Data.Random.Distribution.Triangular:triangularCDF from random-fu-0.2.6.2, B

Percentage Accurate: 89.2% → 98.1%

Time: 4.0s

Alternatives: 14

Speedup: 0.6×

Specification

Math

\[\begin{array}{l} \\ \frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x\_m}{\left(y - z\right) \cdot \left(t - z\right)}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-319}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{t - z}}{y - z}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ x_m (* (- y z) (- t z)))))
   (* x_s (if (<= t_1 -1e-319) t_1 (/ (/ x_m (- t z)) (- y z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / ((y - z) * (t - z));
	double tmp;
	if (t_1 <= -1e-319) {
		tmp = t_1;
	} else {
		tmp = (x_m / (t - z)) / (y - z);
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x_m / ((y - z) * (t - z))
    if (t_1 <= (-1d-319)) then
        tmp = t_1
    else
        tmp = (x_m / (t - z)) / (y - z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / ((y - z) * (t - z));
	double tmp;
	if (t_1 <= -1e-319) {
		tmp = t_1;
	} else {
		tmp = (x_m / (t - z)) / (y - z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	t_1 = x_m / ((y - z) * (t - z))
	tmp = 0
	if t_1 <= -1e-319:
		tmp = t_1
	else:
		tmp = (x_m / (t - z)) / (y - z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m / Float64(Float64(y - z) * Float64(t - z)))
	tmp = 0.0
	if (t_1 <= -1e-319)
		tmp = t_1;
	else
		tmp = Float64(Float64(x_m / Float64(t - z)) / Float64(y - z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m / ((y - z) * (t - z));
	tmp = 0.0;
	if (t_1 <= -1e-319)
		tmp = t_1;
	else
		tmp = (x_m / (t - z)) / (y - z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m / N[(N[(y - z), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, -1e-319], t$95$1, N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 t z))) < -9.99989e-320

   1. Initial program 98.2%

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

   if -9.99989e-320 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 t z)))

   1. Initial program 86.9%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift--.f64 98.0

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

   3. Applied rewrites 98.0%

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

Alternative 2: 93.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{\frac{x\_m}{-z}}{y - z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+135}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+84}:\\ \;\;\;\;\frac{x\_m}{\left(y - z\right) \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ (/ x_m (- z)) (- y z))))
   (*
    x_s
    (if (<= z -1.75e+135)
      t_1
      (if (<= z 5e+84) (/ x_m (* (- y z) (- t z))) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m / -z) / (y - z);
	double tmp;
	if (z <= -1.75e+135) {
		tmp = t_1;
	} else if (z <= 5e+84) {
		tmp = x_m / ((y - z) * (t - z));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x_m / -z) / (y - z)
    if (z <= (-1.75d+135)) then
        tmp = t_1
    else if (z <= 5d+84) then
        tmp = x_m / ((y - z) * (t - z))
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m / -z) / (y - z);
	double tmp;
	if (z <= -1.75e+135) {
		tmp = t_1;
	} else if (z <= 5e+84) {
		tmp = x_m / ((y - z) * (t - z));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	t_1 = (x_m / -z) / (y - z)
	tmp = 0
	if z <= -1.75e+135:
		tmp = t_1
	elif z <= 5e+84:
		tmp = x_m / ((y - z) * (t - z))
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(x_m / Float64(-z)) / Float64(y - z))
	tmp = 0.0
	if (z <= -1.75e+135)
		tmp = t_1;
	elseif (z <= 5e+84)
		tmp = Float64(x_m / Float64(Float64(y - z) * Float64(t - z)));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = (x_m / -z) / (y - z);
	tmp = 0.0;
	if (z <= -1.75e+135)
		tmp = t_1;
	elseif (z <= 5e+84)
		tmp = x_m / ((y - z) * (t - z));
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x$95$m / (-z)), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -1.75e+135], t$95$1, If[LessEqual[z, 5e+84], N[(x$95$m / N[(N[(y - z), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.7500000000000001e135 or 5.0000000000000001e84 < z

   1. Initial program 80.4%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift--.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in z around inf

      \[\leadsto \frac{\frac{x}{\color{blue}{-1 \cdot z}}}{y - z} \]
   5. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\frac{x}{\mathsf{neg}\left(z\right)}}{y - z} \]

