Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3

Percentage Accurate: 84.3% → 96.9%

Time: 11.1s

Alternatives: 10

Speedup: 0.8×

Specification

Math

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

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(Float64(x * 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[(N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]

Initial Program: 84.3% accurate, 1.0× speedup

Math

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

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(Float64(x * 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[(N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]

Alternative 1: 96.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 500000:\\ \;\;\;\;\frac{x\_m \cdot \left(y - z\right)}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{t - z} \cdot \left(y - z\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= x_m 500000.0)
    (/ (* x_m (- y z)) (- t z))
    (* (/ x_m (- t z)) (- y z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 500000.0) {
		tmp = (x_m * (y - z)) / (t - z);
	} else {
		tmp = (x_m / (t - z)) * (y - z);
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_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 (x_m <= 500000.0d0) then
        tmp = (x_m * (y - z)) / (t - z)
    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);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 500000.0) {
		tmp = (x_m * (y - z)) / (t - z);
	} 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)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if x_m <= 500000.0:
		tmp = (x_m * (y - z)) / (t - z)
	else:
		tmp = (x_m / (t - z)) * (y - z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (x_m <= 500000.0)
		tmp = Float64(Float64(x_m * Float64(y - z)) / Float64(t - z));
	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);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (x_m <= 500000.0)
		tmp = (x_m * (y - z)) / (t - z);
	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]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[x$95$m, 500000.0], N[(N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], 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 x < 5e5

   1. Initial program 91.1%

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

   if 5e5 < x

   1. Initial program 75.3%

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

   4. Applied rewrites 98.0%

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

Alternative 2: 58.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{y}{t} \cdot x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -72000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 45:\\ \;\;\;\;\frac{z \cdot x\_m}{z - t}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+151}:\\ \;\;\;\;\frac{y \cdot x\_m}{t}\\ \mathbf{elif}\;y \leq 1.275 \cdot 10^{+185}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \left(z - y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* (/ y t) x_m)))
   (*
    x_s
    (if (<= y -72000000.0)
      t_1
      (if (<= y 45.0)
        (/ (* z x_m) (- z t))
        (if (<= y 1.75e+151)
          (/ (* y x_m) t)
          (if (<= y 1.275e+185) (* (/ x_m z) (- z y)) t_1)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (y / t) * x_m;
	double tmp;
	if (y <= -72000000.0) {
		tmp = t_1;
	} else if (y <= 45.0) {
		tmp = (z * x_m) / (z - t);
	} else if (y <= 1.75e+151) {
		tmp = (y * x_m) / t;
	} else if (y <= 1.275e+185) {
		tmp = (x_m / z) * (z - y);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_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 = (y / t) * x_m
    if (y <= (-72000000.0d0)) then
        tmp = t_1
    else if (y <= 45.0d0) then
        tmp = (z * x_m) / (z - t)
    else if (y <= 1.75d+151) then
        tmp = (y * x_m) / t
    else if (y <= 1.275d+185) then
        tmp = (x_m / z) * (z - 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);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (y / t) * x_m;
	double tmp;
	if (y <= -72000000.0) {
		tmp = t_1;
	} else if (y <= 45.0) {
		tmp = (z * x_m) / (z - t);
	} else if (y <= 1.75e+151) {
		tmp = (y * x_m) / t;
	} else if (y <= 1.275e+185) {
		tmp = (x_m / z) * (z - y);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = (y / t) * x_m
	tmp = 0
	if y <= -72000000.0:
		tmp = t_1
	elif y <= 45.0:
		tmp = (z * x_m) / (z - t)
	elif y <= 1.75e+151:
		tmp = (y * x_m) / t
	elif y <= 1.275e+185:
		tmp = (x_m / z) * (z - y)
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(y / t) * x_m)
	tmp = 0.0
	if (y <= -72000000.0)
		tmp = t_1;
	elseif (y <= 45.0)
		tmp = Float64(Float64(z * x_m) / Float64(z - t));
	elseif (y <= 1.75e+151)
		tmp = Float64(Float64(y * x_m) / t);
	elseif (y <= 1.275e+185)
		tmp = Float64(Float64(x_m / z) * Float64(z - y));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = (y / t) * x_m;
	tmp = 0.0;
	if (y <= -72000000.0)
		tmp = t_1;
	elseif (y <= 45.0)
		tmp = (z * x_m) / (z - t);
	elseif (y <= 1.75e+151)
		tmp = (y * x_m) / t;
	elseif (y <= 1.275e+185)
		tmp = (x_m / z) * (z - 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]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] * x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -72000000.0], t$95$1, If[LessEqual[y, 45.0], N[(N[(z * x$95$m), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.75e+151], N[(N[(y * x$95$m), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[y, 1.275e+185], N[(N[(x$95$m / z), $MachinePrecision] * N[(z - y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if y < -7.2e7 or 1.27499999999999999e185 < y

