Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2

Percentage Accurate: 55.6% → 84.9%

Time: 8.3s

Alternatives: 16

Speedup: 2.5×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (/
  (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t)
  (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
}

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, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = ((((((((x * y) + z) * y) + 27464.7644705d0) * y) + 230661.510616d0) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
}

Python

def code(x, y, z, t, a, b, c, i):
	return ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i)

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + a) * y) + b) * y) + c) * y) + i))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[(N[(N[(N[(N[(N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision] + 27464.7644705), $MachinePrecision] * y), $MachinePrecision] + 230661.510616), $MachinePrecision] * y), $MachinePrecision] + t), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(y + a), $MachinePrecision] * y), $MachinePrecision] + b), $MachinePrecision] * y), $MachinePrecision] + c), $MachinePrecision] * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]

Initial Program: 55.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (/
  (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t)
  (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
}

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, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = ((((((((x * y) + z) * y) + 27464.7644705d0) * y) + 230661.510616d0) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
}

Python

def code(x, y, z, t, a, b, c, i):
	return ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i)

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + a) * y) + b) * y) + c) * y) + i))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[(N[(N[(N[(N[(N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision] + 27464.7644705), $MachinePrecision] * y), $MachinePrecision] + 230661.510616), $MachinePrecision] * y), $MachinePrecision] + t), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(y + a), $MachinePrecision] * y), $MachinePrecision] + b), $MachinePrecision] * y), $MachinePrecision] + c), $MachinePrecision] * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]

Alternative 1: 84.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{z}{y}\\ \mathbf{if}\;y \leq -2.35 \cdot 10^{+48}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+61}:\\ \;\;\;\;\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ x (/ z y))))
   (if (<= y -2.35e+48)
     t_1
     (if (<= y 7e+61)
       (/
        (+
         (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y)
         t)
        (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = x + (z / y);
	double tmp;
	if (y <= -2.35e+48) {
		tmp = t_1;
	} else if (y <= 7e+61) {
		tmp = ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + (z / y)
    if (y <= (-2.35d+48)) then
        tmp = t_1
    else if (y <= 7d+61) then
        tmp = ((((((((x * y) + z) * y) + 27464.7644705d0) * y) + 230661.510616d0) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = x + (z / y);
	double tmp;
	if (y <= -2.35e+48) {
		tmp = t_1;
	} else if (y <= 7e+61) {
		tmp = ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = x + (z / y)
	tmp = 0
	if y <= -2.35e+48:
		tmp = t_1
	elif y <= 7e+61:
		tmp = ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x + Float64(z / y))
	tmp = 0.0
	if (y <= -2.35e+48)
		tmp = t_1;
	elseif (y <= 7e+61)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + a) * y) + b) * y) + c) * y) + i));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = x + (z / y);
	tmp = 0.0;
	if (y <= -2.35e+48)
		tmp = t_1;
	elseif (y <= 7e+61)
		tmp = ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x + N[(z / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.35e+48], t$95$1, If[LessEqual[y, 7e+61], N[(N[(N[(N[(N[(N[(N[(N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision] + 27464.7644705), $MachinePrecision] * y), $MachinePrecision] + 230661.510616), $MachinePrecision] * y), $MachinePrecision] + t), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(y + a), $MachinePrecision] * y), $MachinePrecision] + b), $MachinePrecision] * y), $MachinePrecision] + c), $MachinePrecision] * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.35000000000000006e48 or 7.00000000000000036e61 < y

   1. Initial program 2.7%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 67.2

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

   4. Applied rewrites 67.2%

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

   5. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lift-/.f64 72.4

         \[\leadsto x + \frac{z}{y} \]

   7. Applied rewrites 72.4%

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

   if -2.35000000000000006e48 < y < 7.00000000000000036e61

   1. Initial program 93.3%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 84.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{z}{y}\\ \mathbf{if}\;y \leq -2.35 \cdot 10^{+48}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+61}:\\ \;\;\;\;\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(a + y, y, b\right), c \cdot y\right) + i}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ x (/ z y))))
   (if (<= y -2.35e+48)
     t_1
     (if (<= y 7e+61)
       (/
        (+
         (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y)
         t)
        (+ (fma (* y y) (fma (+ a y) y b) (* c y)) i))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = x + (z / y);
	double tmp;
	if (y <= -2.35e+48) {
		tmp = t_1;
	} else if (y <= 7e+61) {
		tmp = ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (fma((y * y), fma((a + y), y, b), (c * y)) + i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x + Float64(z / y))
	tmp = 0.0
	if (y <= -2.35e+48)
		tmp = t_1;
	elseif (y <= 7e+61)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / Float64(fma(Float64(y * y), fma(Float64(a + y), y, b), Float64(c * y)) + i));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x + N[(z / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.35e+48], t$95$1, If[LessEqual[y, 7e+61], N[(N[(N[(N[(N[(N[(N[(N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision] + 27464.7644705), $MachinePrecision] * y), $MachinePrecision] + 230661.510616), $MachinePrecision] * y), $MachinePrecision] + t), $MachinePrecision] / N[(N[(N[(y * y), $MachinePrecision] * N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] + N[(c * y), $MachinePrecision]), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.35000000000000006e48 or 7.00000000000000036e61 < y

   1. Initial program 2.7%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 67.2

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

   4. Applied rewrites 67.2%

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

   5. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lift-/.f64 72.4

         \[\leadsto x + \frac{z}{y} \]

   7. Applied rewrites 72.4%

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

   if -2.35000000000000006e48 < y < 7.00000000000000036e61

   1. Initial program 93.3%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in c around 0

      \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\color{blue}{\left(c \cdot y + {y}^{2} \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + i} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\left({y}^{2} \cdot \left(b + y \cdot \left(a + y\right)\right) + \color{blue}{c \cdot y}\right) + i} \]

      2. +-commutative N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\left({y}^{2} \cdot \left(b + y \cdot \left(y + a\right)\right) + c \cdot y\right) + i} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\left({y}^{2} \cdot \left(b + \left(y + a\right) \cdot y\right) + c \cdot y\right) + i} \]

      4. +-commutative N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\left({y}^{2} \cdot \left(\left(y + a\right) \cdot y + b\right) + c \cdot y\right) + i} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\mathsf{fma}\left({y}^{2}, \color{blue}{\left(y + a\right) \cdot y + b}, c \cdot y\right) + i} \]

      6. unpow2 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\mathsf{fma}\left(y \cdot y, \color{blue}{\left(y + a\right) \cdot y} + b, c \cdot y\right) + i} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\mathsf{fma}\left(y \cdot y, \color{blue}{\left(y + a\right) \cdot y} + b, c \cdot y\right) + i} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y + a, \color{blue}{y}, b\right), c \cdot y\right) + i} \]

      9. +-commutative N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(a + y, y, b\right), c \cdot y\right) + i} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(a + y, y, b\right), c \cdot y\right) + i} \]

      11. lower-*.f64 92.4

          \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(a + y, y, \color{blue}{b}\right), c \cdot y\right) + i} \]