      2. lift-neg.f64 94.1

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

   6. Applied rewrites 94.1%

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

   if -1.7500000000000001e135 < z < 5.0000000000000001e84

   1. Initial program 93.4%

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

Alternative 3: 82.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x\_m}{t - z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.96 \cdot 10^{+186}:\\ \;\;\;\;\frac{t\_1}{y}\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{x\_m}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-106}:\\ \;\;\;\;\frac{t\_1}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{y - z}}{t}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ x_m (- t z))))
   (*
    x_s
    (if (<= y -1.96e+186)
      (/ t_1 y)
      (if (<= y -7.2e+14)
        (/ x_m (* y (- t z)))
        (if (<= y 1.85e-106) (/ t_1 (- z)) (/ (/ x_m (- y z)) t)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / (t - z);
	double tmp;
	if (y <= -1.96e+186) {
		tmp = t_1 / y;
	} else if (y <= -7.2e+14) {
		tmp = x_m / (y * (t - z));
	} else if (y <= 1.85e-106) {
		tmp = t_1 / -z;
	} else {
		tmp = (x_m / (y - z)) / t;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x_m / (t - z)
    if (y <= (-1.96d+186)) then
        tmp = t_1 / y
    else if (y <= (-7.2d+14)) then
        tmp = x_m / (y * (t - z))
    else if (y <= 1.85d-106) then
        tmp = t_1 / -z
    else
        tmp = (x_m / (y - z)) / t
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / (t - z);
	double tmp;
	if (y <= -1.96e+186) {
		tmp = t_1 / y;
	} else if (y <= -7.2e+14) {
		tmp = x_m / (y * (t - z));
	} else if (y <= 1.85e-106) {
		tmp = t_1 / -z;
	} else {
		tmp = (x_m / (y - z)) / t;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	t_1 = x_m / (t - z)
	tmp = 0
	if y <= -1.96e+186:
		tmp = t_1 / y
	elif y <= -7.2e+14:
		tmp = x_m / (y * (t - z))
	elif y <= 1.85e-106:
		tmp = t_1 / -z
	else:
		tmp = (x_m / (y - z)) / t
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m / Float64(t - z))
	tmp = 0.0
	if (y <= -1.96e+186)
		tmp = Float64(t_1 / y);
	elseif (y <= -7.2e+14)
		tmp = Float64(x_m / Float64(y * Float64(t - z)));
	elseif (y <= 1.85e-106)
		tmp = Float64(t_1 / Float64(-z));
	else
		tmp = Float64(Float64(x_m / Float64(y - z)) / t);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m / (t - z);
	tmp = 0.0;
	if (y <= -1.96e+186)
		tmp = t_1 / y;
	elseif (y <= -7.2e+14)
		tmp = x_m / (y * (t - z));
	elseif (y <= 1.85e-106)
		tmp = t_1 / -z;
	else
		tmp = (x_m / (y - z)) / t;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -1.96e+186], N[(t$95$1 / y), $MachinePrecision], If[LessEqual[y, -7.2e+14], N[(x$95$m / N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.85e-106], N[(t$95$1 / (-z)), $MachinePrecision], N[(N[(x$95$m / N[(y - z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if y < -1.95999999999999994e186

   1. Initial program 83.4%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift--.f64 94.8

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

   3. Applied rewrites 94.8%

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

   4. Taylor expanded in z around inf

      \[\leadsto \frac{\frac{x}{t - z}}{\color{blue}{z \cdot \left(\frac{y}{z} - 1\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{x}{t - z}}{\left(\frac{y}{z} - 1\right) \cdot \color{blue}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{x}{t - z}}{\left(\frac{y}{z} - 1\right) \cdot \color{blue}{z}} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\frac{x}{t - z}}{\left(\frac{y}{z} - 1\right) \cdot z} \]

      4. lower-/.f64 76.2

         \[\leadsto \frac{\frac{x}{t - z}}{\left(\frac{y}{z} - 1\right) \cdot z} \]

   6. Applied rewrites 76.2%

      \[\leadsto \frac{\frac{x}{t - z}}{\color{blue}{\left(\frac{y}{z} - 1\right) \cdot z}} \]

   7. Taylor expanded in y around inf

      \[\leadsto \frac{\frac{x}{t - z}}{\frac{y}{z} \cdot z} \]
   8. Step-by-step derivation

      1. lift-/.f64 74.6

         \[\leadsto \frac{\frac{x}{t - z}}{\frac{y}{z} \cdot z} \]

   9. Applied rewrites 74.6%

      \[\leadsto \frac{\frac{x}{t - z}}{\frac{y}{z} \cdot z} \]

   10. Taylor expanded in y around inf

       \[\leadsto \frac{\frac{x}{t - z}}{y} \]
   11. Step-by-step derivation

   12. Applied rewrites 93.2%

       \[\leadsto \frac{\frac{x}{t - z}}{y} \]

   if -1.95999999999999994e186 < y < -7.2e14

   1. Initial program 89.4%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 81.9%

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

   if -7.2e14 < y < 1.8499999999999999e-106

   1. Initial program 91.6%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift--.f64 97.0

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

   3. Applied rewrites 97.0%

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

   4. Taylor expanded in z around inf

      \[\leadsto \frac{\frac{x}{t - z}}{\color{blue}{z \cdot \left(\frac{y}{z} - 1\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{x}{t - z}}{\left(\frac{y}{z} - 1\right) \cdot \color{blue}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{x}{t - z}}{\left(\frac{y}{z} - 1\right) \cdot \color{blue}{z}} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\frac{x}{t - z}}{\left(\frac{y}{z} - 1\right) \cdot z} \]

      4. lower-/.f64 96.9

         \[\leadsto \frac{\frac{x}{t - z}}{\left(\frac{y}{z} - 1\right) \cdot z} \]

   6. Applied rewrites 96.9%

      \[\leadsto \frac{\frac{x}{t - z}}{\color{blue}{\left(\frac{y}{z} - 1\right) \cdot z}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{x}{t - z}}{-1 \cdot \color{blue}{z}} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\frac{x}{t - z}}{\mathsf{neg}\left(z\right)} \]

      2. lift-neg.f64 77.9

         \[\leadsto \frac{\frac{x}{t - z}}{-z} \]

   9. Applied rewrites 77.9%

      \[\leadsto \frac{\frac{x}{t - z}}{-z} \]

   if 1.8499999999999999e-106 < y

   1. Initial program 88.0%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 80.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift--.f64 86.9