   1. Initial program 79.8%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 64.3

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

   5. Applied rewrites 64.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 62.0%

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

   if -7.2e7 < y < 45

   1. Initial program 91.2%

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

   4. Applied rewrites 78.7%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. *-rgt-identity N/A

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

      4. *-inverses N/A

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

      5. associate-/l* N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. +-commutative N/A

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

      11. distribute-rgt-out N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. distribute-neg-out N/A

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

      14. metadata-eval N/A

          \[\leadsto \frac{x \cdot z}{z \cdot \left(1 + \left(\mathsf{neg}\left(\color{blue}{\frac{t}{z}}\right)\right)\right)} \]

      15. mul-1-neg N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. distribute-rgt-in N/A

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

      20. metadata-eval N/A

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

      21. mul-1-neg N/A

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

   7. Applied rewrites 78.1%

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

   if 45 < y < 1.7500000000000001e151

   1. Initial program 95.1%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 57.1

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

   5. Applied rewrites 57.1%

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

   if 1.7500000000000001e151 < y < 1.27499999999999999e185

   1. Initial program 78.5%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

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

      14. remove-double-neg N/A

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

      15. mul-1-neg N/A

          \[\leadsto \frac{z + -1 \cdot y}{z} \cdot x \]

      16. metadata-eval N/A

          \[\leadsto \frac{z + \left(\mathsf{neg}\left(1\right)\right) \cdot y}{z} \cdot x \]

      17. fp-cancel-sub-sign-inv N/A

          \[\leadsto \frac{z - 1 \cdot y}{z} \cdot x \]

      18. *-lft-identity N/A

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

      19. lower--.f64 89.5

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

   5. Applied rewrites 89.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.7

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

   7. Applied rewrites 99.7%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -72000000:\\ \;\;\;\;\frac{y}{t} \cdot x\\ \mathbf{elif}\;y \leq 45:\\ \;\;\;\;\frac{z \cdot x}{z - t}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+151}:\\ \;\;\;\;\frac{y \cdot x}{t}\\ \mathbf{elif}\;y \leq 1.275 \cdot 10^{+185}:\\ \;\;\;\;\frac{x}{z} \cdot \left(z - y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x\_m}{z}, x\_m\right)\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-28}:\\ \;\;\;\;\frac{y - z}{t} \cdot x\_m\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+174}:\\ \;\;\;\;\frac{z \cdot x\_m}{z - t}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -3e+94)
    (fma t (/ x_m z) x_m)
    (if (<= z 1.75e-28)
      (* (/ (- y z) t) x_m)
      (if (<= z 2.4e+174) (/ (* z x_m) (- z t)) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -3e+94) {
		tmp = fma(t, (x_m / z), x_m);
	} else if (z <= 1.75e-28) {
		tmp = ((y - z) / t) * x_m;
	} else if (z <= 2.4e+174) {
		tmp = (z * x_m) / (z - t);
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -3e+94)
		tmp = fma(t, Float64(x_m / z), x_m);
	elseif (z <= 1.75e-28)
		tmp = Float64(Float64(Float64(y - z) / t) * x_m);
	elseif (z <= 2.4e+174)
		tmp = Float64(Float64(z * x_m) / Float64(z - t));
	else
		tmp = x_m;
	end
	return Float64(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]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -3e+94], N[(t * N[(x$95$m / z), $MachinePrecision] + x$95$m), $MachinePrecision], If[LessEqual[z, 1.75e-28], N[(N[(N[(y - z), $MachinePrecision] / t), $MachinePrecision] * x$95$m), $MachinePrecision], If[LessEqual[z, 2.4e+174], N[(N[(z * x$95$m), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision], x$95$m]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if z < -3.0000000000000001e94

   1. Initial program 77.8%

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

   4. Applied rewrites 66.3%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. *-rgt-identity N/A

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

      4. *-inverses N/A

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

      5. associate-/l* N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. +-commutative N/A

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

      11. distribute-rgt-out N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. distribute-neg-out N/A

