(-z\right)} \cdot \frac{\mathsf{neg}\left(z\right)}{t \cdot t} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \left(-z\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(z\right)}{t \cdot t}} \]

      10. lower-neg.f64 N/A

          \[\leadsto \frac{x \cdot x}{y \cdot y} + \left(-z\right) \cdot \frac{\color{blue}{-z}}{t \cdot t} \]

      11. lower-*.f64 92.9

          \[\leadsto \frac{x \cdot x}{y \cdot y} + \left(-z\right) \cdot \frac{-z}{\color{blue}{t \cdot t}} \]

   6. Applied rewrites 92.9%

      \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\left(-z\right) \cdot \frac{-z}{t \cdot t}} \]

   if 1.9999999999999998e228 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 55.8%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing

   4. Applied rewrites 81.5%

      \[\leadsto \color{blue}{\left(\frac{z}{t} + \frac{x}{y}\right) \cdot \left(\frac{z}{t} - \frac{x}{y}\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(\frac{z}{t} + \frac{x}{y}\right) \cdot \color{blue}{\frac{z}{t}} \]
   6. Step-by-step derivation

      1. lower-/.f64 85.9

         \[\leadsto \left(\frac{z}{t} + \frac{x}{y}\right) \cdot \frac{z}{\color{blue}{t}} \]

   7. Applied rewrites 85.9%

      \[\leadsto \left(\frac{z}{t} + \frac{x}{y}\right) \cdot \color{blue}{\frac{z}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 2 \cdot 10^{-315}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;\frac{z \cdot z}{t \cdot t} \leq 2 \cdot 10^{+228}:\\ \;\;\;\;\frac{x \cdot x}{y \cdot y} + z \cdot \frac{z}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{t} + \frac{x}{y}\right) \cdot \frac{z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ z_m = \left|z\right| \\ t_m = \left|t\right| \\ \begin{array}{l} t_1 := \frac{z\_m \cdot z\_m}{t\_m \cdot t\_m}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-315}:\\ \;\;\;\;\frac{x\_m}{y\_m} \cdot \frac{x\_m}{y\_m}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+228}:\\ \;\;\;\;\frac{x\_m \cdot x\_m}{y\_m \cdot y\_m} + t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z\_m}{t\_m} + \frac{x\_m}{y\_m}\right) \cdot \frac{z\_m}{t\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
z_m = (fabs.f64 z)
t_m = (fabs.f64 t)
(FPCore (x_m y_m z_m t_m)
 :precision binary64
 (let* ((t_1 (/ (* z_m z_m) (* t_m t_m))))
   (if (<= t_1 2e-315)
     (* (/ x_m y_m) (/ x_m y_m))
     (if (<= t_1 2e+228)
       (+ (/ (* x_m x_m) (* y_m y_m)) t_1)
       (* (+ (/ z_m t_m) (/ x_m y_m)) (/ z_m t_m))))))

C

x_m = fabs(x);
y_m = fabs(y);
z_m = fabs(z);
t_m = fabs(t);
double code(double x_m, double y_m, double z_m, double t_m) {
	double t_1 = (z_m * z_m) / (t_m * t_m);
	double tmp;
	if (t_1 <= 2e-315) {
		tmp = (x_m / y_m) * (x_m / y_m);
	} else if (t_1 <= 2e+228) {
		tmp = ((x_m * x_m) / (y_m * y_m)) + t_1;
	} else {
		tmp = ((z_m / t_m) + (x_m / y_m)) * (z_m / t_m);
	}
	return tmp;
}

Fortran

x_m =     private
y_m =     private
z_m =     private
t_m =     private
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(x_m, y_m, z_m, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z_m * z_m) / (t_m * t_m)
    if (t_1 <= 2d-315) then
        tmp = (x_m / y_m) * (x_m / y_m)
    else if (t_1 <= 2d+228) then
        tmp = ((x_m * x_m) / (y_m * y_m)) + t_1
    else
        tmp = ((z_m / t_m) + (x_m / y_m)) * (z_m / t_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
z_m = Math.abs(z);
t_m = Math.abs(t);
public static double code(double x_m, double y_m, double z_m, double t_m) {
	double t_1 = (z_m * z_m) / (t_m * t_m);
	double tmp;
	if (t_1 <= 2e-315) {
		tmp = (x_m / y_m) * (x_m / y_m);
	} else if (t_1 <= 2e+228) {
		tmp = ((x_m * x_m) / (y_m * y_m)) + t_1;
	} else {
		tmp = ((z_m / t_m) + (x_m / y_m)) * (z_m / t_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
z_m = math.fabs(z)
t_m = math.fabs(t)
def code(x_m, y_m, z_m, t_m):
	t_1 = (z_m * z_m) / (t_m * t_m)
	tmp = 0
	if t_1 <= 2e-315:
		tmp = (x_m / y_m) * (x_m / y_m)
	elif t_1 <= 2e+228:
		tmp = ((x_m * x_m) / (y_m * y_m)) + t_1
	else:
		tmp = ((z_m / t_m) + (x_m / y_m)) * (z_m / t_m)
	return tmp

Julia

x_m = abs(x)
y_m = abs(y)
z_m = abs(z)
t_m = abs(t)
function code(x_m, y_m, z_m, t_m)
	t_1 = Float64(Float64(z_m * z_m) / Float64(t_m * t_m))
	tmp = 0.0
	if (t_1 <= 2e-315)
		tmp = Float64(Float64(x_m / y_m) * Float64(x_m / y_m));
	elseif (t_1 <= 2e+228)
		tmp = Float64(Float64(Float64(x_m * x_m) / Float64(y_m * y_m)) + t_1);
	else
		tmp = Float64(Float64(Float64(z_m / t_m) + Float64(x_m / y_m)) * Float64(z_m / t_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
y_m = abs(y);
z_m = abs(z);
t_m = abs(t);
function tmp_2 = code(x_m, y_m, z_m, t_m)
	t_1 = (z_m * z_m) / (t_m * t_m);
	tmp = 0.0;
	if (t_1 <= 2e-315)
		tmp = (x_m / y_m) * (x_m / y_m);
	elseif (t_1 <= 2e+228)
		tmp = ((x_m * x_m) / (y_m * y_m)) + t_1;
	else
		tmp = ((z_m / t_m) + (x_m / y_m)) * (z_m / t_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[x$95$m_, y$95$m_, z$95$m_, t$95$m_] := Block[{t$95$1 = N[(N[(z$95$m * z$95$m), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-315], N[(N[(x$95$m / y$95$m), $MachinePrecision] * N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+228], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(N[(z$95$m / t$95$m), $MachinePrecision] + N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision] * N[(z$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 2.0000000019e-315