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

   6. Applied rewrites 86.9%

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

Alternative 4: 82.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.96 \cdot 10^{+186}:\\ \;\;\;\;\frac{\frac{x\_m}{t - z}}{y}\\ \mathbf{elif}\;y \leq -5400000000000:\\ \;\;\;\;\frac{x\_m}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{x\_m}{-z}}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{y - z}}{t}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= y -1.96e+186)
    (/ (/ x_m (- t z)) y)
    (if (<= y -5400000000000.0)
      (/ x_m (* y (- t z)))
      (if (<= y 8e-107) (/ (/ x_m (- z)) (- t z)) (/ (/ x_m (- y z)) t))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -1.96e+186) {
		tmp = (x_m / (t - z)) / y;
	} else if (y <= -5400000000000.0) {
		tmp = x_m / (y * (t - z));
	} else if (y <= 8e-107) {
		tmp = (x_m / -z) / (t - z);
	} else {
		tmp = (x_m / (y - z)) / t;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-1.96d+186)) then
        tmp = (x_m / (t - z)) / y
    else if (y <= (-5400000000000.0d0)) then
        tmp = x_m / (y * (t - z))
    else if (y <= 8d-107) then
        tmp = (x_m / -z) / (t - z)
    else
        tmp = (x_m / (y - z)) / t
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -1.96e+186) {
		tmp = (x_m / (t - z)) / y;
	} else if (y <= -5400000000000.0) {
		tmp = x_m / (y * (t - z));
	} else if (y <= 8e-107) {
		tmp = (x_m / -z) / (t - z);
	} else {
		tmp = (x_m / (y - z)) / t;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if y <= -1.96e+186:
		tmp = (x_m / (t - z)) / y
	elif y <= -5400000000000.0:
		tmp = x_m / (y * (t - z))
	elif y <= 8e-107:
		tmp = (x_m / -z) / (t - z)
	else:
		tmp = (x_m / (y - z)) / t
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (y <= -1.96e+186)
		tmp = Float64(Float64(x_m / Float64(t - z)) / y);
	elseif (y <= -5400000000000.0)
		tmp = Float64(x_m / Float64(y * Float64(t - z)));
	elseif (y <= 8e-107)
		tmp = Float64(Float64(x_m / Float64(-z)) / Float64(t - z));
	else
		tmp = Float64(Float64(x_m / Float64(y - z)) / t);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (y <= -1.96e+186)
		tmp = (x_m / (t - z)) / y;
	elseif (y <= -5400000000000.0)
		tmp = x_m / (y * (t - z));
	elseif (y <= 8e-107)
		tmp = (x_m / -z) / (t - z);
	else
		tmp = (x_m / (y - z)) / t;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[y, -1.96e+186], N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, -5400000000000.0], N[(x$95$m / N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8e-107], N[(N[(x$95$m / (-z)), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(y - z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if y < -1.95999999999999994e186

   1. Initial program 83.4%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift--.f64 94.8

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

   3. Applied rewrites 94.8%

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

   4. Taylor expanded in z around inf

      \[\leadsto \frac{\frac{x}{t - z}}{\color{blue}{z \cdot \left(\frac{y}{z} - 1\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{x}{t - z}}{\left(\frac{y}{z} - 1\right) \cdot \color{blue}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{x}{t - z}}{\left(\frac{y}{z} - 1\right) \cdot \color{blue}{z}} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\frac{x}{t - z}}{\left(\frac{y}{z} - 1\right) \cdot z} \]

      4. lower-/.f64 76.2

         \[\leadsto \frac{\frac{x}{t - z}}{\left(\frac{y}{z} - 1\right) \cdot z} \]

   6. Applied rewrites 76.2%

      \[\leadsto \frac{\frac{x}{t - z}}{\color{blue}{\left(\frac{y}{z} - 1\right) \cdot z}} \]

   7. Taylor expanded in y around inf

      \[\leadsto \frac{\frac{x}{t - z}}{\frac{y}{z} \cdot z} \]
   8. Step-by-step derivation

      1. lift-/.f64 74.6

         \[\leadsto \frac{\frac{x}{t - z}}{\frac{y}{z} \cdot z} \]

   9. Applied rewrites 74.6%

      \[\leadsto \frac{\frac{x}{t - z}}{\frac{y}{z} \cdot z} \]

   10. Taylor expanded in y around inf

       \[\leadsto \frac{\frac{x}{t - z}}{y} \]
   11. Step-by-step derivation

   12. Applied rewrites 93.2%

       \[\leadsto \frac{\frac{x}{t - z}}{y} \]

   if -1.95999999999999994e186 < y < -5.4e12

   1. Initial program 89.4%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 81.7%

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

   if -5.4e12 < y < 8e-107

   1. Initial program 91.6%

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

   2. Taylor expanded in y around 0

      \[\leadsto \frac{x}{\color{blue}{\left(-1 \cdot z\right)} \cdot \left(t - z\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 71.8

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

   4. Applied rewrites 71.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{-z}}{t - z}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{-z}}{t - z}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{-z}}}{t - z} \]

      7. lift--.f64 77.2

         \[\leadsto \frac{\frac{x}{-z}}{\color{blue}{t - z}} \]

   6. Applied rewrites 77.2%

      \[\leadsto \color{blue}{\frac{\frac{x}{-z}}{t - z}} \]

   if 8e-107 < y

   1. Initial program 88.0%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 80.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift--.f64 86.9