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

      14. metadata-eval N/A

          \[\leadsto \frac{x \cdot z}{z \cdot \left(1 + \left(\mathsf{neg}\left(\color{blue}{\frac{t}{z}}\right)\right)\right)} \]

      15. mul-1-neg N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. distribute-rgt-in N/A

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

      20. metadata-eval N/A

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

      21. mul-1-neg N/A

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

   7. Applied rewrites 69.3%

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

   8. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 81.8

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

   10. Applied rewrites 81.8%

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

   if -3.0000000000000001e94 < z < 1.75e-28

   1. Initial program 93.9%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 75.5

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

   5. Applied rewrites 75.5%

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

   if 1.75e-28 < z < 2.3999999999999998e174

   1. Initial program 89.3%

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

   4. Applied rewrites 88.0%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. *-rgt-identity N/A

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

      4. *-inverses N/A

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

      5. associate-/l* N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. +-commutative N/A

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

      11. distribute-rgt-out N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. distribute-neg-out N/A

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

      14. metadata-eval N/A

          \[\leadsto \frac{x \cdot z}{z \cdot \left(1 + \left(\mathsf{neg}\left(\color{blue}{\frac{t}{z}}\right)\right)\right)} \]

      15. mul-1-neg N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. distribute-rgt-in N/A

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

      20. metadata-eval N/A

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

      21. mul-1-neg N/A

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

   7. Applied rewrites 67.2%

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

   if 2.3999999999999998e174 < z

   1. Initial program 69.7%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 82.4%

      \[\leadsto \color{blue}{x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x}{z}, x\right)\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-28}:\\ \;\;\;\;\frac{y - z}{t} \cdot x\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+174}:\\ \;\;\;\;\frac{z \cdot x}{z - t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x\_m}{z}, x\_m\right)\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-90}:\\ \;\;\;\;\frac{y}{t} \cdot x\_m\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+174}:\\ \;\;\;\;\frac{z \cdot x\_m}{z - t}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -2.8e+94)
    (fma t (/ x_m z) x_m)
    (if (<= z 6e-90)
      (* (/ y t) x_m)
      (if (<= z 2.4e+174) (/ (* z x_m) (- z t)) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -2.8e+94) {
		tmp = fma(t, (x_m / z), x_m);
	} else if (z <= 6e-90) {
		tmp = (y / t) * x_m;
	} else if (z <= 2.4e+174) {
		tmp = (z * x_m) / (z - t);
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -2.8e+94)
		tmp = fma(t, Float64(x_m / z), x_m);
	elseif (z <= 6e-90)
		tmp = Float64(Float64(y / t) * x_m);
	elseif (z <= 2.4e+174)
		tmp = Float64(Float64(z * x_m) / Float64(z - t));
	else
		tmp = x_m;
	end
	return Float64(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]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -2.8e+94], N[(t * N[(x$95$m / z), $MachinePrecision] + x$95$m), $MachinePrecision], If[LessEqual[z, 6e-90], N[(N[(y / t), $MachinePrecision] * x$95$m), $MachinePrecision], If[LessEqual[z, 2.4e+174], N[(N[(z * x$95$m), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision], x$95$m]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if z < -2.79999999999999998e94

   1. Initial program 77.8%

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

   4. Applied rewrites 66.3%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. *-rgt-identity N/A

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

      4. *-inverses N/A

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

      5. associate-/l* N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. +-commutative N/A

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

      11. distribute-rgt-out N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. distribute-neg-out N/A

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

      14. metadata-eval N/A

          \[\leadsto \frac{x \cdot z}{z \cdot \left(1 + \left(\mathsf{neg}\left(\color{blue}{\frac{t}{z}}\right)\right)\right)} \]

      15. mul-1-neg N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. distribute-rgt-in N/A

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

      20. metadata-eval N/A

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

      21. mul-1-neg N/A

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

   7. Applied rewrites 69.3%

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

   8. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 81.8

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

   10. Applied rewrites 81.8%

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

   if -2.79999999999999998e94 < z < 6.00000000000000041e-90

   1. Initial program 93.6%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 77.7

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

   5. Applied rewrites 77.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 66.1%

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

   if 6.00000000000000041e-90 < z < 2.3999999999999998e174

   1. Initial program 91.2%

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

   4. Applied rewrites 91.7%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. *-rgt-identity N/A

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

      4. *-inverses N/A

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

      5. associate-/l* N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. +-commutative N/A