   4. Applied rewrites 92.4%

      \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(a + y, y, b\right), c \cdot y\right)} + i} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 84.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)\\ \mathbf{if}\;\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{t\_1}, \frac{t}{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (fma (fma (+ a y) y b) y c) y i)))
   (if (<=
        (/
         (+
          (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y)
          t)
         (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i))
        INFINITY)
     (fma
      y
      (/ (fma (fma (fma y x z) y 27464.7644705) y 230661.510616) t_1)
      (/ t t_1))
     (+ x (/ z y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(fma(fma((a + y), y, b), y, c), y, i);
	double tmp;
	if ((((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i)) <= ((double) INFINITY)) {
		tmp = fma(y, (fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616) / t_1), (t / t_1));
	} else {
		tmp = x + (z / y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(fma(fma(Float64(a + y), y, b), y, c), y, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + a) * y) + b) * y) + c) * y) + i)) <= Inf)
		tmp = fma(y, Float64(fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616) / t_1), Float64(t / t_1));
	else
		tmp = Float64(x + Float64(z / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision] * y + i), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(N[(N[(N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision] + 27464.7644705), $MachinePrecision] * y), $MachinePrecision] + 230661.510616), $MachinePrecision] * y), $MachinePrecision] + t), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(y + a), $MachinePrecision] * y), $MachinePrecision] + b), $MachinePrecision] * y), $MachinePrecision] + c), $MachinePrecision] * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], Infinity], N[(y * N[(N[(N[(N[(y * x + z), $MachinePrecision] * y + 27464.7644705), $MachinePrecision] * y + 230661.510616), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(t / t$95$1), $MachinePrecision]), $MachinePrecision], N[(x + N[(z / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i)) < +inf.0

   1. Initial program 90.2%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   3. Applied rewrites 91.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)} \]

   if +inf.0 < (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i))

   1. Initial program 0.0%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 69.1

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

   4. Applied rewrites 69.1%

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

   5. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lift-/.f64 74.7

         \[\leadsto x + \frac{z}{y} \]

   7. Applied rewrites 74.7%

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

Alternative 4: 79.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right)\\ t_2 := x + \frac{z}{y}\\ t_3 := \mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right)\\ \mathbf{if}\;y \leq -2.6 \cdot 10^{+48}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -1.5 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_1, y, t\right)}{y}}{t\_3}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{\mathsf{fma}\left(230661.510616, y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t\_1}{\mathsf{fma}\left(t\_3, y, i\right)}, \frac{t}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (fma (fma y x z) y 27464.7644705) y 230661.510616))
        (t_2 (+ x (/ z y)))
        (t_3 (fma (fma (+ a y) y b) y c)))
   (if (<= y -2.6e+48)
     t_2
     (if (<= y -1.5e-62)
       (/ (/ (fma t_1 y t) y) t_3)
       (if (<= y 1.8e-29)
         (/
          (fma 230661.510616 y t)
          (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i))
         (if (<= y 2.6e+58) (fma y (/ t_1 (fma t_3 y i)) (/ t i)) t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616);
	double t_2 = x + (z / y);
	double t_3 = fma(fma((a + y), y, b), y, c);
	double tmp;
	if (y <= -2.6e+48) {
		tmp = t_2;
	} else if (y <= -1.5e-62) {
		tmp = (fma(t_1, y, t) / y) / t_3;
	} else if (y <= 1.8e-29) {
		tmp = fma(230661.510616, y, t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
	} else if (y <= 2.6e+58) {
		tmp = fma(y, (t_1 / fma(t_3, y, i)), (t / i));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616)
	t_2 = Float64(x + Float64(z / y))
	t_3 = fma(fma(Float64(a + y), y, b), y, c)
	tmp = 0.0
	if (y <= -2.6e+48)
		tmp = t_2;
	elseif (y <= -1.5e-62)
		tmp = Float64(Float64(fma(t_1, y, t) / y) / t_3);
	elseif (y <= 1.8e-29)
		tmp = Float64(fma(230661.510616, y, t) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + a) * y) + b) * y) + c) * y) + i));
	elseif (y <= 2.6e+58)
		tmp = fma(y, Float64(t_1 / fma(t_3, y, i)), Float64(t / i));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(y * x + z), $MachinePrecision] * y + 27464.7644705), $MachinePrecision] * y + 230661.510616), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(z / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision]}, If[LessEqual[y, -2.6e+48], t$95$2, If[LessEqual[y, -1.5e-62], N[(N[(N[(t$95$1 * y + t), $MachinePrecision] / y), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[y, 1.8e-29], N[(N[(230661.510616 * y + t), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(y + a), $MachinePrecision] * y), $MachinePrecision] + b), $MachinePrecision] * y), $MachinePrecision] + c), $MachinePrecision] * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.6e+58], N[(y * N[(t$95$1 / N[(t$95$3 * y + i), $MachinePrecision]), $MachinePrecision] + N[(t / i), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.59999999999999995e48 or 2.59999999999999988e58 < y

   1. Initial program 2.8%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 66.9

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

   4. Applied rewrites 66.9%

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

   5. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lift-/.f64 72.1

         \[\leadsto x + \frac{z}{y} \]

   7. Applied rewrites 72.1%

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

   if -2.59999999999999995e48 < y < -1.5000000000000001e-62

   1. Initial program 83.8%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{y}}{\color{blue}{c + y \cdot \left(b + y \cdot \left(a + y\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{y}}{\color{blue}{c + y \cdot \left(b + y \cdot \left(a + y\right)\right)}} \]

   4. Applied rewrites 60.5%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{y}}{\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right)}} \]

   if -1.5000000000000001e-62 < y < 1.79999999999999987e-29

   1. Initial program 99.8%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{t + \frac{28832688827}{125000} \cdot y}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{28832688827}{125000} \cdot y + \color{blue}{t}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      2. lower-fma.f64 94.9

         \[\leadsto \frac{\mathsf{fma}\left(230661.510616, \color{blue}{y}, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   4. Applied rewrites 94.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(230661.510616, y, t\right)}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   if 1.79999999999999987e-29 < y < 2.59999999999999988e58

   1. Initial program 70.8%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   3. Applied rewrites 73.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)} \]