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

   6. Applied rewrites 86.9%

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

Alternative 5: 82.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.96 \cdot 10^{+186}:\\ \;\;\;\;\frac{\frac{x\_m}{t - z}}{y}\\ \mathbf{elif}\;y \leq -5400000000000:\\ \;\;\;\;\frac{x\_m}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-106}:\\ \;\;\;\;\frac{x\_m}{\left(-z\right) \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{y - z}}{t}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= y -1.96e+186)
    (/ (/ x_m (- t z)) y)
    (if (<= y -5400000000000.0)
      (/ x_m (* y (- t z)))
      (if (<= y 1.25e-106) (/ x_m (* (- z) (- t z))) (/ (/ x_m (- y z)) t))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -1.96e+186) {
		tmp = (x_m / (t - z)) / y;
	} else if (y <= -5400000000000.0) {
		tmp = x_m / (y * (t - z));
	} else if (y <= 1.25e-106) {
		tmp = x_m / (-z * (t - z));
	} else {
		tmp = (x_m / (y - z)) / t;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-1.96d+186)) then
        tmp = (x_m / (t - z)) / y
    else if (y <= (-5400000000000.0d0)) then
        tmp = x_m / (y * (t - z))
    else if (y <= 1.25d-106) then
        tmp = x_m / (-z * (t - z))
    else
        tmp = (x_m / (y - z)) / t
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -1.96e+186) {
		tmp = (x_m / (t - z)) / y;
	} else if (y <= -5400000000000.0) {
		tmp = x_m / (y * (t - z));
	} else if (y <= 1.25e-106) {
		tmp = x_m / (-z * (t - z));
	} else {
		tmp = (x_m / (y - z)) / t;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if y <= -1.96e+186:
		tmp = (x_m / (t - z)) / y
	elif y <= -5400000000000.0:
		tmp = x_m / (y * (t - z))
	elif y <= 1.25e-106:
		tmp = x_m / (-z * (t - z))
	else:
		tmp = (x_m / (y - z)) / t
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (y <= -1.96e+186)
		tmp = Float64(Float64(x_m / Float64(t - z)) / y);
	elseif (y <= -5400000000000.0)
		tmp = Float64(x_m / Float64(y * Float64(t - z)));
	elseif (y <= 1.25e-106)
		tmp = Float64(x_m / Float64(Float64(-z) * Float64(t - z)));
	else
		tmp = Float64(Float64(x_m / Float64(y - z)) / t);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (y <= -1.96e+186)
		tmp = (x_m / (t - z)) / y;
	elseif (y <= -5400000000000.0)
		tmp = x_m / (y * (t - z));
	elseif (y <= 1.25e-106)
		tmp = x_m / (-z * (t - z));
	else
		tmp = (x_m / (y - z)) / t;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[y, -1.96e+186], N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, -5400000000000.0], N[(x$95$m / N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.25e-106], N[(x$95$m / N[((-z) * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(y - z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if y < -1.95999999999999994e186

   1. Initial program 83.4%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift--.f64 94.8

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

   3. Applied rewrites 94.8%

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

   4. Taylor expanded in z around inf

      \[\leadsto \frac{\frac{x}{t - z}}{\color{blue}{z \cdot \left(\frac{y}{z} - 1\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{x}{t - z}}{\left(\frac{y}{z} - 1\right) \cdot \color{blue}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{x}{t - z}}{\left(\frac{y}{z} - 1\right) \cdot \color{blue}{z}} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\frac{x}{t - z}}{\left(\frac{y}{z} - 1\right) \cdot z} \]

      4. lower-/.f64 76.2

         \[\leadsto \frac{\frac{x}{t - z}}{\left(\frac{y}{z} - 1\right) \cdot z} \]

   6. Applied rewrites 76.2%

      \[\leadsto \frac{\frac{x}{t - z}}{\color{blue}{\left(\frac{y}{z} - 1\right) \cdot z}} \]

   7. Taylor expanded in y around inf

      \[\leadsto \frac{\frac{x}{t - z}}{\frac{y}{z} \cdot z} \]
   8. Step-by-step derivation

      1. lift-/.f64 74.6

         \[\leadsto \frac{\frac{x}{t - z}}{\frac{y}{z} \cdot z} \]

   9. Applied rewrites 74.6%

      \[\leadsto \frac{\frac{x}{t - z}}{\frac{y}{z} \cdot z} \]

   10. Taylor expanded in y around inf

       \[\leadsto \frac{\frac{x}{t - z}}{y} \]
   11. Step-by-step derivation

   12. Applied rewrites 93.2%

       \[\leadsto \frac{\frac{x}{t - z}}{y} \]

   if -1.95999999999999994e186 < y < -5.4e12

   1. Initial program 89.4%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 81.7%

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

   if -5.4e12 < y < 1.24999999999999996e-106

   1. Initial program 91.6%

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

   2. Taylor expanded in y around 0

      \[\leadsto \frac{x}{\color{blue}{\left(-1 \cdot z\right)} \cdot \left(t - z\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 71.8

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

   4. Applied rewrites 71.8%

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

   if 1.24999999999999996e-106 < y

   1. Initial program 88.0%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 80.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift--.f64 86.9

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

   6. Applied rewrites 86.9%

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

Alternative 6: 79.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{-113}:\\ \;\;\;\;\frac{\frac{x\_m}{y}}{t - z}\\ \mathbf{elif}\;t \leq 6.3 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{x\_m}{-z}}{y - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y - z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= t -1.6e-113)
    (/ (/ x_m y) (- t z))
    (if (<= t 6.3e-52) (/ (/ x_m (- z)) (- y z)) (/ (/ x_m t) (- y z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= -1.6e-113) {
		tmp = (x_m / y) / (t - z);
	} else if (t <= 6.3e-52) {
		tmp = (x_m / -z) / (y - z);
	} else {
		tmp = (x_m / t) / (y - z);
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.6d-113)) then
        tmp = (x_m / y) / (t - z)
    else if (t <= 6.3d-52) then
        tmp = (x_m / -z) / (y - z)
    else
        tmp = (x_m / t) / (y - z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= -1.6e-113) {
		tmp = (x_m / y) / (t - z);
	} else if (t <= 6.3e-52) {
		tmp = (x_m / -z) / (y - z);
	} else {
		tmp = (x_m / t) / (y - z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if t <= -1.6e-113:
		tmp = (x_m / y) / (t - z)
	elif t <= 6.3e-52:
		tmp = (x_m / -z) / (y - z)
	else:
		tmp = (x_m / t) / (y - z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (t <= -1.6e-113)
		tmp = Float64(Float64(x_m / y) / Float64(t - z));
	elseif (t <= 6.3e-52)
		tmp = Float64(Float64(x_m / Float64(-z)) / Float64(y - z));
	else
		tmp = Float64(Float64(x_m / t) / Float64(y - z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (t <= -1.6e-113)
		tmp = (x_m / y) / (t - z);
	elseif (t <= 6.3e-52)
		tmp = (x_m / -z) / (y - z);
	else
		tmp = (x_m / t) / (y - z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[t, -1.6e-113], N[(N[(x$95$m / y), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.3e-52], N[(N[(x$95$m / (-z)), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -1.6000000000000001e-113