   1. Initial program 70.3%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z \cdot z}{t \cdot t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{\color{blue}{z \cdot z}}{t \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{\color{blue}{t \cdot t}} \]

      4. times-frac N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{t}} \cdot \frac{z}{t} \]

      7. lower-/.f64 74.6

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{z}{t} \cdot \color{blue}{\frac{z}{t}} \]

   4. Applied rewrites 74.6%

      \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y} + \frac{z}{t} \cdot \frac{z}{t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y}} + \frac{z}{t} \cdot \frac{z}{t} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{y \cdot y}} + \frac{z}{t} \cdot \frac{z}{t} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x \cdot x}{y}}{y}} + \frac{z}{t} \cdot \frac{z}{t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot x}}{y}}{y} + \frac{z}{t} \cdot \frac{z}{t} \]

      7. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x}}{y} + \frac{z}{t} \cdot \frac{z}{t} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y}} \cdot x}{y} + \frac{z}{t} \cdot \frac{z}{t} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x}}{y} + \frac{z}{t} \cdot \frac{z}{t} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{\frac{x}{y} \cdot x}{y} + \frac{z}{t} \cdot \color{blue}{\frac{z}{t}} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{\frac{x}{y} \cdot x}{y} + \color{blue}{\frac{\frac{z}{t} \cdot z}{t}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\frac{x}{y} \cdot x}{y} + \frac{\color{blue}{\frac{z}{t} \cdot z}}{t} \]

      13. frac-add N/A

          \[\leadsto \color{blue}{\frac{\left(\frac{x}{y} \cdot x\right) \cdot t + y \cdot \left(\frac{z}{t} \cdot z\right)}{y \cdot t}} \]

      14. *-commutative N/A

          \[\leadsto \frac{\color{blue}{t \cdot \left(\frac{x}{y} \cdot x\right)} + y \cdot \left(\frac{z}{t} \cdot z\right)}{y \cdot t} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{t \cdot \left(\frac{x}{y} \cdot x\right)} + y \cdot \left(\frac{z}{t} \cdot z\right)}{y \cdot t} \]

      16. *-commutative N/A

          \[\leadsto \frac{t \cdot \left(\frac{x}{y} \cdot x\right) + \color{blue}{\left(\frac{z}{t} \cdot z\right) \cdot y}}{y \cdot t} \]

      17. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{z}{t} \cdot z\right) \cdot y + t \cdot \left(\frac{x}{y} \cdot x\right)}}{y \cdot t} \]

      18. lift-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t} \cdot z, y, t \cdot \left(\frac{x}{y} \cdot x\right)\right)}}{y \cdot t} \]

   6. Applied rewrites 86.4%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\frac{z}{t} \cdot z, y, \left(\frac{x}{y} \cdot x\right) \cdot t\right)}{y}}{t}} \]

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{{x}^{2}}{{y}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{x \cdot x}{{\color{blue}{y}}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot \color{blue}{y}} \]

      3. times-frac N/A

         \[\leadsto \frac{x}{y} \cdot \color{blue}{\frac{x}{y}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{x}{y} \cdot \color{blue}{\frac{x}{y}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{x}{y} \cdot \frac{\color{blue}{x}}{y} \]

      6. lower-/.f64 95.2

         \[\leadsto \frac{x}{y} \cdot \frac{x}{\color{blue}{y}} \]

   9. Applied rewrites 95.2%

      \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} \]

   if 2.0000000019e-315 < (/.f64 (*.f64 z z) (*.f64 t t)) < 1.9999999999999998e228

   1. Initial program 92.7%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing

   if 1.9999999999999998e228 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 55.8%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing

   4. Applied rewrites 81.5%

      \[\leadsto \color{blue}{\left(\frac{z}{t} + \frac{x}{y}\right) \cdot \left(\frac{z}{t} - \frac{x}{y}\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(\frac{z}{t} + \frac{x}{y}\right) \cdot \color{blue}{\frac{z}{t}} \]
   6. Step-by-step derivation

      1. lower-/.f64 85.9

         \[\leadsto \left(\frac{z}{t} + \frac{x}{y}\right) \cdot \frac{z}{\color{blue}{t}} \]

   7. Applied rewrites 85.9%

      \[\leadsto \left(\frac{z}{t} + \frac{x}{y}\right) \cdot \color{blue}{\frac{z}{t}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 86.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ z_m = \left|z\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{z\_m \cdot z\_m}{t\_m \cdot t\_m} \leq 5 \cdot 10^{-285}:\\ \;\;\;\;\frac{x\_m}{y\_m} \cdot \frac{x\_m}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z\_m}{t\_m} + \frac{x\_m}{y\_m}\right) \cdot \frac{z\_m}{t\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
z_m = (fabs.f64 z)
t_m = (fabs.f64 t)
(FPCore (x_m y_m z_m t_m)
 :precision binary64
 (if (<= (/ (* z_m z_m) (* t_m t_m)) 5e-285)
   (* (/ x_m y_m) (/ x_m y_m))
   (* (+ (/ z_m t_m) (/ x_m y_m)) (/ z_m t_m))))

C

x_m = fabs(x);
y_m = fabs(y);
z_m = fabs(z);
t_m = fabs(t);
double code(double x_m, double y_m, double z_m, double t_m) {
	double tmp;
	if (((z_m * z_m) / (t_m * t_m)) <= 5e-285) {
		tmp = (x_m / y_m) * (x_m / y_m);
	} else {
		tmp = ((z_m / t_m) + (x_m / y_m)) * (z_m / t_m);
	}
	return tmp;
}

Fortran

x_m =     private
y_m =     private
z_m =     private
t_m =     private
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(x_m, y_m, z_m, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (((z_m * z_m) / (t_m * t_m)) <= 5d-285) then
        tmp = (x_m / y_m) * (x_m / y_m)
    else
        tmp = ((z_m / t_m) + (x_m / y_m)) * (z_m / t_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
z_m = Math.abs(z);
t_m = Math.abs(t);
public static double code(double x_m, double y_m, double z_m, double t_m) {
	double tmp;
	if (((z_m * z_m) / (t_m * t_m)) <= 5e-285) {
		tmp = (x_m / y_m) * (x_m / y_m);
	} else {
		tmp = ((z_m / t_m) + (x_m / y_m)) * (z_m / t_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
z_m = math.fabs(z)
t_m = math.fabs(t)
def code(x_m, y_m, z_m, t_m):
	tmp = 0
	if ((z_m * z_m) / (t_m * t_m)) <= 5e-285:
		tmp = (x_m / y_m) * (x_m / y_m)
	else:
		tmp = ((z_m / t_m) + (x_m / y_m)) * (z_m / t_m)
	return tmp