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

      11. distribute-rgt-out N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. distribute-neg-out N/A

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

      14. metadata-eval N/A

          \[\leadsto \frac{x \cdot z}{z \cdot \left(1 + \left(\mathsf{neg}\left(\color{blue}{\frac{t}{z}}\right)\right)\right)} \]

      15. mul-1-neg N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. distribute-rgt-in N/A

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

      20. metadata-eval N/A

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

      21. mul-1-neg N/A

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

   7. Applied rewrites 61.4%

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

   if 2.3999999999999998e174 < z

   1. Initial program 69.7%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 82.4%

      \[\leadsto \color{blue}{x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x}{z}, x\right)\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-90}:\\ \;\;\;\;\frac{y}{t} \cdot x\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+174}:\\ \;\;\;\;\frac{z \cdot x}{z - t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+206}:\\ \;\;\;\;\frac{z - y}{z} \cdot x\_m\\ \mathbf{elif}\;z \leq 5.3 \cdot 10^{+181}:\\ \;\;\;\;\frac{x\_m}{t - z} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m - \frac{x\_m}{z} \cdot \left(y - t\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -3.8e+206)
    (* (/ (- z y) z) x_m)
    (if (<= z 5.3e+181)
      (* (/ x_m (- t z)) (- y z))
      (- x_m (* (/ x_m z) (- y t)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -3.8e+206) {
		tmp = ((z - y) / z) * x_m;
	} else if (z <= 5.3e+181) {
		tmp = (x_m / (t - z)) * (y - z);
	} else {
		tmp = x_m - ((x_m / z) * (y - t));
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_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 (z <= (-3.8d+206)) then
        tmp = ((z - y) / z) * x_m
    else if (z <= 5.3d+181) then
        tmp = (x_m / (t - z)) * (y - z)
    else
        tmp = x_m - ((x_m / z) * (y - t))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -3.8e+206) {
		tmp = ((z - y) / z) * x_m;
	} else if (z <= 5.3e+181) {
		tmp = (x_m / (t - z)) * (y - z);
	} else {
		tmp = x_m - ((x_m / z) * (y - t));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -3.8e+206:
		tmp = ((z - y) / z) * x_m
	elif z <= 5.3e+181:
		tmp = (x_m / (t - z)) * (y - z)
	else:
		tmp = x_m - ((x_m / z) * (y - t))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -3.8e+206)
		tmp = Float64(Float64(Float64(z - y) / z) * x_m);
	elseif (z <= 5.3e+181)
		tmp = Float64(Float64(x_m / Float64(t - z)) * Float64(y - z));
	else
		tmp = Float64(x_m - Float64(Float64(x_m / z) * Float64(y - t)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -3.8e+206)
		tmp = ((z - y) / z) * x_m;
	elseif (z <= 5.3e+181)
		tmp = (x_m / (t - z)) * (y - z);
	else
		tmp = x_m - ((x_m / z) * (y - 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]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -3.8e+206], N[(N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision] * x$95$m), $MachinePrecision], If[LessEqual[z, 5.3e+181], N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision], N[(x$95$m - N[(N[(x$95$m / z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -3.7999999999999999e206

   1. Initial program 79.7%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

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

      14. remove-double-neg N/A

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

      15. mul-1-neg N/A

          \[\leadsto \frac{z + -1 \cdot y}{z} \cdot x \]

      16. metadata-eval N/A

          \[\leadsto \frac{z + \left(\mathsf{neg}\left(1\right)\right) \cdot y}{z} \cdot x \]

      17. fp-cancel-sub-sign-inv N/A

          \[\leadsto \frac{z - 1 \cdot y}{z} \cdot x \]

      18. *-lft-identity N/A

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

      19. lower--.f64 94.8

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

   5. Applied rewrites 94.8%

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

   if -3.7999999999999999e206 < z < 5.2999999999999996e181

   1. Initial program 90.8%

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

   4. Applied rewrites 90.5%

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

   if 5.2999999999999996e181 < z

   1. Initial program 72.0%

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

   3. Taylor expanded in z around inf

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. associate-+l- N/A

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

      8. div-sub N/A

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

      9. lower--.f64 N/A

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

      10. div-sub N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. associate-/l* N/A

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

      14. distribute-rgt-out-- N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 98.6