   1. Initial program 86.9%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 81.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 85.1

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

   6. Applied rewrites 85.1%

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

   if -1.6000000000000001e-113 < t < 6.3000000000000003e-52

   1. Initial program 90.6%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift--.f64 96.2

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

   3. Applied rewrites 96.2%

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

   4. Taylor expanded in z around inf

      \[\leadsto \frac{\frac{x}{\color{blue}{-1 \cdot z}}}{y - z} \]
   5. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\frac{x}{\mathsf{neg}\left(z\right)}}{y - z} \]

      2. lift-neg.f64 81.3

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

   6. Applied rewrites 81.3%

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

   if 6.3000000000000003e-52 < t

   1. Initial program 88.6%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift--.f64 96.7

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

   3. Applied rewrites 96.7%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 81.9%

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

Alternative 7: 73.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq 1.15 \cdot 10^{-234}:\\ \;\;\;\;\frac{\frac{x\_m}{t - z}}{y}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-53}:\\ \;\;\;\;\frac{\frac{x\_m}{-z}}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y - z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= t 1.15e-234)
    (/ (/ x_m (- t z)) y)
    (if (<= t 3.2e-53) (/ (/ x_m (- z)) (- z)) (/ (/ x_m t) (- y z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= 1.15e-234) {
		tmp = (x_m / (t - z)) / y;
	} else if (t <= 3.2e-53) {
		tmp = (x_m / -z) / -z;
	} else {
		tmp = (x_m / t) / (y - z);
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 1.15d-234) then
        tmp = (x_m / (t - z)) / y
    else if (t <= 3.2d-53) then
        tmp = (x_m / -z) / -z
    else
        tmp = (x_m / t) / (y - z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= 1.15e-234) {
		tmp = (x_m / (t - z)) / y;
	} else if (t <= 3.2e-53) {
		tmp = (x_m / -z) / -z;
	} else {
		tmp = (x_m / t) / (y - z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if t <= 1.15e-234:
		tmp = (x_m / (t - z)) / y
	elif t <= 3.2e-53:
		tmp = (x_m / -z) / -z
	else:
		tmp = (x_m / t) / (y - z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (t <= 1.15e-234)
		tmp = Float64(Float64(x_m / Float64(t - z)) / y);
	elseif (t <= 3.2e-53)
		tmp = Float64(Float64(x_m / Float64(-z)) / Float64(-z));
	else
		tmp = Float64(Float64(x_m / t) / Float64(y - z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (t <= 1.15e-234)
		tmp = (x_m / (t - z)) / y;
	elseif (t <= 3.2e-53)
		tmp = (x_m / -z) / -z;
	else
		tmp = (x_m / t) / (y - z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[t, 1.15e-234], N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t, 3.2e-53], N[(N[(x$95$m / (-z)), $MachinePrecision] / (-z)), $MachinePrecision], N[(N[(x$95$m / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.14999999999999995e-234

   1. Initial program 89.2%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift--.f64 96.3

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

   3. Applied rewrites 96.3%

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

   4. Taylor expanded in z around inf

      \[\leadsto \frac{\frac{x}{t - z}}{\color{blue}{z \cdot \left(\frac{y}{z} - 1\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{x}{t - z}}{\left(\frac{y}{z} - 1\right) \cdot \color{blue}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{x}{t - z}}{\left(\frac{y}{z} - 1\right) \cdot \color{blue}{z}} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\frac{x}{t - z}}{\left(\frac{y}{z} - 1\right) \cdot z} \]

      4. lower-/.f64 85.8

         \[\leadsto \frac{\frac{x}{t - z}}{\left(\frac{y}{z} - 1\right) \cdot z} \]

   6. Applied rewrites 85.8%

      \[\leadsto \frac{\frac{x}{t - z}}{\color{blue}{\left(\frac{y}{z} - 1\right) \cdot z}} \]

   7. Taylor expanded in y around inf

      \[\leadsto \frac{\frac{x}{t - z}}{\frac{y}{z} \cdot z} \]
   8. Step-by-step derivation

      1. lift-/.f64 58.8

         \[\leadsto \frac{\frac{x}{t - z}}{\frac{y}{z} \cdot z} \]

   9. Applied rewrites 58.8%

      \[\leadsto \frac{\frac{x}{t - z}}{\frac{y}{z} \cdot z} \]

   10. Taylor expanded in y around inf

       \[\leadsto \frac{\frac{x}{t - z}}{y} \]
   11. Step-by-step derivation

   12. Applied rewrites 71.0%

       \[\leadsto \frac{\frac{x}{t - z}}{y} \]

   if 1.14999999999999995e-234 < t < 3.2000000000000001e-53

   1. Initial program 90.6%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift--.f64 97.5

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

   3. Applied rewrites 97.5%

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

   4. Taylor expanded in z around inf

      \[\leadsto \frac{\frac{x}{\color{blue}{-1 \cdot z}}}{y - z} \]
   5. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\frac{x}{\mathsf{neg}\left(z\right)}}{y - z} \]

      2. lift-neg.f64 76.2

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

   6. Applied rewrites 76.2%

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

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{x}{-z}}{\color{blue}{-1 \cdot z}} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\frac{x}{-z}}{\mathsf{neg}\left(z\right)} \]