   4. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, \frac{54929528941}{2000000}\right), y, \frac{28832688827}{125000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, \frac{t}{\color{blue}{i}}\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 37.8%

      \[\leadsto \mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, \frac{t}{\color{blue}{i}}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 78.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{y}}{\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right)}\\ t_2 := x + \frac{z}{y}\\ \mathbf{if}\;y \leq -2.6 \cdot 10^{+48}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -1.5 \cdot 10^{-62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 210:\\ \;\;\;\;\frac{\mathsf{fma}\left(230661.510616, y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{+64}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (/
          (/
           (fma (fma (fma (fma y x z) y 27464.7644705) y 230661.510616) y t)
           y)
          (fma (fma (+ a y) y b) y c)))
        (t_2 (+ x (/ z y))))
   (if (<= y -2.6e+48)
     t_2
     (if (<= y -1.5e-62)
       t_1
       (if (<= y 210.0)
         (/
          (fma 230661.510616 y t)
          (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i))
         (if (<= y 7.4e+64) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (fma(fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616), y, t) / y) / fma(fma((a + y), y, b), y, c);
	double t_2 = x + (z / y);
	double tmp;
	if (y <= -2.6e+48) {
		tmp = t_2;
	} else if (y <= -1.5e-62) {
		tmp = t_1;
	} else if (y <= 210.0) {
		tmp = fma(230661.510616, y, t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
	} else if (y <= 7.4e+64) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(fma(fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616), y, t) / y) / fma(fma(Float64(a + y), y, b), y, c))
	t_2 = Float64(x + Float64(z / y))
	tmp = 0.0
	if (y <= -2.6e+48)
		tmp = t_2;
	elseif (y <= -1.5e-62)
		tmp = t_1;
	elseif (y <= 210.0)
		tmp = Float64(fma(230661.510616, y, t) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + a) * y) + b) * y) + c) * y) + i));
	elseif (y <= 7.4e+64)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(y * x + z), $MachinePrecision] * y + 27464.7644705), $MachinePrecision] * y + 230661.510616), $MachinePrecision] * y + t), $MachinePrecision] / y), $MachinePrecision] / N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(z / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.6e+48], t$95$2, If[LessEqual[y, -1.5e-62], t$95$1, If[LessEqual[y, 210.0], N[(N[(230661.510616 * y + t), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(y + a), $MachinePrecision] * y), $MachinePrecision] + b), $MachinePrecision] * y), $MachinePrecision] + c), $MachinePrecision] * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.4e+64], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.59999999999999995e48 or 7.39999999999999966e64 < y

   1. Initial program 2.4%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 67.5

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

   4. Applied rewrites 67.5%

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

   5. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lift-/.f64 72.8

         \[\leadsto x + \frac{z}{y} \]

   7. Applied rewrites 72.8%

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

   if -2.59999999999999995e48 < y < -1.5000000000000001e-62 or 210 < y < 7.39999999999999966e64

   1. Initial program 71.5%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{y}}{\color{blue}{c + y \cdot \left(b + y \cdot \left(a + y\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{y}}{\color{blue}{c + y \cdot \left(b + y \cdot \left(a + y\right)\right)}} \]

   4. Applied rewrites 55.6%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{y}}{\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right)}} \]

   if -1.5000000000000001e-62 < y < 210

   1. Initial program 99.7%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{t + \frac{28832688827}{125000} \cdot y}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{28832688827}{125000} \cdot y + \color{blue}{t}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      2. lower-fma.f64 92.2

         \[\leadsto \frac{\mathsf{fma}\left(230661.510616, \color{blue}{y}, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   4. Applied rewrites 92.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(230661.510616, y, t\right)}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 74.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right)\\ t_2 := x + \frac{z}{y}\\ \mathbf{if}\;y \leq -2.15 \cdot 10^{+31}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -1.18 \cdot 10^{-51}:\\ \;\;\;\;\frac{{y}^{4} \cdot x}{\mathsf{fma}\left(t\_1, y, i\right)}\\ \mathbf{elif}\;y \leq 1600:\\ \;\;\;\;\frac{\mathsf{fma}\left(230661.510616, y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\\ \mathbf{elif}\;y \leq 1.62 \cdot 10^{+65}:\\ \;\;\;\;\frac{x \cdot {y}^{3}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (fma (+ a y) y b) y c)) (t_2 (+ x (/ z y))))
   (if (<= y -2.15e+31)
     t_2
     (if (<= y -1.18e-51)
       (/ (* (pow y 4.0) x) (fma t_1 y i))
       (if (<= y 1600.0)
         (/
          (fma 230661.510616 y t)
          (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i))
         (if (<= y 1.62e+65) (/ (* x (pow y 3.0)) t_1) t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(fma((a + y), y, b), y, c);
	double t_2 = x + (z / y);
	double tmp;
	if (y <= -2.15e+31) {
		tmp = t_2;
	} else if (y <= -1.18e-51) {
		tmp = (pow(y, 4.0) * x) / fma(t_1, y, i);
	} else if (y <= 1600.0) {
		tmp = fma(230661.510616, y, t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
	} else if (y <= 1.62e+65) {
		tmp = (x * pow(y, 3.0)) / t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(fma(Float64(a + y), y, b), y, c)
	t_2 = Float64(x + Float64(z / y))
	tmp = 0.0
	if (y <= -2.15e+31)
		tmp = t_2;
	elseif (y <= -1.18e-51)
		tmp = Float64(Float64((y ^ 4.0) * x) / fma(t_1, y, i));
	elseif (y <= 1600.0)
		tmp = Float64(fma(230661.510616, y, t) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + a) * y) + b) * y) + c) * y) + i));
	elseif (y <= 1.62e+65)
		tmp = Float64(Float64(x * (y ^ 3.0)) / t_1);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(z / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.15e+31], t$95$2, If[LessEqual[y, -1.18e-51], N[(N[(N[Power[y, 4.0], $MachinePrecision] * x), $MachinePrecision] / N[(t$95$1 * y + i), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1600.0], N[(N[(230661.510616 * y + t), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(y + a), $MachinePrecision] * y), $MachinePrecision] + b), $MachinePrecision] * y), $MachinePrecision] + c), $MachinePrecision] * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.62e+65], N[(N[(x * N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.14999999999999994e31 or 1.61999999999999997e65 < y

   1. Initial program 3.9%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 66.0

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

   4. Applied rewrites 66.0%

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

   5. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lift-/.f64 71.1

         \[\leadsto x + \frac{z}{y} \]

   7. Applied rewrites 71.1%

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

   if -2.14999999999999994e31 < y < -1.18000000000000004e-51

   1. Initial program 88.7%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x \cdot {y}^{4}}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot {y}^{4}}{\color{blue}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{{y}^{4} \cdot x}{\color{blue}{i} + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{y}^{4} \cdot x}{\color{blue}{i} + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{y}^{4} \cdot x}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{{y}^{4} \cdot x}{y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right) + \color{blue}{i}} \]

      6. *-commutative N/A

         \[\leadsto \frac{{y}^{4} \cdot x}{\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right) \cdot y + i} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{{y}^{4} \cdot x}{\mathsf{fma}\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right), \color{blue}{y}, i\right)} \]

   4. Applied rewrites 23.4%

      \[\leadsto \color{blue}{\frac{{y}^{4} \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}} \]

   if -1.18000000000000004e-51 < y < 1600

   1. Initial program 99.7%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{t + \frac{28832688827}{125000} \cdot y}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{28832688827}{125000} \cdot y + \color{blue}{t}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      2. lower-fma.f64 91.6

         \[\leadsto \frac{\mathsf{fma}\left(230661.510616, \color{blue}{y}, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   4. Applied rewrites 91.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(230661.510616, y, t\right)}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   if 1600 < y < 1.61999999999999997e65

   1. Initial program 50.6%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{y}}{\color{blue}{c + y \cdot \left(b + y \cdot \left(a + y\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{y}}{\color{blue}{c + y \cdot \left(b + y \cdot \left(a + y\right)\right)}} \]