Julia

x_m = abs(x)
y_m = abs(y)
z_m = abs(z)
t_m = abs(t)
function code(x_m, y_m, z_m, t_m)
	tmp = 0.0
	if (Float64(Float64(z_m * z_m) / Float64(t_m * t_m)) <= 5e-285)
		tmp = Float64(Float64(x_m / y_m) * Float64(x_m / y_m));
	else
		tmp = Float64(Float64(Float64(z_m / t_m) + Float64(x_m / y_m)) * Float64(z_m / t_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
y_m = abs(y);
z_m = abs(z);
t_m = abs(t);
function tmp_2 = code(x_m, y_m, z_m, t_m)
	tmp = 0.0;
	if (((z_m * z_m) / (t_m * t_m)) <= 5e-285)
		tmp = (x_m / y_m) * (x_m / y_m);
	else
		tmp = ((z_m / t_m) + (x_m / y_m)) * (z_m / t_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[x$95$m_, y$95$m_, z$95$m_, t$95$m_] := If[LessEqual[N[(N[(z$95$m * z$95$m), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision], 5e-285], N[(N[(x$95$m / y$95$m), $MachinePrecision] * N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z$95$m / t$95$m), $MachinePrecision] + N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision] * N[(z$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 5.00000000000000018e-285

   1. Initial program 70.6%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z \cdot z}{t \cdot t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{\color{blue}{z \cdot z}}{t \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{\color{blue}{t \cdot t}} \]

      4. times-frac N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{t}} \cdot \frac{z}{t} \]

      7. lower-/.f64 74.9

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{z}{t} \cdot \color{blue}{\frac{z}{t}} \]

   4. Applied rewrites 74.9%

      \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y} + \frac{z}{t} \cdot \frac{z}{t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y}} + \frac{z}{t} \cdot \frac{z}{t} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{y \cdot y}} + \frac{z}{t} \cdot \frac{z}{t} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x \cdot x}{y}}{y}} + \frac{z}{t} \cdot \frac{z}{t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot x}}{y}}{y} + \frac{z}{t} \cdot \frac{z}{t} \]

      7. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x}}{y} + \frac{z}{t} \cdot \frac{z}{t} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y}} \cdot x}{y} + \frac{z}{t} \cdot \frac{z}{t} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x}}{y} + \frac{z}{t} \cdot \frac{z}{t} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{\frac{x}{y} \cdot x}{y} + \frac{z}{t} \cdot \color{blue}{\frac{z}{t}} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{\frac{x}{y} \cdot x}{y} + \color{blue}{\frac{\frac{z}{t} \cdot z}{t}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\frac{x}{y} \cdot x}{y} + \frac{\color{blue}{\frac{z}{t} \cdot z}}{t} \]

      13. frac-add N/A

          \[\leadsto \color{blue}{\frac{\left(\frac{x}{y} \cdot x\right) \cdot t + y \cdot \left(\frac{z}{t} \cdot z\right)}{y \cdot t}} \]

      14. *-commutative N/A

          \[\leadsto \frac{\color{blue}{t \cdot \left(\frac{x}{y} \cdot x\right)} + y \cdot \left(\frac{z}{t} \cdot z\right)}{y \cdot t} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{t \cdot \left(\frac{x}{y} \cdot x\right)} + y \cdot \left(\frac{z}{t} \cdot z\right)}{y \cdot t} \]

      16. *-commutative N/A

          \[\leadsto \frac{t \cdot \left(\frac{x}{y} \cdot x\right) + \color{blue}{\left(\frac{z}{t} \cdot z\right) \cdot y}}{y \cdot t} \]

      17. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{z}{t} \cdot z\right) \cdot y + t \cdot \left(\frac{x}{y} \cdot x\right)}}{y \cdot t} \]

      18. lift-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t} \cdot z, y, t \cdot \left(\frac{x}{y} \cdot x\right)\right)}}{y \cdot t} \]

   6. Applied rewrites 86.5%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\frac{z}{t} \cdot z, y, \left(\frac{x}{y} \cdot x\right) \cdot t\right)}{y}}{t}} \]

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{{x}^{2}}{{y}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{x \cdot x}{{\color{blue}{y}}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot \color{blue}{y}} \]

      3. times-frac N/A

         \[\leadsto \frac{x}{y} \cdot \color{blue}{\frac{x}{y}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{x}{y} \cdot \color{blue}{\frac{x}{y}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{x}{y} \cdot \frac{\color{blue}{x}}{y} \]

      6. lower-/.f64 95.3

         \[\leadsto \frac{x}{y} \cdot \frac{x}{\color{blue}{y}} \]

   9. Applied rewrites 95.3%

      \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} \]

   if 5.00000000000000018e-285 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 65.5%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing

   4. Applied rewrites 77.0%

      \[\leadsto \color{blue}{\left(\frac{z}{t} + \frac{x}{y}\right) \cdot \left(\frac{z}{t} - \frac{x}{y}\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(\frac{z}{t} + \frac{x}{y}\right) \cdot \color{blue}{\frac{z}{t}} \]
   6. Step-by-step derivation

      1. lower-/.f64 81.9

         \[\leadsto \left(\frac{z}{t} + \frac{x}{y}\right) \cdot \frac{z}{\color{blue}{t}} \]

   7. Applied rewrites 81.9%

      \[\leadsto \left(\frac{z}{t} + \frac{x}{y}\right) \cdot \color{blue}{\frac{z}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 80.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ z_m = \left|z\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{z\_m \cdot z\_m}{t\_m \cdot t\_m} \leq 5 \cdot 10^{-285}:\\ \;\;\;\;\frac{x\_m}{y\_m} \cdot \frac{x\_m}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{z\_m}{t\_m} \cdot \frac{z\_m}{t\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
z_m = (fabs.f64 z)
t_m = (fabs.f64 t)
(FPCore (x_m y_m z_m t_m)
 :precision binary64
 (if (<= (/ (* z_m z_m) (* t_m t_m)) 5e-285)
   (* (/ x_m y_m) (/ x_m y_m))
   (* (/ z_m t_m) (/ z_m t_m))))