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

   5. Applied rewrites 98.6%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+206}:\\ \;\;\;\;\frac{z - y}{z} \cdot x\\ \mathbf{elif}\;z \leq 5.3 \cdot 10^{+181}:\\ \;\;\;\;\frac{x}{t - z} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{z} \cdot \left(y - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+53} \lor \neg \left(z \leq 1.15 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{z - y}{z} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{t} \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= z -4e+53) (not (<= z 1.15e-5)))
    (* (/ (- z y) z) x_m)
    (* (/ (- y z) t) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -4e+53) || !(z <= 1.15e-5)) {
		tmp = ((z - y) / z) * x_m;
	} else {
		tmp = ((y - z) / t) * x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_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 ((z <= (-4d+53)) .or. (.not. (z <= 1.15d-5))) then
        tmp = ((z - y) / z) * x_m
    else
        tmp = ((y - z) / t) * x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -4e+53) || !(z <= 1.15e-5)) {
		tmp = ((z - y) / z) * x_m;
	} else {
		tmp = ((y - z) / t) * x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (z <= -4e+53) or not (z <= 1.15e-5):
		tmp = ((z - y) / z) * x_m
	else:
		tmp = ((y - z) / t) * x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((z <= -4e+53) || !(z <= 1.15e-5))
		tmp = Float64(Float64(Float64(z - y) / z) * x_m);
	else
		tmp = Float64(Float64(Float64(y - z) / t) * x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((z <= -4e+53) || ~((z <= 1.15e-5)))
		tmp = ((z - y) / z) * x_m;
	else
		tmp = ((y - z) / t) * x_m;
	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]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[z, -4e+53], N[Not[LessEqual[z, 1.15e-5]], $MachinePrecision]], N[(N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(y - z), $MachinePrecision] / t), $MachinePrecision] * x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -4e53 or 1.15e-5 < z

   1. Initial program 79.9%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

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

      14. remove-double-neg N/A

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

      15. mul-1-neg N/A

          \[\leadsto \frac{z + -1 \cdot y}{z} \cdot x \]

      16. metadata-eval N/A

          \[\leadsto \frac{z + \left(\mathsf{neg}\left(1\right)\right) \cdot y}{z} \cdot x \]

      17. fp-cancel-sub-sign-inv N/A

          \[\leadsto \frac{z - 1 \cdot y}{z} \cdot x \]

      18. *-lft-identity N/A

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

      19. lower--.f64 81.0

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

   5. Applied rewrites 81.0%

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

   if -4e53 < z < 1.15e-5

   1. Initial program 94.4%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 77.3

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

   5. Applied rewrites 77.3%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+53} \lor \neg \left(z \leq 1.15 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{z - y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{t} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x\_m}{z}, x\_m\right)\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-28}:\\ \;\;\;\;\frac{y}{t} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -2.8e+94)
    (fma t (/ x_m z) x_m)
    (if (<= z 2.4e-28) (* (/ y t) x_m) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -2.8e+94) {
		tmp = fma(t, (x_m / z), x_m);
	} else if (z <= 2.4e-28) {
		tmp = (y / t) * x_m;
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -2.8e+94)
		tmp = fma(t, Float64(x_m / z), x_m);
	elseif (z <= 2.4e-28)
		tmp = Float64(Float64(y / t) * x_m);
	else
		tmp = x_m;
	end
	return Float64(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]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -2.8e+94], N[(t * N[(x$95$m / z), $MachinePrecision] + x$95$m), $MachinePrecision], If[LessEqual[z, 2.4e-28], N[(N[(y / t), $MachinePrecision] * x$95$m), $MachinePrecision], x$95$m]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -2.79999999999999998e94

   1. Initial program 77.8%

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

   4. Applied rewrites 66.3%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. *-rgt-identity N/A

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

      4. *-inverses N/A

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

      5. associate-/l* N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. +-commutative N/A

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

      11. distribute-rgt-out N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. distribute-neg-out N/A

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

      14. metadata-eval N/A

          \[\leadsto \frac{x \cdot z}{z \cdot \left(1 + \left(\mathsf{neg}\left(\color{blue}{\frac{t}{z}}\right)\right)\right)} \]

      15. mul-1-neg N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. distribute-rgt-in N/A