   4. Applied rewrites 47.1%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{y}}{\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right)}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{x \cdot {y}^{3}}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(a + y, y, b\right)}, y, c\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot {y}^{3}}{\mathsf{fma}\left(\mathsf{fma}\left(a + y, \color{blue}{y}, b\right), y, c\right)} \]

      2. lift-pow.f64 19.8

         \[\leadsto \frac{x \cdot {y}^{3}}{\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right)} \]

   7. Applied rewrites 19.8%

      \[\leadsto \frac{x \cdot {y}^{3}}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(a + y, y, b\right)}, y, c\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 69.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right)\\ t_2 := x + \frac{z}{y}\\ \mathbf{if}\;y \leq -2.15 \cdot 10^{+31}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -1.5 \cdot 10^{-62}:\\ \;\;\;\;\frac{{y}^{4} \cdot x}{\mathsf{fma}\left(t\_1, y, i\right)}\\ \mathbf{elif}\;y \leq 900:\\ \;\;\;\;\frac{230661.510616 \cdot y + t}{\frac{c \cdot y}{a} \cdot a + i}\\ \mathbf{elif}\;y \leq 1.62 \cdot 10^{+65}:\\ \;\;\;\;\frac{x \cdot {y}^{3}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (fma (+ a y) y b) y c)) (t_2 (+ x (/ z y))))
   (if (<= y -2.15e+31)
     t_2
     (if (<= y -1.5e-62)
       (/ (* (pow y 4.0) x) (fma t_1 y i))
       (if (<= y 900.0)
         (/ (+ (* 230661.510616 y) t) (+ (* (/ (* c y) a) a) i))
         (if (<= y 1.62e+65) (/ (* x (pow y 3.0)) t_1) t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(fma((a + y), y, b), y, c);
	double t_2 = x + (z / y);
	double tmp;
	if (y <= -2.15e+31) {
		tmp = t_2;
	} else if (y <= -1.5e-62) {
		tmp = (pow(y, 4.0) * x) / fma(t_1, y, i);
	} else if (y <= 900.0) {
		tmp = ((230661.510616 * y) + t) / ((((c * y) / a) * a) + i);
	} else if (y <= 1.62e+65) {
		tmp = (x * pow(y, 3.0)) / t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(fma(Float64(a + y), y, b), y, c)
	t_2 = Float64(x + Float64(z / y))
	tmp = 0.0
	if (y <= -2.15e+31)
		tmp = t_2;
	elseif (y <= -1.5e-62)
		tmp = Float64(Float64((y ^ 4.0) * x) / fma(t_1, y, i));
	elseif (y <= 900.0)
		tmp = Float64(Float64(Float64(230661.510616 * y) + t) / Float64(Float64(Float64(Float64(c * y) / a) * a) + i));
	elseif (y <= 1.62e+65)
		tmp = Float64(Float64(x * (y ^ 3.0)) / t_1);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(z / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.15e+31], t$95$2, If[LessEqual[y, -1.5e-62], N[(N[(N[Power[y, 4.0], $MachinePrecision] * x), $MachinePrecision] / N[(t$95$1 * y + i), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 900.0], N[(N[(N[(230661.510616 * y), $MachinePrecision] + t), $MachinePrecision] / N[(N[(N[(N[(c * y), $MachinePrecision] / a), $MachinePrecision] * a), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.62e+65], N[(N[(x * N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.14999999999999994e31 or 1.61999999999999997e65 < y

   1. Initial program 3.9%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 66.0

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

   4. Applied rewrites 66.0%

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

   5. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lift-/.f64 71.1

         \[\leadsto x + \frac{z}{y} \]

   7. Applied rewrites 71.1%

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

   if -2.14999999999999994e31 < y < -1.5000000000000001e-62

   1. Initial program 90.0%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x \cdot {y}^{4}}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot {y}^{4}}{\color{blue}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{{y}^{4} \cdot x}{\color{blue}{i} + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{y}^{4} \cdot x}{\color{blue}{i} + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{y}^{4} \cdot x}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{{y}^{4} \cdot x}{y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right) + \color{blue}{i}} \]

      6. *-commutative N/A

         \[\leadsto \frac{{y}^{4} \cdot x}{\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right) \cdot y + i} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{{y}^{4} \cdot x}{\mathsf{fma}\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right), \color{blue}{y}, i\right)} \]

   4. Applied rewrites 22.4%

      \[\leadsto \color{blue}{\frac{{y}^{4} \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}} \]

   if -1.5000000000000001e-62 < y < 900

   1. Initial program 99.7%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in a around inf

      \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\color{blue}{a \cdot \left(\frac{y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)}{a} + {y}^{3}\right)} + i} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\left(\frac{y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)}{a} + {y}^{3}\right) \cdot \color{blue}{a} + i} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\left(\frac{y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)}{a} + {y}^{3}\right) \cdot \color{blue}{a} + i} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\left(y \cdot \frac{c + y \cdot \left(b + {y}^{2}\right)}{a} + {y}^{3}\right) \cdot a + i} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\mathsf{fma}\left(y, \frac{c + y \cdot \left(b + {y}^{2}\right)}{a}, {y}^{3}\right) \cdot a + i} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\mathsf{fma}\left(y, \frac{c + y \cdot \left(b + {y}^{2}\right)}{a}, {y}^{3}\right) \cdot a + i} \]

      6. +-commutative N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\mathsf{fma}\left(y, \frac{y \cdot \left(b + {y}^{2}\right) + c}{a}, {y}^{3}\right) \cdot a + i} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\mathsf{fma}\left(y, \frac{\left(b + {y}^{2}\right) \cdot y + c}{a}, {y}^{3}\right) \cdot a + i} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(b + {y}^{2}, y, c\right)}{a}, {y}^{3}\right) \cdot a + i} \]

      9. +-commutative N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left({y}^{2} + b, y, c\right)}{a}, {y}^{3}\right) \cdot a + i} \]

      10. unpow2 N/A

          \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(y \cdot y + b, y, c\right)}{a}, {y}^{3}\right) \cdot a + i} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y, b\right), y, c\right)}{a}, {y}^{3}\right) \cdot a + i} \]

      12. lower-pow.f64 85.0

          \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y, b\right), y, c\right)}{a}, {y}^{3}\right) \cdot a + i} \]

   4. Applied rewrites 85.0%

      \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y, b\right), y, c\right)}{a}, {y}^{3}\right) \cdot a} + i} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\frac{c \cdot y}{a} \cdot a + i} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\frac{c \cdot y}{a} \cdot a + i} \]

      2. lower-*.f64 84.4

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\frac{c \cdot y}{a} \cdot a + i} \]

   7. Applied rewrites 84.4%

      \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\frac{c \cdot y}{a} \cdot a + i} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{\frac{28832688827}{125000}} \cdot y + t}{\frac{c \cdot y}{a} \cdot a + i} \]
   9. Step-by-step derivation

   10. Applied rewrites 80.0%

       \[\leadsto \frac{\color{blue}{230661.510616} \cdot y + t}{\frac{c \cdot y}{a} \cdot a + i} \]

   if 900 < y < 1.61999999999999997e65

   1. Initial program 50.7%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{y}}{\color{blue}{c + y \cdot \left(b + y \cdot \left(a + y\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{y}}{\color{blue}{c + y \cdot \left(b + y \cdot \left(a + y\right)\right)}} \]