C

x_m = fabs(x);
y_m = fabs(y);
z_m = fabs(z);
t_m = fabs(t);
double code(double x_m, double y_m, double z_m, double t_m) {
	double tmp;
	if (((z_m * z_m) / (t_m * t_m)) <= 5e-285) {
		tmp = (x_m / y_m) * (x_m / y_m);
	} else {
		tmp = (z_m / t_m) * (z_m / t_m);
	}
	return tmp;
}

Fortran

x_m =     private
y_m =     private
z_m =     private
t_m =     private
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(x_m, y_m, z_m, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (((z_m * z_m) / (t_m * t_m)) <= 5d-285) then
        tmp = (x_m / y_m) * (x_m / y_m)
    else
        tmp = (z_m / t_m) * (z_m / t_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
z_m = Math.abs(z);
t_m = Math.abs(t);
public static double code(double x_m, double y_m, double z_m, double t_m) {
	double tmp;
	if (((z_m * z_m) / (t_m * t_m)) <= 5e-285) {
		tmp = (x_m / y_m) * (x_m / y_m);
	} else {
		tmp = (z_m / t_m) * (z_m / t_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
z_m = math.fabs(z)
t_m = math.fabs(t)
def code(x_m, y_m, z_m, t_m):
	tmp = 0
	if ((z_m * z_m) / (t_m * t_m)) <= 5e-285:
		tmp = (x_m / y_m) * (x_m / y_m)
	else:
		tmp = (z_m / t_m) * (z_m / t_m)
	return tmp

Julia

x_m = abs(x)
y_m = abs(y)
z_m = abs(z)
t_m = abs(t)
function code(x_m, y_m, z_m, t_m)
	tmp = 0.0
	if (Float64(Float64(z_m * z_m) / Float64(t_m * t_m)) <= 5e-285)
		tmp = Float64(Float64(x_m / y_m) * Float64(x_m / y_m));
	else
		tmp = Float64(Float64(z_m / t_m) * Float64(z_m / t_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
y_m = abs(y);
z_m = abs(z);
t_m = abs(t);
function tmp_2 = code(x_m, y_m, z_m, t_m)
	tmp = 0.0;
	if (((z_m * z_m) / (t_m * t_m)) <= 5e-285)
		tmp = (x_m / y_m) * (x_m / y_m);
	else
		tmp = (z_m / t_m) * (z_m / t_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[x$95$m_, y$95$m_, z$95$m_, t$95$m_] := If[LessEqual[N[(N[(z$95$m * z$95$m), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision], 5e-285], N[(N[(x$95$m / y$95$m), $MachinePrecision] * N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(z$95$m / t$95$m), $MachinePrecision] * N[(z$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 5.00000000000000018e-285

   1. Initial program 70.6%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z \cdot z}{t \cdot t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{\color{blue}{z \cdot z}}{t \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{\color{blue}{t \cdot t}} \]

      4. times-frac N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{t}} \cdot \frac{z}{t} \]

      7. lower-/.f64 74.9

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{z}{t} \cdot \color{blue}{\frac{z}{t}} \]

   4. Applied rewrites 74.9%

      \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y} + \frac{z}{t} \cdot \frac{z}{t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y}} + \frac{z}{t} \cdot \frac{z}{t} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{y \cdot y}} + \frac{z}{t} \cdot \frac{z}{t} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x \cdot x}{y}}{y}} + \frac{z}{t} \cdot \frac{z}{t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot x}}{y}}{y} + \frac{z}{t} \cdot \frac{z}{t} \]

      7. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x}}{y} + \frac{z}{t} \cdot \frac{z}{t} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y}} \cdot x}{y} + \frac{z}{t} \cdot \frac{z}{t} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x}}{y} + \frac{z}{t} \cdot \frac{z}{t} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{\frac{x}{y} \cdot x}{y} + \frac{z}{t} \cdot \color{blue}{\frac{z}{t}} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{\frac{x}{y} \cdot x}{y} + \color{blue}{\frac{\frac{z}{t} \cdot z}{t}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\frac{x}{y} \cdot x}{y} + \frac{\color{blue}{\frac{z}{t} \cdot z}}{t} \]

      13. frac-add N/A

          \[\leadsto \color{blue}{\frac{\left(\frac{x}{y} \cdot x\right) \cdot t + y \cdot \left(\frac{z}{t} \cdot z\right)}{y \cdot t}} \]

      14. *-commutative N/A

          \[\leadsto \frac{\color{blue}{t \cdot \left(\frac{x}{y} \cdot x\right)} + y \cdot \left(\frac{z}{t} \cdot z\right)}{y \cdot t} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{t \cdot \left(\frac{x}{y} \cdot x\right)} + y \cdot \left(\frac{z}{t} \cdot z\right)}{y \cdot t} \]

      16. *-commutative N/A

          \[\leadsto \frac{t \cdot \left(\frac{x}{y} \cdot x\right) + \color{blue}{\left(\frac{z}{t} \cdot z\right) \cdot y}}{y \cdot t} \]

      17. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{z}{t} \cdot z\right) \cdot y + t \cdot \left(\frac{x}{y} \cdot x\right)}}{y \cdot t} \]

      18. lift-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t} \cdot z, y, t \cdot \left(\frac{x}{y} \cdot x\right)\right)}}{y \cdot t} \]

   6. Applied rewrites 86.5%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\frac{z}{t} \cdot z, y, \left(\frac{x}{y} \cdot x\right) \cdot t\right)}{y}}{t}} \]

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{{x}^{2}}{{y}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{x \cdot x}{{\color{blue}{y}}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot \color{blue}{y}} \]

      3. times-frac N/A

         \[\leadsto \frac{x}{y} \cdot \color{blue}{\frac{x}{y}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{x}{y} \cdot \color{blue}{\frac{x}{y}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{x}{y} \cdot \frac{\color{blue}{x}}{y} \]

      6. lower-/.f64 95.3

         \[\leadsto \frac{x}{y} \cdot \frac{x}{\color{blue}{y}} \]

   9. Applied rewrites 95.3%

      \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} \]

   if 5.00000000000000018e-285 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 65.5%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{z \cdot z}{t \cdot t} + \frac{x \cdot x}{y \cdot y}} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z \cdot z}{t \cdot t}} + \frac{x \cdot x}{y \cdot y} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{z \cdot z}{\color{blue}{t \cdot t}} + \frac{x \cdot x}{y \cdot y} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{z \cdot z}{t}}{t}} + \frac{x \cdot x}{y \cdot y} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\frac{z \cdot z}{t}}{t} + \color{blue}{\frac{x \cdot x}{y \cdot y}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{z \cdot z}{t}}{t} + \frac{x \cdot x}{\color{blue}{y \cdot y}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{\frac{z \cdot z}{t}}{t} + \color{blue}{\frac{\frac{x \cdot x}{y}}{y}} \]