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

      20. metadata-eval N/A

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

      21. mul-1-neg N/A

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

   7. Applied rewrites 69.3%

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

   8. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 81.8

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

   10. Applied rewrites 81.8%

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

   if -2.79999999999999998e94 < z < 2.4000000000000002e-28

   1. Initial program 93.9%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 75.7

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 62.5%

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

   if 2.4000000000000002e-28 < z

   1. Initial program 81.5%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 63.5%

      \[\leadsto \color{blue}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x}{z}, x\right)\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-28}:\\ \;\;\;\;\frac{y}{t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+94}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-28}:\\ \;\;\;\;\frac{y}{t} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (* x_s (if (<= z -2.8e+94) x_m (if (<= z 2.4e-28) (* (/ y t) x_m) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -2.8e+94) {
		tmp = x_m;
	} else if (z <= 2.4e-28) {
		tmp = (y / t) * x_m;
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_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 (z <= (-2.8d+94)) then
        tmp = x_m
    else if (z <= 2.4d-28) then
        tmp = (y / t) * x_m
    else
        tmp = x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -2.8e+94) {
		tmp = x_m;
	} else if (z <= 2.4e-28) {
		tmp = (y / t) * x_m;
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -2.8e+94:
		tmp = x_m
	elif z <= 2.4e-28:
		tmp = (y / t) * x_m
	else:
		tmp = x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -2.8e+94)
		tmp = x_m;
	elseif (z <= 2.4e-28)
		tmp = Float64(Float64(y / t) * x_m);
	else
		tmp = x_m;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -2.8e+94)
		tmp = x_m;
	elseif (z <= 2.4e-28)
		tmp = (y / t) * x_m;
	else
		tmp = x_m;
	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]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -2.8e+94], x$95$m, If[LessEqual[z, 2.4e-28], N[(N[(y / t), $MachinePrecision] * x$95$m), $MachinePrecision], x$95$m]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -2.79999999999999998e94 or 2.4000000000000002e-28 < z

   1. Initial program 80.2%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 69.8%

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

   if -2.79999999999999998e94 < z < 2.4000000000000002e-28

   1. Initial program 93.9%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 75.7

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 62.5%

      \[\leadsto \frac{y}{t} \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+94}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-28}:\\ \;\;\;\;\frac{y}{t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 37.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.65 \cdot 10^{-176}:\\ \;\;\;\;\frac{z \cdot x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (* x_s (if (<= x_m 2.65e-176) (/ (* z x_m) z) x_m)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 2.65e-176) {
		tmp = (z * x_m) / z;
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_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 (x_m <= 2.65d-176) then
        tmp = (z * x_m) / z
    else
        tmp = x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 2.65e-176) {
		tmp = (z * x_m) / z;
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if x_m <= 2.65e-176:
		tmp = (z * x_m) / z
	else:
		tmp = x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (x_m <= 2.65e-176)
		tmp = Float64(Float64(z * x_m) / z);
	else
		tmp = x_m;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (x_m <= 2.65e-176)
		tmp = (z * x_m) / z;
	else
		tmp = x_m;
	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]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[x$95$m, 2.65e-176], N[(N[(z * x$95$m), $MachinePrecision] / z), $MachinePrecision], x$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.65000000000000006e-176

   1. Initial program 90.0%

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

   4. Applied rewrites 77.9%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. *-rgt-identity N/A

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

      4. *-inverses N/A

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

      5. associate-/l* N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. +-commutative N/A

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

      11. distribute-rgt-out N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. distribute-neg-out N/A

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

      14. metadata-eval N/A

          \[\leadsto \frac{x \cdot z}{z \cdot \left(1 + \left(\mathsf{neg}\left(\color{blue}{\frac{t}{z}}\right)\right)\right)} \]

      15. mul-1-neg N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. distribute-rgt-in N/A

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

      20. metadata-eval N/A

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

      21. mul-1-neg N/A

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

   7. Applied rewrites 50.6%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 36.5%

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

   if 2.65000000000000006e-176 < x

   1. Initial program 84.3%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 36.8%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.65 \cdot 10^{-176}:\\ \;\;\;\;\frac{z \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 35.1% accurate, 23.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot x\_m \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t) :precision binary64 (* x_s x_m))

C

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

Fortran

x\_m =     private
x\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_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
end function

Java

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

Python

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

Julia

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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z, t)
	tmp = x_s * x_m;
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]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * x$95$m), $MachinePrecision]

Derivation

1. Initial program 88.0%

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

3. Taylor expanded in z around inf

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

5. Applied rewrites 36.1%

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

6. Final simplification 36.1%

   \[\leadsto x \]
7. Add Preprocessing

Developer Target 1: 96.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return x / ((t - z) / (y - 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 / ((t - z) / (y - z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025026 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3"
  :precision binary64

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

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