   4. Applied rewrites 47.3%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{y}}{\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right)}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{x \cdot {y}^{3}}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(a + y, y, b\right)}, y, c\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot {y}^{3}}{\mathsf{fma}\left(\mathsf{fma}\left(a + y, \color{blue}{y}, b\right), y, c\right)} \]

      2. lift-pow.f64 19.7

         \[\leadsto \frac{x \cdot {y}^{3}}{\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right)} \]

   7. Applied rewrites 19.7%

      \[\leadsto \frac{x \cdot {y}^{3}}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(a + y, y, b\right)}, y, c\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 68.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot {y}^{3}}{\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right)}\\ t_2 := x + \frac{z}{y}\\ \mathbf{if}\;y \leq -1.05 \cdot 10^{+33}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -1.46 \cdot 10^{-43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 900:\\ \;\;\;\;\frac{230661.510616 \cdot y + t}{\frac{c \cdot y}{a} \cdot a + i}\\ \mathbf{elif}\;y \leq 1.62 \cdot 10^{+65}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (/ (* x (pow y 3.0)) (fma (fma (+ a y) y b) y c)))
        (t_2 (+ x (/ z y))))
   (if (<= y -1.05e+33)
     t_2
     (if (<= y -1.46e-43)
       t_1
       (if (<= y 900.0)
         (/ (+ (* 230661.510616 y) t) (+ (* (/ (* c y) a) a) i))
         (if (<= y 1.62e+65) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (x * pow(y, 3.0)) / fma(fma((a + y), y, b), y, c);
	double t_2 = x + (z / y);
	double tmp;
	if (y <= -1.05e+33) {
		tmp = t_2;
	} else if (y <= -1.46e-43) {
		tmp = t_1;
	} else if (y <= 900.0) {
		tmp = ((230661.510616 * y) + t) / ((((c * y) / a) * a) + i);
	} else if (y <= 1.62e+65) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(x * (y ^ 3.0)) / fma(fma(Float64(a + y), y, b), y, c))
	t_2 = Float64(x + Float64(z / y))
	tmp = 0.0
	if (y <= -1.05e+33)
		tmp = t_2;
	elseif (y <= -1.46e-43)
		tmp = t_1;
	elseif (y <= 900.0)
		tmp = Float64(Float64(Float64(230661.510616 * y) + t) / Float64(Float64(Float64(Float64(c * y) / a) * a) + i));
	elseif (y <= 1.62e+65)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(x * N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(z / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.05e+33], t$95$2, If[LessEqual[y, -1.46e-43], t$95$1, If[LessEqual[y, 900.0], N[(N[(N[(230661.510616 * y), $MachinePrecision] + t), $MachinePrecision] / N[(N[(N[(N[(c * y), $MachinePrecision] / a), $MachinePrecision] * a), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.62e+65], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.05e33 or 1.61999999999999997e65 < y

   1. Initial program 3.8%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 66.1

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

   4. Applied rewrites 66.1%

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

   5. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lift-/.f64 71.2

         \[\leadsto x + \frac{z}{y} \]

   7. Applied rewrites 71.2%

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

   if -1.05e33 < y < -1.45999999999999997e-43 or 900 < y < 1.61999999999999997e65

   1. Initial program 70.4%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{y}}{\color{blue}{c + y \cdot \left(b + y \cdot \left(a + y\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{y}}{\color{blue}{c + y \cdot \left(b + y \cdot \left(a + y\right)\right)}} \]

   4. Applied rewrites 57.3%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{y}}{\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right)}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{x \cdot {y}^{3}}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(a + y, y, b\right)}, y, c\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot {y}^{3}}{\mathsf{fma}\left(\mathsf{fma}\left(a + y, \color{blue}{y}, b\right), y, c\right)} \]

      2. lift-pow.f64 20.4

         \[\leadsto \frac{x \cdot {y}^{3}}{\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right)} \]

   7. Applied rewrites 20.4%

      \[\leadsto \frac{x \cdot {y}^{3}}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(a + y, y, b\right)}, y, c\right)} \]

   if -1.45999999999999997e-43 < y < 900

   1. Initial program 99.7%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in a around inf

      \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\color{blue}{a \cdot \left(\frac{y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)}{a} + {y}^{3}\right)} + i} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\left(\frac{y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)}{a} + {y}^{3}\right) \cdot \color{blue}{a} + i} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\left(\frac{y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)}{a} + {y}^{3}\right) \cdot \color{blue}{a} + i} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\left(y \cdot \frac{c + y \cdot \left(b + {y}^{2}\right)}{a} + {y}^{3}\right) \cdot a + i} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\mathsf{fma}\left(y, \frac{c + y \cdot \left(b + {y}^{2}\right)}{a}, {y}^{3}\right) \cdot a + i} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\mathsf{fma}\left(y, \frac{c + y \cdot \left(b + {y}^{2}\right)}{a}, {y}^{3}\right) \cdot a + i} \]

      6. +-commutative N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\mathsf{fma}\left(y, \frac{y \cdot \left(b + {y}^{2}\right) + c}{a}, {y}^{3}\right) \cdot a + i} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\mathsf{fma}\left(y, \frac{\left(b + {y}^{2}\right) \cdot y + c}{a}, {y}^{3}\right) \cdot a + i} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(b + {y}^{2}, y, c\right)}{a}, {y}^{3}\right) \cdot a + i} \]

      9. +-commutative N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left({y}^{2} + b, y, c\right)}{a}, {y}^{3}\right) \cdot a + i} \]

      10. unpow2 N/A

          \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(y \cdot y + b, y, c\right)}{a}, {y}^{3}\right) \cdot a + i} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y, b\right), y, c\right)}{a}, {y}^{3}\right) \cdot a + i} \]

      12. lower-pow.f64 85.0

          \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y, b\right), y, c\right)}{a}, {y}^{3}\right) \cdot a + i} \]

   4. Applied rewrites 85.0%

      \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y, b\right), y, c\right)}{a}, {y}^{3}\right) \cdot a} + i} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\frac{c \cdot y}{a} \cdot a + i} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\frac{c \cdot y}{a} \cdot a + i} \]

      2. lower-*.f64 83.9

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\frac{c \cdot y}{a} \cdot a + i} \]

   7. Applied rewrites 83.9%

      \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\frac{c \cdot y}{a} \cdot a + i} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{\frac{28832688827}{125000}} \cdot y + t}{\frac{c \cdot y}{a} \cdot a + i} \]
   9. Step-by-step derivation