      9. frac-add N/A

         \[\leadsto \color{blue}{\frac{\frac{z \cdot z}{t} \cdot y + t \cdot \frac{x \cdot x}{y}}{t \cdot y}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{z \cdot z}{t} \cdot y + t \cdot \frac{x \cdot x}{y}}{t \cdot y}} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z \cdot z}{t}, y, t \cdot \frac{x \cdot x}{y}\right)}}{t \cdot y} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{z \cdot z}}{t}, y, t \cdot \frac{x \cdot x}{y}\right)}{t \cdot y} \]

      13. associate-*l/ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{z}{t} \cdot z}, y, t \cdot \frac{x \cdot x}{y}\right)}{t \cdot y} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{z}{t} \cdot z}, y, t \cdot \frac{x \cdot x}{y}\right)}{t \cdot y} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{z}{t}} \cdot z, y, t \cdot \frac{x \cdot x}{y}\right)}{t \cdot y} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t} \cdot z, y, \color{blue}{t \cdot \frac{x \cdot x}{y}}\right)}{t \cdot y} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t} \cdot z, y, t \cdot \frac{\color{blue}{x \cdot x}}{y}\right)}{t \cdot y} \]

      18. associate-*l/ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t} \cdot z, y, t \cdot \color{blue}{\left(\frac{x}{y} \cdot x\right)}\right)}{t \cdot y} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t} \cdot z, y, t \cdot \color{blue}{\left(\frac{x}{y} \cdot x\right)}\right)}{t \cdot y} \]

      20. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t} \cdot z, y, t \cdot \left(\color{blue}{\frac{x}{y}} \cdot x\right)\right)}{t \cdot y} \]

      21. lower-*.f64 82.1

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t} \cdot z, y, t \cdot \left(\frac{x}{y} \cdot x\right)\right)}{\color{blue}{t \cdot y}} \]

   4. Applied rewrites 82.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{z}{t} \cdot z, y, t \cdot \left(\frac{x}{y} \cdot x\right)\right)}{t \cdot y}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{z}{t} \cdot z\right) \cdot y + t \cdot \left(\frac{x}{y} \cdot x\right)}}{t \cdot y} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{z}{t} \cdot z\right)} + t \cdot \left(\frac{x}{y} \cdot x\right)}{t \cdot y} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(\frac{z}{t} \cdot z\right)} + t \cdot \left(\frac{x}{y} \cdot x\right)}{t \cdot y} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot \frac{z}{t}\right) \cdot z} + t \cdot \left(\frac{x}{y} \cdot x\right)}{t \cdot y} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y \cdot \frac{z}{t}, z, t \cdot \left(\frac{x}{y} \cdot x\right)\right)}}{t \cdot y} \]

      6. lower-*.f64 84.4

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{y \cdot \frac{z}{t}}, z, t \cdot \left(\frac{x}{y} \cdot x\right)\right)}{t \cdot y} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y \cdot \frac{z}{t}, z, \color{blue}{t \cdot \left(\frac{x}{y} \cdot x\right)}\right)}{t \cdot y} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y \cdot \frac{z}{t}, z, \color{blue}{\left(\frac{x}{y} \cdot x\right) \cdot t}\right)}{t \cdot y} \]

      9. lower-*.f64 84.4

         \[\leadsto \frac{\mathsf{fma}\left(y \cdot \frac{z}{t}, z, \color{blue}{\left(\frac{x}{y} \cdot x\right) \cdot t}\right)}{t \cdot y} \]

   6. Applied rewrites 84.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y \cdot \frac{z}{t}, z, \left(\frac{x}{y} \cdot x\right) \cdot t\right)}}{t \cdot y} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}}} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{{\color{blue}{z}}^{2}}{{t}^{2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{{z}^{2}}{{t}^{2}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{{z}^{2}}{{t}^{2}} \]

      4. *-commutative N/A

         \[\leadsto \frac{{z}^{2}}{{t}^{2}} \]

      5. associate-*l* N/A

         \[\leadsto \frac{{z}^{2}}{{t}^{2}} \]

      6. +-commutative N/A

         \[\leadsto \frac{{\color{blue}{z}}^{2}}{{t}^{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{{z}^{\color{blue}{2}}}{{t}^{2}} \]

      8. associate-/l/ N/A

         \[\leadsto \frac{\color{blue}{{z}^{2}}}{{t}^{2}} \]

   9. Applied rewrites 81.5%

      \[\leadsto \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 5 \cdot 10^{-285}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ z_m = \left|z\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{z\_m \cdot z\_m}{t\_m \cdot t\_m} \leq 5 \cdot 10^{-285}:\\ \;\;\;\;\frac{\frac{x\_m}{y\_m}}{y\_m} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{z\_m}{t\_m} \cdot \frac{z\_m}{t\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
z_m = (fabs.f64 z)
t_m = (fabs.f64 t)
(FPCore (x_m y_m z_m t_m)
 :precision binary64
 (if (<= (/ (* z_m z_m) (* t_m t_m)) 5e-285)
   (* (/ (/ x_m y_m) y_m) x_m)
   (* (/ z_m t_m) (/ z_m t_m))))

C

x_m = fabs(x);
y_m = fabs(y);
z_m = fabs(z);
t_m = fabs(t);
double code(double x_m, double y_m, double z_m, double t_m) {
	double tmp;
	if (((z_m * z_m) / (t_m * t_m)) <= 5e-285) {
		tmp = ((x_m / y_m) / y_m) * x_m;
	} else {
		tmp = (z_m / t_m) * (z_m / t_m);
	}
	return tmp;
}

Fortran

x_m =     private
y_m =     private
z_m =     private
t_m =     private
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(x_m, y_m, z_m, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (((z_m * z_m) / (t_m * t_m)) <= 5d-285) then
        tmp = ((x_m / y_m) / y_m) * x_m
    else
        tmp = (z_m / t_m) * (z_m / t_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
z_m = Math.abs(z);
t_m = Math.abs(t);
public static double code(double x_m, double y_m, double z_m, double t_m) {
	double tmp;
	if (((z_m * z_m) / (t_m * t_m)) <= 5e-285) {
		tmp = ((x_m / y_m) / y_m) * x_m;
	} else {
		tmp = (z_m / t_m) * (z_m / t_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
z_m = math.fabs(z)
t_m = math.fabs(t)
def code(x_m, y_m, z_m, t_m):
	tmp = 0
	if ((z_m * z_m) / (t_m * t_m)) <= 5e-285:
		tmp = ((x_m / y_m) / y_m) * x_m
	else:
		tmp = (z_m / t_m) * (z_m / t_m)
	return tmp