      2. lift-neg.f64 57.3

         \[\leadsto \frac{\frac{x}{-z}}{-z} \]

   9. Applied rewrites 57.3%

      \[\leadsto \frac{\frac{x}{-z}}{\color{blue}{-z}} \]

   if 3.2000000000000001e-53 < t

   1. Initial program 88.6%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift--.f64 96.7

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

   3. Applied rewrites 96.7%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 81.7%

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

Alternative 8: 73.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{-234}:\\ \;\;\;\;\frac{x\_m}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-53}:\\ \;\;\;\;\frac{\frac{x\_m}{-z}}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y - z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= t 5.5e-234)
    (/ x_m (* y (- t z)))
    (if (<= t 3.2e-53) (/ (/ x_m (- z)) (- z)) (/ (/ x_m t) (- y z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= 5.5e-234) {
		tmp = x_m / (y * (t - z));
	} else if (t <= 3.2e-53) {
		tmp = (x_m / -z) / -z;
	} else {
		tmp = (x_m / t) / (y - z);
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 5.5d-234) then
        tmp = x_m / (y * (t - z))
    else if (t <= 3.2d-53) then
        tmp = (x_m / -z) / -z
    else
        tmp = (x_m / t) / (y - z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= 5.5e-234) {
		tmp = x_m / (y * (t - z));
	} else if (t <= 3.2e-53) {
		tmp = (x_m / -z) / -z;
	} else {
		tmp = (x_m / t) / (y - z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if t <= 5.5e-234:
		tmp = x_m / (y * (t - z))
	elif t <= 3.2e-53:
		tmp = (x_m / -z) / -z
	else:
		tmp = (x_m / t) / (y - z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (t <= 5.5e-234)
		tmp = Float64(x_m / Float64(y * Float64(t - z)));
	elseif (t <= 3.2e-53)
		tmp = Float64(Float64(x_m / Float64(-z)) / Float64(-z));
	else
		tmp = Float64(Float64(x_m / t) / Float64(y - z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (t <= 5.5e-234)
		tmp = x_m / (y * (t - z));
	elseif (t <= 3.2e-53)
		tmp = (x_m / -z) / -z;
	else
		tmp = (x_m / t) / (y - z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[t, 5.5e-234], N[(x$95$m / N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.2e-53], N[(N[(x$95$m / (-z)), $MachinePrecision] / (-z)), $MachinePrecision], N[(N[(x$95$m / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 5.5e-234

   1. Initial program 89.2%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 69.0%

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

   if 5.5e-234 < t < 3.2000000000000001e-53

   1. Initial program 90.6%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift--.f64 97.5

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

   3. Applied rewrites 97.5%

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

   4. Taylor expanded in z around inf

      \[\leadsto \frac{\frac{x}{\color{blue}{-1 \cdot z}}}{y - z} \]
   5. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\frac{x}{\mathsf{neg}\left(z\right)}}{y - z} \]

      2. lift-neg.f64 76.2

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

   6. Applied rewrites 76.2%

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

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{x}{-z}}{\color{blue}{-1 \cdot z}} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\frac{x}{-z}}{\mathsf{neg}\left(z\right)} \]

      2. lift-neg.f64 57.3

         \[\leadsto \frac{\frac{x}{-z}}{-z} \]

   9. Applied rewrites 57.3%

      \[\leadsto \frac{\frac{x}{-z}}{\color{blue}{-z}} \]

   if 3.2000000000000001e-53 < t

   1. Initial program 88.6%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift--.f64 96.7

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

   3. Applied rewrites 96.7%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 81.7%

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

Alternative 9: 73.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{-234}:\\ \;\;\;\;\frac{x\_m}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;t \leq 2.45 \cdot 10^{-54}:\\ \;\;\;\;\frac{\frac{x\_m}{-z}}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\left(y - z\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= t 5.5e-234)
    (/ x_m (* y (- t z)))
    (if (<= t 2.45e-54) (/ (/ x_m (- z)) (- z)) (/ x_m (* (- y z) t))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= 5.5e-234) {
		tmp = x_m / (y * (t - z));
	} else if (t <= 2.45e-54) {
		tmp = (x_m / -z) / -z;
	} else {
		tmp = x_m / ((y - z) * t);
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 5.5d-234) then
        tmp = x_m / (y * (t - z))
    else if (t <= 2.45d-54) then
        tmp = (x_m / -z) / -z
    else
        tmp = x_m / ((y - z) * t)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= 5.5e-234) {
		tmp = x_m / (y * (t - z));
	} else if (t <= 2.45e-54) {
		tmp = (x_m / -z) / -z;
	} else {
		tmp = x_m / ((y - z) * t);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if t <= 5.5e-234:
		tmp = x_m / (y * (t - z))
	elif t <= 2.45e-54:
		tmp = (x_m / -z) / -z
	else:
		tmp = x_m / ((y - z) * t)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (t <= 5.5e-234)
		tmp = Float64(x_m / Float64(y * Float64(t - z)));
	elseif (t <= 2.45e-54)
		tmp = Float64(Float64(x_m / Float64(-z)) / Float64(-z));
	else
		tmp = Float64(x_m / Float64(Float64(y - z) * t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (t <= 5.5e-234)
		tmp = x_m / (y * (t - z));
	elseif (t <= 2.45e-54)
		tmp = (x_m / -z) / -z;
	else
		tmp = x_m / ((y - z) * t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[t, 5.5e-234], N[(x$95$m / N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.45e-54], N[(N[(x$95$m / (-z)), $MachinePrecision] / (-z)), $MachinePrecision], N[(x$95$m / N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 5.5e-234