   10. Applied rewrites 79.1%

       \[\leadsto \frac{\color{blue}{230661.510616} \cdot y + t}{\frac{c \cdot y}{a} \cdot a + i} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 68.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right)\\ t_2 := \frac{x \cdot {y}^{3}}{t\_1}\\ t_3 := x + \frac{z}{y}\\ \mathbf{if}\;y \leq -1.05 \cdot 10^{+33}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y \leq -4.6 \cdot 10^{-12}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-29}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(t\_1, y, i\right)}\\ \mathbf{elif}\;y \leq 240000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{i}\\ \mathbf{elif}\;y \leq 1.62 \cdot 10^{+65}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (fma (+ a y) y b) y c))
        (t_2 (/ (* x (pow y 3.0)) t_1))
        (t_3 (+ x (/ z y))))
   (if (<= y -1.05e+33)
     t_3
     (if (<= y -4.6e-12)
       t_2
       (if (<= y 2e-29)
         (/ t (fma t_1 y i))
         (if (<= y 240000.0)
           (/
            (fma (fma (fma (fma y x z) y 27464.7644705) y 230661.510616) y t)
            i)
           (if (<= y 1.62e+65) t_2 t_3)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(fma((a + y), y, b), y, c);
	double t_2 = (x * pow(y, 3.0)) / t_1;
	double t_3 = x + (z / y);
	double tmp;
	if (y <= -1.05e+33) {
		tmp = t_3;
	} else if (y <= -4.6e-12) {
		tmp = t_2;
	} else if (y <= 2e-29) {
		tmp = t / fma(t_1, y, i);
	} else if (y <= 240000.0) {
		tmp = fma(fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616), y, t) / i;
	} else if (y <= 1.62e+65) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(fma(Float64(a + y), y, b), y, c)
	t_2 = Float64(Float64(x * (y ^ 3.0)) / t_1)
	t_3 = Float64(x + Float64(z / y))
	tmp = 0.0
	if (y <= -1.05e+33)
		tmp = t_3;
	elseif (y <= -4.6e-12)
		tmp = t_2;
	elseif (y <= 2e-29)
		tmp = Float64(t / fma(t_1, y, i));
	elseif (y <= 240000.0)
		tmp = Float64(fma(fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616), y, t) / i);
	elseif (y <= 1.62e+65)
		tmp = t_2;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(x + N[(z / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.05e+33], t$95$3, If[LessEqual[y, -4.6e-12], t$95$2, If[LessEqual[y, 2e-29], N[(t / N[(t$95$1 * y + i), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 240000.0], N[(N[(N[(N[(N[(y * x + z), $MachinePrecision] * y + 27464.7644705), $MachinePrecision] * y + 230661.510616), $MachinePrecision] * y + t), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[y, 1.62e+65], t$95$2, t$95$3]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.05e33 or 1.61999999999999997e65 < y

   1. Initial program 3.8%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 66.1

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

   4. Applied rewrites 66.1%

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

   5. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lift-/.f64 71.2

         \[\leadsto x + \frac{z}{y} \]

   7. Applied rewrites 71.2%

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

   if -1.05e33 < y < -4.59999999999999979e-12 or 2.4e5 < y < 1.61999999999999997e65

   1. Initial program 60.6%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{y}}{\color{blue}{c + y \cdot \left(b + y \cdot \left(a + y\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{y}}{\color{blue}{c + y \cdot \left(b + y \cdot \left(a + y\right)\right)}} \]

   4. Applied rewrites 55.5%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{y}}{\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right)}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{x \cdot {y}^{3}}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(a + y, y, b\right)}, y, c\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot {y}^{3}}{\mathsf{fma}\left(\mathsf{fma}\left(a + y, \color{blue}{y}, b\right), y, c\right)} \]

      2. lift-pow.f64 21.6

         \[\leadsto \frac{x \cdot {y}^{3}}{\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right)} \]

   7. Applied rewrites 21.6%

      \[\leadsto \frac{x \cdot {y}^{3}}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(a + y, y, b\right)}, y, c\right)} \]

   if -4.59999999999999979e-12 < y < 1.99999999999999989e-29

   1. Initial program 99.7%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t}{\color{blue}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{t}{y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right) + \color{blue}{i}} \]

      3. *-commutative N/A

         \[\leadsto \frac{t}{\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right) \cdot y + i} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{t}{\mathsf{fma}\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right), \color{blue}{y}, i\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{t}{\mathsf{fma}\left(y \cdot \left(b + y \cdot \left(a + y\right)\right) + c, y, i\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{t}{\mathsf{fma}\left(\left(b + y \cdot \left(a + y\right)\right) \cdot y + c, y, i\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{t}{\mathsf{fma}\left(\left(b + y \cdot \left(y + a\right)\right) \cdot y + c, y, i\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{t}{\mathsf{fma}\left(\left(b + \left(y + a\right) \cdot y\right) \cdot y + c, y, i\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{t}{\mathsf{fma}\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c, y, i\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\left(y + a\right) \cdot y + b, y, c\right), y, i\right)} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right)} \]

      12. +-commutative N/A

          \[\leadsto \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)} \]

      13. lower-+.f64 77.7

          \[\leadsto \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)} \]

   4. Applied rewrites 77.7%

      \[\leadsto \color{blue}{\frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}} \]

   if 1.99999999999999989e-29 < y < 2.4e5

   1. Initial program 97.8%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{i}} \]

   4. Applied rewrites 30.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{i}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 10: 67.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{z}{y}\\ \mathbf{if}\;y \leq -1.05 \cdot 10^{+33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -1.46 \cdot 10^{-43}:\\ \;\;\;\;\frac{x \cdot {y}^{3}}{\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right)}\\ \mathbf{elif}\;y \leq 2.75 \cdot 10^{+38}:\\ \;\;\;\;\frac{{y}^{3} \cdot z + t}{c \cdot y + i}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ x (/ z y))))
   (if (<= y -1.05e+33)
     t_1
     (if (<= y -1.46e-43)
       (/ (* x (pow y 3.0)) (fma (fma (+ a y) y b) y c))
       (if (<= y 2.75e+38) (/ (+ (* (pow y 3.0) z) t) (+ (* c y) i)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = x + (z / y);
	double tmp;
	if (y <= -1.05e+33) {
		tmp = t_1;
	} else if (y <= -1.46e-43) {
		tmp = (x * pow(y, 3.0)) / fma(fma((a + y), y, b), y, c);
	} else if (y <= 2.75e+38) {
		tmp = ((pow(y, 3.0) * z) + t) / ((c * y) + i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x + Float64(z / y))
	tmp = 0.0
	if (y <= -1.05e+33)
		tmp = t_1;
	elseif (y <= -1.46e-43)
		tmp = Float64(Float64(x * (y ^ 3.0)) / fma(fma(Float64(a + y), y, b), y, c));
	elseif (y <= 2.75e+38)
		tmp = Float64(Float64(Float64((y ^ 3.0) * z) + t) / Float64(Float64(c * y) + i));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x + N[(z / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.05e+33], t$95$1, If[LessEqual[y, -1.46e-43], N[(N[(x * N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.75e+38], N[(N[(N[(N[Power[y, 3.0], $MachinePrecision] * z), $MachinePrecision] + t), $MachinePrecision] / N[(N[(c * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.05e33 or 2.7500000000000002e38 < y

   1. Initial program 5.2%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 63.8

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

   4. Applied rewrites 63.8%

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

   5. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lift-/.f64 68.7

         \[\leadsto x + \frac{z}{y} \]