Julia

x_m = abs(x)
y_m = abs(y)
z_m = abs(z)
t_m = abs(t)
function code(x_m, y_m, z_m, t_m)
	tmp = 0.0
	if (Float64(Float64(z_m * z_m) / Float64(t_m * t_m)) <= 5e-285)
		tmp = Float64(Float64(Float64(x_m / y_m) / y_m) * x_m);
	else
		tmp = Float64(Float64(z_m / t_m) * Float64(z_m / t_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
y_m = abs(y);
z_m = abs(z);
t_m = abs(t);
function tmp_2 = code(x_m, y_m, z_m, t_m)
	tmp = 0.0;
	if (((z_m * z_m) / (t_m * t_m)) <= 5e-285)
		tmp = ((x_m / y_m) / y_m) * x_m;
	else
		tmp = (z_m / t_m) * (z_m / t_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[x$95$m_, y$95$m_, z$95$m_, t$95$m_] := If[LessEqual[N[(N[(z$95$m * z$95$m), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision], 5e-285], N[(N[(N[(x$95$m / y$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(z$95$m / t$95$m), $MachinePrecision] * N[(z$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 5.00000000000000018e-285

   1. Initial program 70.6%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{{x}^{2}}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{x \cdot x}{{\color{blue}{y}}^{2}} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{x}{{y}^{2}} \cdot \color{blue}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{{y}^{2}} \cdot \color{blue}{x} \]

      4. unpow2 N/A

         \[\leadsto \frac{x}{y \cdot y} \cdot x \]

      5. associate-/r* N/A

         \[\leadsto \frac{\frac{x}{y}}{y} \cdot x \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y}}{y} \cdot x \]

      7. lower-/.f64 87.4

         \[\leadsto \frac{\frac{x}{y}}{y} \cdot x \]

   5. Applied rewrites 87.4%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y} \cdot x} \]

   if 5.00000000000000018e-285 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 65.5%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{z \cdot z}{t \cdot t} + \frac{x \cdot x}{y \cdot y}} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z \cdot z}{t \cdot t}} + \frac{x \cdot x}{y \cdot y} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{z \cdot z}{\color{blue}{t \cdot t}} + \frac{x \cdot x}{y \cdot y} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{z \cdot z}{t}}{t}} + \frac{x \cdot x}{y \cdot y} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\frac{z \cdot z}{t}}{t} + \color{blue}{\frac{x \cdot x}{y \cdot y}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{z \cdot z}{t}}{t} + \frac{x \cdot x}{\color{blue}{y \cdot y}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{\frac{z \cdot z}{t}}{t} + \color{blue}{\frac{\frac{x \cdot x}{y}}{y}} \]

      9. frac-add N/A

         \[\leadsto \color{blue}{\frac{\frac{z \cdot z}{t} \cdot y + t \cdot \frac{x \cdot x}{y}}{t \cdot y}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{z \cdot z}{t} \cdot y + t \cdot \frac{x \cdot x}{y}}{t \cdot y}} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z \cdot z}{t}, y, t \cdot \frac{x \cdot x}{y}\right)}}{t \cdot y} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{z \cdot z}}{t}, y, t \cdot \frac{x \cdot x}{y}\right)}{t \cdot y} \]

      13. associate-*l/ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{z}{t} \cdot z}, y, t \cdot \frac{x \cdot x}{y}\right)}{t \cdot y} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{z}{t} \cdot z}, y, t \cdot \frac{x \cdot x}{y}\right)}{t \cdot y} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{z}{t}} \cdot z, y, t \cdot \frac{x \cdot x}{y}\right)}{t \cdot y} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t} \cdot z, y, \color{blue}{t \cdot \frac{x \cdot x}{y}}\right)}{t \cdot y} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t} \cdot z, y, t \cdot \frac{\color{blue}{x \cdot x}}{y}\right)}{t \cdot y} \]

      18. associate-*l/ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t} \cdot z, y, t \cdot \color{blue}{\left(\frac{x}{y} \cdot x\right)}\right)}{t \cdot y} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t} \cdot z, y, t \cdot \color{blue}{\left(\frac{x}{y} \cdot x\right)}\right)}{t \cdot y} \]

      20. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t} \cdot z, y, t \cdot \left(\color{blue}{\frac{x}{y}} \cdot x\right)\right)}{t \cdot y} \]

      21. lower-*.f64 82.1

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t} \cdot z, y, t \cdot \left(\frac{x}{y} \cdot x\right)\right)}{\color{blue}{t \cdot y}} \]

   4. Applied rewrites 82.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{z}{t} \cdot z, y, t \cdot \left(\frac{x}{y} \cdot x\right)\right)}{t \cdot y}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{z}{t} \cdot z\right) \cdot y + t \cdot \left(\frac{x}{y} \cdot x\right)}}{t \cdot y} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{z}{t} \cdot z\right)} + t \cdot \left(\frac{x}{y} \cdot x\right)}{t \cdot y} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(\frac{z}{t} \cdot z\right)} + t \cdot \left(\frac{x}{y} \cdot x\right)}{t \cdot y} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot \frac{z}{t}\right) \cdot z} + t \cdot \left(\frac{x}{y} \cdot x\right)}{t \cdot y} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y \cdot \frac{z}{t}, z, t \cdot \left(\frac{x}{y} \cdot x\right)\right)}}{t \cdot y} \]

      6. lower-*.f64 84.4

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{y \cdot \frac{z}{t}}, z, t \cdot \left(\frac{x}{y} \cdot x\right)\right)}{t \cdot y} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y \cdot \frac{z}{t}, z, \color{blue}{t \cdot \left(\frac{x}{y} \cdot x\right)}\right)}{t \cdot y} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y \cdot \frac{z}{t}, z, \color{blue}{\left(\frac{x}{y} \cdot x\right) \cdot t}\right)}{t \cdot y} \]

      9. lower-*.f64 84.4

         \[\leadsto \frac{\mathsf{fma}\left(y \cdot \frac{z}{t}, z, \color{blue}{\left(\frac{x}{y} \cdot x\right) \cdot t}\right)}{t \cdot y} \]