   1. Initial program 89.2%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 69.0%

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

   if 5.5e-234 < t < 2.4500000000000001e-54

   1. Initial program 90.5%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift--.f64 97.4

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

   3. Applied rewrites 97.4%

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

   4. Taylor expanded in z around inf

      \[\leadsto \frac{\frac{x}{\color{blue}{-1 \cdot z}}}{y - z} \]
   5. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\frac{x}{\mathsf{neg}\left(z\right)}}{y - z} \]

      2. lift-neg.f64 76.2

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

   6. Applied rewrites 76.2%

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

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{x}{-z}}{\color{blue}{-1 \cdot z}} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\frac{x}{-z}}{\mathsf{neg}\left(z\right)} \]

      2. lift-neg.f64 57.2

         \[\leadsto \frac{\frac{x}{-z}}{-z} \]

   9. Applied rewrites 57.2%

      \[\leadsto \frac{\frac{x}{-z}}{\color{blue}{-z}} \]

   if 2.4500000000000001e-54 < t

   1. Initial program 88.6%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 79.6%

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

Alternative 10: 72.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{\frac{x\_m}{-z}}{-z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -5000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+20}:\\ \;\;\;\;\frac{x\_m}{y \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ (/ x_m (- z)) (- z))))
   (*
    x_s
    (if (<= z -5000000000.0)
      t_1
      (if (<= z 2.7e+20) (/ x_m (* y (- t z))) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m / -z) / -z;
	double tmp;
	if (z <= -5000000000.0) {
		tmp = t_1;
	} else if (z <= 2.7e+20) {
		tmp = x_m / (y * (t - z));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x_m / -z) / -z
    if (z <= (-5000000000.0d0)) then
        tmp = t_1
    else if (z <= 2.7d+20) then
        tmp = x_m / (y * (t - z))
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m / -z) / -z;
	double tmp;
	if (z <= -5000000000.0) {
		tmp = t_1;
	} else if (z <= 2.7e+20) {
		tmp = x_m / (y * (t - z));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	t_1 = (x_m / -z) / -z
	tmp = 0
	if z <= -5000000000.0:
		tmp = t_1
	elif z <= 2.7e+20:
		tmp = x_m / (y * (t - z))
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(x_m / Float64(-z)) / Float64(-z))
	tmp = 0.0
	if (z <= -5000000000.0)
		tmp = t_1;
	elseif (z <= 2.7e+20)
		tmp = Float64(x_m / Float64(y * Float64(t - z)));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = (x_m / -z) / -z;
	tmp = 0.0;
	if (z <= -5000000000.0)
		tmp = t_1;
	elseif (z <= 2.7e+20)
		tmp = x_m / (y * (t - z));
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x$95$m / (-z)), $MachinePrecision] / (-z)), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -5000000000.0], t$95$1, If[LessEqual[z, 2.7e+20], N[(x$95$m / N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5e9 or 2.7e20 < z

   1. Initial program 83.8%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift--.f64 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in z around inf

      \[\leadsto \frac{\frac{x}{\color{blue}{-1 \cdot z}}}{y - z} \]
   5. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\frac{x}{\mathsf{neg}\left(z\right)}}{y - z} \]

      2. lift-neg.f64 86.9

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

   6. Applied rewrites 86.9%

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

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{x}{-z}}{\color{blue}{-1 \cdot z}} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\frac{x}{-z}}{\mathsf{neg}\left(z\right)} \]

      2. lift-neg.f64 75.9

         \[\leadsto \frac{\frac{x}{-z}}{-z} \]

   9. Applied rewrites 75.9%

      \[\leadsto \frac{\frac{x}{-z}}{\color{blue}{-z}} \]

   if -5e9 < z < 2.7e20

   1. Initial program 94.1%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 70.8%

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

Alternative 11: 67.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{\frac{x\_m}{-z}}{-z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2300000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{+20}:\\ \;\;\;\;\frac{\frac{x\_m}{y}}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ (/ x_m (- z)) (- z))))
   (*
    x_s
    (if (<= z -2300000.0) t_1 (if (<= z 1.75e+20) (/ (/ x_m y) t) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m / -z) / -z;
	double tmp;
	if (z <= -2300000.0) {
		tmp = t_1;
	} else if (z <= 1.75e+20) {
		tmp = (x_m / y) / t;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x_m / -z) / -z
    if (z <= (-2300000.0d0)) then
        tmp = t_1
    else if (z <= 1.75d+20) then
        tmp = (x_m / y) / t
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m / -z) / -z;
	double tmp;
	if (z <= -2300000.0) {
		tmp = t_1;
	} else if (z <= 1.75e+20) {
		tmp = (x_m / y) / t;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	t_1 = (x_m / -z) / -z
	tmp = 0
	if z <= -2300000.0:
		tmp = t_1
	elif z <= 1.75e+20:
		tmp = (x_m / y) / t
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(x_m / Float64(-z)) / Float64(-z))
	tmp = 0.0
	if (z <= -2300000.0)
		tmp = t_1;
	elseif (z <= 1.75e+20)
		tmp = Float64(Float64(x_m / y) / t);
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = (x_m / -z) / -z;
	tmp = 0.0;
	if (z <= -2300000.0)
		tmp = t_1;
	elseif (z <= 1.75e+20)
		tmp = (x_m / y) / t;
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x$95$m / (-z)), $MachinePrecision] / (-z)), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -2300000.0], t$95$1, If[LessEqual[z, 1.75e+20], N[(N[(x$95$m / y), $MachinePrecision] / t), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.3e6 or 1.75e20 < z

   1. Initial program 83.8%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift--.f64 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in z around inf

      \[\leadsto \frac{\frac{x}{\color{blue}{-1 \cdot z}}}{y - z} \]
   5. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\frac{x}{\mathsf{neg}\left(z\right)}}{y - z} \]