   7. Applied rewrites 68.7%

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

   if -1.05e33 < y < -1.45999999999999997e-43

   1. Initial program 87.4%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{y}}{\color{blue}{c + y \cdot \left(b + y \cdot \left(a + y\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{y}}{\color{blue}{c + y \cdot \left(b + y \cdot \left(a + y\right)\right)}} \]

   4. Applied rewrites 66.0%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{y}}{\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right)}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{x \cdot {y}^{3}}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(a + y, y, b\right)}, y, c\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot {y}^{3}}{\mathsf{fma}\left(\mathsf{fma}\left(a + y, \color{blue}{y}, b\right), y, c\right)} \]

      2. lift-pow.f64 20.9

         \[\leadsto \frac{x \cdot {y}^{3}}{\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right)} \]

   7. Applied rewrites 20.9%

      \[\leadsto \frac{x \cdot {y}^{3}}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(a + y, y, b\right)}, y, c\right)} \]

   if -1.45999999999999997e-43 < y < 2.7500000000000002e38

   1. Initial program 97.5%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in z around inf

      \[\leadsto \frac{\color{blue}{{y}^{3} \cdot z} + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{{y}^{3} \cdot \color{blue}{z} + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      2. lower-pow.f64 79.2

         \[\leadsto \frac{{y}^{3} \cdot z + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   4. Applied rewrites 79.2%

      \[\leadsto \frac{\color{blue}{{y}^{3} \cdot z} + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{{y}^{3} \cdot z + t}{\color{blue}{c} \cdot y + i} \]
   6. Step-by-step derivation

   7. Applied rewrites 70.7%

      \[\leadsto \frac{{y}^{3} \cdot z + t}{\color{blue}{c} \cdot y + i} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 66.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{z}{y}\\ \mathbf{if}\;y \leq -4.2 \cdot 10^{+30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.75 \cdot 10^{+38}:\\ \;\;\;\;\frac{{y}^{3} \cdot z + t}{c \cdot y + i}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ x (/ z y))))
   (if (<= y -4.2e+30)
     t_1
     (if (<= y 2.75e+38) (/ (+ (* (pow y 3.0) z) t) (+ (* c y) i)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = x + (z / y);
	double tmp;
	if (y <= -4.2e+30) {
		tmp = t_1;
	} else if (y <= 2.75e+38) {
		tmp = ((pow(y, 3.0) * z) + t) / ((c * y) + i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + (z / y)
    if (y <= (-4.2d+30)) then
        tmp = t_1
    else if (y <= 2.75d+38) then
        tmp = (((y ** 3.0d0) * z) + t) / ((c * y) + i)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = x + (z / y);
	double tmp;
	if (y <= -4.2e+30) {
		tmp = t_1;
	} else if (y <= 2.75e+38) {
		tmp = ((Math.pow(y, 3.0) * z) + t) / ((c * y) + i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = x + (z / y)
	tmp = 0
	if y <= -4.2e+30:
		tmp = t_1
	elif y <= 2.75e+38:
		tmp = ((math.pow(y, 3.0) * z) + t) / ((c * y) + i)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x + Float64(z / y))
	tmp = 0.0
	if (y <= -4.2e+30)
		tmp = t_1;
	elseif (y <= 2.75e+38)
		tmp = Float64(Float64(Float64((y ^ 3.0) * z) + t) / Float64(Float64(c * y) + i));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = x + (z / y);
	tmp = 0.0;
	if (y <= -4.2e+30)
		tmp = t_1;
	elseif (y <= 2.75e+38)
		tmp = (((y ^ 3.0) * z) + t) / ((c * y) + i);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x + N[(z / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.2e+30], t$95$1, If[LessEqual[y, 2.75e+38], N[(N[(N[(N[Power[y, 3.0], $MachinePrecision] * z), $MachinePrecision] + t), $MachinePrecision] / N[(N[(c * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.2e30 or 2.7500000000000002e38 < y

   1. Initial program 5.4%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 63.6

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

   4. Applied rewrites 63.6%

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

   5. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lift-/.f64 68.5

         \[\leadsto x + \frac{z}{y} \]

   7. Applied rewrites 68.5%

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

   if -4.2e30 < y < 2.7500000000000002e38

   1. Initial program 96.5%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in z around inf

      \[\leadsto \frac{\color{blue}{{y}^{3} \cdot z} + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{{y}^{3} \cdot \color{blue}{z} + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      2. lower-pow.f64 76.3

         \[\leadsto \frac{{y}^{3} \cdot z + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   4. Applied rewrites 76.3%

      \[\leadsto \frac{\color{blue}{{y}^{3} \cdot z} + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{{y}^{3} \cdot z + t}{\color{blue}{c} \cdot y + i} \]
   6. Step-by-step derivation

   7. Applied rewrites 66.3%

      \[\leadsto \frac{{y}^{3} \cdot z + t}{\color{blue}{c} \cdot y + i} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 59.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{z}{y}\\ \mathbf{if}\;y \leq -8 \cdot 10^{+44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+42}:\\ \;\;\;\;\frac{{y}^{3} \cdot z + t}{{y}^{4} + i}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ x (/ z y))))
   (if (<= y -8e+44)
     t_1
     (if (<= y 1.6e+42) (/ (+ (* (pow y 3.0) z) t) (+ (pow y 4.0) i)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = x + (z / y);
	double tmp;
	if (y <= -8e+44) {
		tmp = t_1;
	} else if (y <= 1.6e+42) {
		tmp = ((pow(y, 3.0) * z) + t) / (pow(y, 4.0) + i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + (z / y)
    if (y <= (-8d+44)) then
        tmp = t_1
    else if (y <= 1.6d+42) then
        tmp = (((y ** 3.0d0) * z) + t) / ((y ** 4.0d0) + i)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = x + (z / y);
	double tmp;
	if (y <= -8e+44) {
		tmp = t_1;
	} else if (y <= 1.6e+42) {
		tmp = ((Math.pow(y, 3.0) * z) + t) / (Math.pow(y, 4.0) + i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = x + (z / y)
	tmp = 0
	if y <= -8e+44:
		tmp = t_1
	elif y <= 1.6e+42:
		tmp = ((math.pow(y, 3.0) * z) + t) / (math.pow(y, 4.0) + i)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x + Float64(z / y))
	tmp = 0.0
	if (y <= -8e+44)
		tmp = t_1;
	elseif (y <= 1.6e+42)
		tmp = Float64(Float64(Float64((y ^ 3.0) * z) + t) / Float64((y ^ 4.0) + i));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = x + (z / y);
	tmp = 0.0;
	if (y <= -8e+44)
		tmp = t_1;
	elseif (y <= 1.6e+42)
		tmp = (((y ^ 3.0) * z) + t) / ((y ^ 4.0) + i);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x + N[(z / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8e+44], t$95$1, If[LessEqual[y, 1.6e+42], N[(N[(N[(N[Power[y, 3.0], $MachinePrecision] * z), $MachinePrecision] + t), $MachinePrecision] / N[(N[Power[y, 4.0], $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.0000000000000007e44 or 1.60000000000000001e42 < y

   1. Initial program 4.0%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 65.2

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

   4. Applied rewrites 65.2%

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

   5. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lift-/.f64 70.2

         \[\leadsto x + \frac{z}{y} \]