   6. Applied rewrites 84.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y \cdot \frac{z}{t}, z, \left(\frac{x}{y} \cdot x\right) \cdot t\right)}}{t \cdot y} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}}} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{{\color{blue}{z}}^{2}}{{t}^{2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{{z}^{2}}{{t}^{2}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{{z}^{2}}{{t}^{2}} \]

      4. *-commutative N/A

         \[\leadsto \frac{{z}^{2}}{{t}^{2}} \]

      5. associate-*l* N/A

         \[\leadsto \frac{{z}^{2}}{{t}^{2}} \]

      6. +-commutative N/A

         \[\leadsto \frac{{\color{blue}{z}}^{2}}{{t}^{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{{z}^{\color{blue}{2}}}{{t}^{2}} \]

      8. associate-/l/ N/A

         \[\leadsto \frac{\color{blue}{{z}^{2}}}{{t}^{2}} \]

   9. Applied rewrites 81.5%

      \[\leadsto \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 5 \cdot 10^{-285}:\\ \;\;\;\;\frac{\frac{x}{y}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ z_m = \left|z\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{z\_m \cdot z\_m}{t\_m \cdot t\_m} \leq 5 \cdot 10^{-285}:\\ \;\;\;\;\frac{\frac{x\_m}{y\_m}}{y\_m} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{z\_m}{t\_m}}{t\_m} \cdot z\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
z_m = (fabs.f64 z)
t_m = (fabs.f64 t)
(FPCore (x_m y_m z_m t_m)
 :precision binary64
 (if (<= (/ (* z_m z_m) (* t_m t_m)) 5e-285)
   (* (/ (/ x_m y_m) y_m) x_m)
   (* (/ (/ z_m t_m) t_m) z_m)))

C

x_m = fabs(x);
y_m = fabs(y);
z_m = fabs(z);
t_m = fabs(t);
double code(double x_m, double y_m, double z_m, double t_m) {
	double tmp;
	if (((z_m * z_m) / (t_m * t_m)) <= 5e-285) {
		tmp = ((x_m / y_m) / y_m) * x_m;
	} else {
		tmp = ((z_m / t_m) / t_m) * z_m;
	}
	return tmp;
}

Fortran

x_m =     private
y_m =     private
z_m =     private
t_m =     private
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(x_m, y_m, z_m, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (((z_m * z_m) / (t_m * t_m)) <= 5d-285) then
        tmp = ((x_m / y_m) / y_m) * x_m
    else
        tmp = ((z_m / t_m) / t_m) * z_m
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
z_m = Math.abs(z);
t_m = Math.abs(t);
public static double code(double x_m, double y_m, double z_m, double t_m) {
	double tmp;
	if (((z_m * z_m) / (t_m * t_m)) <= 5e-285) {
		tmp = ((x_m / y_m) / y_m) * x_m;
	} else {
		tmp = ((z_m / t_m) / t_m) * z_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
z_m = math.fabs(z)
t_m = math.fabs(t)
def code(x_m, y_m, z_m, t_m):
	tmp = 0
	if ((z_m * z_m) / (t_m * t_m)) <= 5e-285:
		tmp = ((x_m / y_m) / y_m) * x_m
	else:
		tmp = ((z_m / t_m) / t_m) * z_m
	return tmp

Julia

x_m = abs(x)
y_m = abs(y)
z_m = abs(z)
t_m = abs(t)
function code(x_m, y_m, z_m, t_m)
	tmp = 0.0
	if (Float64(Float64(z_m * z_m) / Float64(t_m * t_m)) <= 5e-285)
		tmp = Float64(Float64(Float64(x_m / y_m) / y_m) * x_m);
	else
		tmp = Float64(Float64(Float64(z_m / t_m) / t_m) * z_m);
	end
	return tmp
end

MATLAB

x_m = abs(x);
y_m = abs(y);
z_m = abs(z);
t_m = abs(t);
function tmp_2 = code(x_m, y_m, z_m, t_m)
	tmp = 0.0;
	if (((z_m * z_m) / (t_m * t_m)) <= 5e-285)
		tmp = ((x_m / y_m) / y_m) * x_m;
	else
		tmp = ((z_m / t_m) / t_m) * z_m;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[x$95$m_, y$95$m_, z$95$m_, t$95$m_] := If[LessEqual[N[(N[(z$95$m * z$95$m), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision], 5e-285], N[(N[(N[(x$95$m / y$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(z$95$m / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * z$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 5.00000000000000018e-285

   1. Initial program 70.6%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{{x}^{2}}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{x \cdot x}{{\color{blue}{y}}^{2}} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{x}{{y}^{2}} \cdot \color{blue}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{{y}^{2}} \cdot \color{blue}{x} \]

      4. unpow2 N/A

         \[\leadsto \frac{x}{y \cdot y} \cdot x \]

      5. associate-/r* N/A

         \[\leadsto \frac{\frac{x}{y}}{y} \cdot x \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y}}{y} \cdot x \]

      7. lower-/.f64 87.4

         \[\leadsto \frac{\frac{x}{y}}{y} \cdot x \]

   5. Applied rewrites 87.4%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y} \cdot x} \]

   if 5.00000000000000018e-285 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 65.5%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{z \cdot z}{{\color{blue}{t}}^{2}} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{z}{{t}^{2}} \cdot \color{blue}{z} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{z}{{t}^{2}} \cdot \color{blue}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{z}{t \cdot t} \cdot z \]

      5. associate-/r* N/A

         \[\leadsto \frac{\frac{z}{t}}{t} \cdot z \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\frac{z}{t}}{t} \cdot z \]

      7. lower-/.f64 78.7

         \[\leadsto \frac{\frac{z}{t}}{t} \cdot z \]

   5. Applied rewrites 78.7%

      \[\leadsto \color{blue}{\frac{\frac{z}{t}}{t} \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 5 \cdot 10^{-285}:\\ \;\;\;\;\frac{\frac{x}{y}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{z}{t}}{t} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ z_m = \left|z\right| \\ t_m = \left|t\right| \\ \frac{\frac{x\_m}{y\_m}}{y\_m} \cdot x\_m \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
z_m = (fabs.f64 z)
t_m = (fabs.f64 t)
(FPCore (x_m y_m z_m t_m) :precision binary64 (* (/ (/ x_m y_m) y_m) x_m))