      2. lift-neg.f64 86.8

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

   6. Applied rewrites 86.8%

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

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{x}{-z}}{\color{blue}{-1 \cdot z}} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\frac{x}{-z}}{\mathsf{neg}\left(z\right)} \]

      2. lift-neg.f64 75.6

         \[\leadsto \frac{\frac{x}{-z}}{-z} \]

   9. Applied rewrites 75.6%

      \[\leadsto \frac{\frac{x}{-z}}{\color{blue}{-z}} \]

   if -2.3e6 < z < 1.75e20

   1. Initial program 94.1%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 71.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift--.f64 73.3

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

   6. Applied rewrites 73.3%

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

   7. Taylor expanded in y around inf

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

      1. lower-/.f64 59.3

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

   9. Applied rewrites 59.3%

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

Alternative 12: 63.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x\_m}{z \cdot z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{+20}:\\ \;\;\;\;\frac{\frac{x\_m}{y}}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ x_m (* z z))))
   (* x_s (if (<= z -5e-20) t_1 (if (<= z 1.75e+20) (/ (/ x_m y) t) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / (z * z);
	double tmp;
	if (z <= -5e-20) {
		tmp = t_1;
	} else if (z <= 1.75e+20) {
		tmp = (x_m / y) / t;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x_m / (z * z)
    if (z <= (-5d-20)) then
        tmp = t_1
    else if (z <= 1.75d+20) then
        tmp = (x_m / y) / t
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / (z * z);
	double tmp;
	if (z <= -5e-20) {
		tmp = t_1;
	} else if (z <= 1.75e+20) {
		tmp = (x_m / y) / t;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	t_1 = x_m / (z * z)
	tmp = 0
	if z <= -5e-20:
		tmp = t_1
	elif z <= 1.75e+20:
		tmp = (x_m / y) / t
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m / Float64(z * z))
	tmp = 0.0
	if (z <= -5e-20)
		tmp = t_1;
	elseif (z <= 1.75e+20)
		tmp = Float64(Float64(x_m / y) / t);
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m / (z * z);
	tmp = 0.0;
	if (z <= -5e-20)
		tmp = t_1;
	elseif (z <= 1.75e+20)
		tmp = (x_m / y) / t;
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -5e-20], t$95$1, If[LessEqual[z, 1.75e+20], N[(N[(x$95$m / y), $MachinePrecision] / t), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.9999999999999999e-20 or 1.75e20 < z

   1. Initial program 84.2%

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

   2. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 66.4

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

   4. Applied rewrites 66.4%

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

   if -4.9999999999999999e-20 < z < 1.75e20

   1. Initial program 94.0%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 72.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift--.f64 74.1

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

   6. Applied rewrites 74.1%

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

   7. Taylor expanded in y around inf

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

      1. lower-/.f64 60.2

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

   9. Applied rewrites 60.2%

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

Alternative 13: 61.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x\_m}{z \cdot z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+19}:\\ \;\;\;\;\frac{x\_m}{t \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ x_m (* z z))))
   (* x_s (if (<= z -4.8e-20) t_1 (if (<= z 5.4e+19) (/ x_m (* t y)) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / (z * z);
	double tmp;
	if (z <= -4.8e-20) {
		tmp = t_1;
	} else if (z <= 5.4e+19) {
		tmp = x_m / (t * y);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x_m / (z * z)
    if (z <= (-4.8d-20)) then
        tmp = t_1
    else if (z <= 5.4d+19) then
        tmp = x_m / (t * y)
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / (z * z);
	double tmp;
	if (z <= -4.8e-20) {
		tmp = t_1;
	} else if (z <= 5.4e+19) {
		tmp = x_m / (t * y);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	t_1 = x_m / (z * z)
	tmp = 0
	if z <= -4.8e-20:
		tmp = t_1
	elif z <= 5.4e+19:
		tmp = x_m / (t * y)
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m / Float64(z * z))
	tmp = 0.0
	if (z <= -4.8e-20)
		tmp = t_1;
	elseif (z <= 5.4e+19)
		tmp = Float64(x_m / Float64(t * y));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m / (z * z);
	tmp = 0.0;
	if (z <= -4.8e-20)
		tmp = t_1;
	elseif (z <= 5.4e+19)
		tmp = x_m / (t * y);
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -4.8e-20], t$95$1, If[LessEqual[z, 5.4e+19], N[(x$95$m / N[(t * y), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.79999999999999986e-20 or 5.4e19 < z

   1. Initial program 84.2%

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

   2. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 66.4

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

   4. Applied rewrites 66.4%

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

   if -4.79999999999999986e-20 < z < 5.4e19

   1. Initial program 94.0%

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

   2. Taylor expanded in z around 0

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

      1. lower-*.f64 57.2

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

   4. Applied rewrites 57.2%

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

Alternative 14: 39.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \frac{x\_m}{t \cdot y} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t) :precision binary64 (* x_s (/ x_m (* t y))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * (x_m / (t * y));
}

Fortran

x\_m =     private
x\_s =     private
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * (x_m / (t * y));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	return x_s * (x_m / (t * y))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	return Float64(x_s * Float64(x_m / Float64(t * y)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp = code(x_s, x_m, y, z, t)
	tmp = x_s * (x_m / (t * y));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * N[(x$95$m / N[(t * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.2%

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

2. Taylor expanded in z around 0

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

   1. lower-*.f64 39.4

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

4. Applied rewrites 39.4%

   \[\leadsto \frac{x}{\color{blue}{t \cdot y}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025110 
(FPCore (x y z t)
  :name "Data.Random.Distribution.Triangular:triangularCDF from random-fu-0.2.6.2, B"
  :precision binary64
  (/ x (* (- y z) (- t z))))