   7. Applied rewrites 70.2%

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

   if -8.0000000000000007e44 < y < 1.60000000000000001e42

   1. Initial program 95.4%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in z around inf

      \[\leadsto \frac{\color{blue}{{y}^{3} \cdot z} + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{{y}^{3} \cdot \color{blue}{z} + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      2. lower-pow.f64 75.2

         \[\leadsto \frac{{y}^{3} \cdot z + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   4. Applied rewrites 75.2%

      \[\leadsto \frac{\color{blue}{{y}^{3} \cdot z} + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   5. Taylor expanded in y around inf

      \[\leadsto \frac{{y}^{3} \cdot z + t}{\color{blue}{{y}^{4}} + i} \]
   6. Step-by-step derivation

      1. lower-pow.f64 50.8

         \[\leadsto \frac{{y}^{3} \cdot z + t}{{y}^{\color{blue}{4}} + i} \]

   7. Applied rewrites 50.8%

      \[\leadsto \frac{{y}^{3} \cdot z + t}{\color{blue}{{y}^{4}} + i} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 57.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{z}{y}\\ \mathbf{if}\;y \leq -4.2 \cdot 10^{+30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.75 \cdot 10^{+38}:\\ \;\;\;\;\frac{t}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ x (/ z y))))
   (if (<= y -4.2e+30) t_1 (if (<= y 2.75e+38) (/ t i) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = x + (z / y);
	double tmp;
	if (y <= -4.2e+30) {
		tmp = t_1;
	} else if (y <= 2.75e+38) {
		tmp = t / i;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + (z / y)
    if (y <= (-4.2d+30)) then
        tmp = t_1
    else if (y <= 2.75d+38) then
        tmp = t / i
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = x + (z / y);
	double tmp;
	if (y <= -4.2e+30) {
		tmp = t_1;
	} else if (y <= 2.75e+38) {
		tmp = t / i;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = x + (z / y)
	tmp = 0
	if y <= -4.2e+30:
		tmp = t_1
	elif y <= 2.75e+38:
		tmp = t / i
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x + Float64(z / y))
	tmp = 0.0
	if (y <= -4.2e+30)
		tmp = t_1;
	elseif (y <= 2.75e+38)
		tmp = Float64(t / i);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = x + (z / y);
	tmp = 0.0;
	if (y <= -4.2e+30)
		tmp = t_1;
	elseif (y <= 2.75e+38)
		tmp = t / i;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x + N[(z / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.2e+30], t$95$1, If[LessEqual[y, 2.75e+38], N[(t / i), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.2e30 or 2.7500000000000002e38 < y

   1. Initial program 5.4%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 63.6

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

   4. Applied rewrites 63.6%

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

   5. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lift-/.f64 68.5

         \[\leadsto x + \frac{z}{y} \]

   7. Applied rewrites 68.5%

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

   if -4.2e30 < y < 2.7500000000000002e38

   1. Initial program 96.5%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{t}{i}} \]
   3. Step-by-step derivation

      1. lower-/.f64 48.8

         \[\leadsto \frac{t}{\color{blue}{i}} \]

   4. Applied rewrites 48.8%

      \[\leadsto \color{blue}{\frac{t}{i}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 50.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-23}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+41}:\\ \;\;\;\;\frac{t}{i}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -2.6e-23) x (if (<= y 1.35e+41) (/ t i) x)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= -2.6e-23) {
		tmp = x;
	} else if (y <= 1.35e+41) {
		tmp = t / i;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if (y <= (-2.6d-23)) then
        tmp = x
    else if (y <= 1.35d+41) then
        tmp = t / i
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= -2.6e-23) {
		tmp = x;
	} else if (y <= 1.35e+41) {
		tmp = t / i;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if y <= -2.6e-23:
		tmp = x
	elif y <= 1.35e+41:
		tmp = t / i
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= -2.6e-23)
		tmp = x;
	elseif (y <= 1.35e+41)
		tmp = Float64(t / i);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (y <= -2.6e-23)
		tmp = x;
	elseif (y <= 1.35e+41)
		tmp = t / i;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, -2.6e-23], x, If[LessEqual[y, 1.35e+41], N[(t / i), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -2.6e-23 or 1.35e41 < y

   1. Initial program 12.1%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 49.5%

      \[\leadsto \color{blue}{x} \]

   if -2.6e-23 < y < 1.35e41

   1. Initial program 97.4%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{t}{i}} \]
   3. Step-by-step derivation

      1. lower-/.f64 51.9

         \[\leadsto \frac{t}{\color{blue}{i}} \]

   4. Applied rewrites 51.9%

      \[\leadsto \color{blue}{\frac{t}{i}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 26.6% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 5.6 \cdot 10^{+196}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i) :precision binary64 (if (<= z 5.6e+196) x (/ z y)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (z <= 5.6e+196) {
		tmp = x;
	} else {
		tmp = z / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if (z <= 5.6d+196) then
        tmp = x
    else
        tmp = z / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (z <= 5.6e+196) {
		tmp = x;
	} else {
		tmp = z / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if z <= 5.6e+196:
		tmp = x
	else:
		tmp = z / y
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (z <= 5.6e+196)
		tmp = x;
	else
		tmp = Float64(z / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (z <= 5.6e+196)
		tmp = x;
	else
		tmp = z / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[z, 5.6e+196], x, N[(z / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5.6000000000000004e196

   1. Initial program 56.0%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 27.2%

      \[\leadsto \color{blue}{x} \]

   if 5.6000000000000004e196 < z

   1. Initial program 51.4%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{{y}^{3} \cdot z}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto {y}^{3} \cdot \color{blue}{\frac{z}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto {y}^{3} \cdot \color{blue}{\frac{z}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      3. lower-pow.f64 N/A

         \[\leadsto {y}^{3} \cdot \frac{\color{blue}{z}}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto {y}^{3} \cdot \frac{z}{\color{blue}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      5. +-commutative N/A

         \[\leadsto {y}^{3} \cdot \frac{z}{y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right) + \color{blue}{i}} \]

      6. *-commutative N/A

         \[\leadsto {y}^{3} \cdot \frac{z}{\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right) \cdot y + i} \]

      7. lower-fma.f64 N/A

         \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right), \color{blue}{y}, i\right)} \]

   4. Applied rewrites 20.3%

      \[\leadsto \color{blue}{{y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}} \]

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 21.5

         \[\leadsto \frac{z}{y} \]

   7. Applied rewrites 21.5%

      \[\leadsto \frac{z}{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 26.2% accurate, 33.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z t a b c i) :precision binary64 x)

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return x;
}

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, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = x
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return x;
}

Python

def code(x, y, z, t, a, b, c, i):
	return x

Julia

function code(x, y, z, t, a, b, c, i)
	return x
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = x;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := x

Derivation

1. Initial program 55.6%

   \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

2. Taylor expanded in y around inf

   \[\leadsto \color{blue}{x} \]
3. Step-by-step derivation

4. Applied rewrites 26.2%

   \[\leadsto \color{blue}{x} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025101 
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2"
  :precision binary64
  (/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))