C

x_m = fabs(x);
y_m = fabs(y);
z_m = fabs(z);
t_m = fabs(t);
double code(double x_m, double y_m, double z_m, double t_m) {
	return ((x_m / y_m) / y_m) * x_m;
}

Fortran

x_m =     private
y_m =     private
z_m =     private
t_m =     private
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(x_m, y_m, z_m, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t_m
    code = ((x_m / y_m) / y_m) * x_m
end function

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
z_m = Math.abs(z);
t_m = Math.abs(t);
public static double code(double x_m, double y_m, double z_m, double t_m) {
	return ((x_m / y_m) / y_m) * x_m;
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
z_m = math.fabs(z)
t_m = math.fabs(t)
def code(x_m, y_m, z_m, t_m):
	return ((x_m / y_m) / y_m) * x_m

Julia

x_m = abs(x)
y_m = abs(y)
z_m = abs(z)
t_m = abs(t)
function code(x_m, y_m, z_m, t_m)
	return Float64(Float64(Float64(x_m / y_m) / y_m) * x_m)
end

MATLAB

x_m = abs(x);
y_m = abs(y);
z_m = abs(z);
t_m = abs(t);
function tmp = code(x_m, y_m, z_m, t_m)
	tmp = ((x_m / y_m) / y_m) * x_m;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[x$95$m_, y$95$m_, z$95$m_, t$95$m_] := N[(N[(N[(x$95$m / y$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]

Derivation

1. Initial program 67.4%

   \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{{x}^{2}}{{y}^{2}}} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{x \cdot x}{{\color{blue}{y}}^{2}} \]

   2. associate-*l/ N/A

      \[\leadsto \frac{x}{{y}^{2}} \cdot \color{blue}{x} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{x}{{y}^{2}} \cdot \color{blue}{x} \]

   4. unpow2 N/A

      \[\leadsto \frac{x}{y \cdot y} \cdot x \]

   5. associate-/r* N/A

      \[\leadsto \frac{\frac{x}{y}}{y} \cdot x \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\frac{x}{y}}{y} \cdot x \]

   7. lower-/.f64 49.2

      \[\leadsto \frac{\frac{x}{y}}{y} \cdot x \]

5. Applied rewrites 49.2%

   \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y} \cdot x} \]

6. Final simplification 49.2%

   \[\leadsto \frac{\frac{x}{y}}{y} \cdot x \]
7. Add Preprocessing

Alternative 9: 53.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ z_m = \left|z\right| \\ t_m = \left|t\right| \\ \frac{x\_m}{y\_m \cdot y\_m} \cdot x\_m \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
z_m = (fabs.f64 z)
t_m = (fabs.f64 t)
(FPCore (x_m y_m z_m t_m) :precision binary64 (* (/ x_m (* y_m y_m)) x_m))

C

x_m = fabs(x);
y_m = fabs(y);
z_m = fabs(z);
t_m = fabs(t);
double code(double x_m, double y_m, double z_m, double t_m) {
	return (x_m / (y_m * y_m)) * x_m;
}

Fortran

x_m =     private
y_m =     private
z_m =     private
t_m =     private
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(x_m, y_m, z_m, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t_m
    code = (x_m / (y_m * y_m)) * x_m
end function

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
z_m = Math.abs(z);
t_m = Math.abs(t);
public static double code(double x_m, double y_m, double z_m, double t_m) {
	return (x_m / (y_m * y_m)) * x_m;
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
z_m = math.fabs(z)
t_m = math.fabs(t)
def code(x_m, y_m, z_m, t_m):
	return (x_m / (y_m * y_m)) * x_m

Julia

x_m = abs(x)
y_m = abs(y)
z_m = abs(z)
t_m = abs(t)
function code(x_m, y_m, z_m, t_m)
	return Float64(Float64(x_m / Float64(y_m * y_m)) * x_m)
end

MATLAB

x_m = abs(x);
y_m = abs(y);
z_m = abs(z);
t_m = abs(t);
function tmp = code(x_m, y_m, z_m, t_m)
	tmp = (x_m / (y_m * y_m)) * x_m;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[x$95$m_, y$95$m_, z$95$m_, t$95$m_] := N[(N[(x$95$m / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]

Derivation

1. Initial program 67.4%

   \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{{x}^{2}}{{y}^{2}}} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{x \cdot x}{{\color{blue}{y}}^{2}} \]

   2. associate-*l/ N/A

      \[\leadsto \frac{x}{{y}^{2}} \cdot \color{blue}{x} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{x}{{y}^{2}} \cdot \color{blue}{x} \]

   4. unpow2 N/A

      \[\leadsto \frac{x}{y \cdot y} \cdot x \]

   5. associate-/r* N/A

      \[\leadsto \frac{\frac{x}{y}}{y} \cdot x \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\frac{x}{y}}{y} \cdot x \]

   7. lower-/.f64 49.2

      \[\leadsto \frac{\frac{x}{y}}{y} \cdot x \]

5. Applied rewrites 49.2%

   \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y} \cdot x} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\frac{x}{y}}{y} \cdot x \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\frac{x}{y}}{y} \cdot x \]

   3. associate-/l/ N/A

      \[\leadsto \frac{x}{y \cdot y} \cdot x \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{x}{y \cdot y} \cdot x \]

   5. lower-/.f64 44.6

      \[\leadsto \frac{x}{y \cdot y} \cdot x \]

7. Applied rewrites 44.6%

   \[\leadsto \frac{x}{y \cdot y} \cdot x \]

8. Final simplification 44.6%

   \[\leadsto \frac{x}{y \cdot y} \cdot x \]
9. Add Preprocessing

Developer Target 1: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (+ (pow (/ x y) 2.0) (pow (/ z t) 2.0)))

C

double code(double x, double y, double z, double t) {
	return pow((x / y), 2.0) + pow((z / t), 2.0);
}

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(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x / y) ** 2.0d0) + ((z / t) ** 2.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return Math.pow((x / y), 2.0) + Math.pow((z / t), 2.0);
}

Python

def code(x, y, z, t):
	return math.pow((x / y), 2.0) + math.pow((z / t), 2.0)

Julia

function code(x, y, z, t)
	return Float64((Float64(x / y) ^ 2.0) + (Float64(z / t) ^ 2.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x / y) ^ 2.0) + ((z / t) ^ 2.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[Power[N[(x / y), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(z / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025026 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Svg.PathConverter:arcToSegments from rasterific-svg-0.2.3.1"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (pow (/ x y) 2) (pow (/ z t) 2)))

  (+ (/ (* x x) (* y y)) (/ (* z z) (* t t))))