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

Percentage Accurate: 83.7% → 97.2%

Time: 3.8s

Alternatives: 11

Speedup: 1.0×

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: 83.7% 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: 97.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 6 \cdot 10^{+195}:\\ \;\;\;\;x\_m \cdot \frac{z - y}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x\_m}{t - z}\\ \end{array} \end{array} \]

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 6e+195)
    (* x_m (/ (- z y) (- z t)))
    (* (- y z) (/ x_m (- t 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 <= 6e+195) {
		tmp = x_m * ((z - y) / (z - t));
	} else {
		tmp = (y - z) * (x_m / (t - 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 <= 6d+195) then
        tmp = x_m * ((z - y) / (z - t))
    else
        tmp = (y - z) * (x_m / (t - 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 <= 6e+195) {
		tmp = x_m * ((z - y) / (z - t));
	} else {
		tmp = (y - z) * (x_m / (t - 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 <= 6e+195:
		tmp = x_m * ((z - y) / (z - t))
	else:
		tmp = (y - z) * (x_m / (t - 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 <= 6e+195)
		tmp = Float64(x_m * Float64(Float64(z - y) / Float64(z - t)));
	else
		tmp = Float64(Float64(y - z) * Float64(x_m / Float64(t - 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 <= 6e+195)
		tmp = x_m * ((z - y) / (z - t));
	else
		tmp = (y - z) * (x_m / (t - 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, 6e+195], N[(x$95$m * N[(N[(z - y), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y - z), $MachinePrecision] * N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 6.0000000000000001e195

   1. Initial program 83.7%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. lower--.f64 97.2

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

   3. Applied rewrites 97.2%

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

   if 6.0000000000000001e195 < x

   1. Initial program 83.7%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 84.8

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

   3. Applied rewrites 84.8%

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

Alternative 2: 96.9% accurate, 1.0× speedup

Math

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

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 (/ (- z y) (- z 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) {
	return x_s * (x_m * ((z - y) / (z - t)));
}

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 * ((z - y) / (z - t)))
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 * ((z - y) / (z - t)));
}

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

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	return Float64(x_s * Float64(x_m * Float64(Float64(z - y) / Float64(z - t))))
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 * ((z - y) / (z - t)));
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 * N[(x$95$m * N[(N[(z - y), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.7%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. frac-2neg N/A

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

   8. lower-/.f64 N/A

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

   9. sub-negate-rev N/A

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

   10. lower--.f64 N/A

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

   11. sub-negate-rev N/A

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

   12. lower--.f64 97.2

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

3. Applied rewrites 97.2%

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

Alternative 3: 75.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m - x\_m \cdot \frac{y}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-22}:\\ \;\;\;\;\frac{x\_m \cdot \left(y - z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (- x_m (* x_m (/ y z)))))
   (*
    x_s
    (if (<= z -1.5e+22) t_1 (if (<= z 1.45e-22) (/ (* x_m (- y z)) t) 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 = x_m - (x_m * (y / z));
	double tmp;
	if (z <= -1.5e+22) {
		tmp = t_1;
	} else if (z <= 1.45e-22) {
		tmp = (x_m * (y - z)) / t;
	} 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 = x_m - (x_m * (y / z))
    if (z <= (-1.5d+22)) then
        tmp = t_1
    else if (z <= 1.45d-22) then
        tmp = (x_m * (y - z)) / t
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m - (x_m * (y / z));
	double tmp;
	if (z <= -1.5e+22) {
		tmp = t_1;
	} else if (z <= 1.45e-22) {
		tmp = (x_m * (y - z)) / t;
	} 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 = x_m - (x_m * (y / z))
	tmp = 0
	if z <= -1.5e+22:
		tmp = t_1
	elif z <= 1.45e-22:
		tmp = (x_m * (y - z)) / t
	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(x_m - Float64(x_m * Float64(y / z)))
	tmp = 0.0
	if (z <= -1.5e+22)
		tmp = t_1;
	elseif (z <= 1.45e-22)
		tmp = Float64(Float64(x_m * Float64(y - z)) / t);
	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 = x_m - (x_m * (y / z));
	tmp = 0.0;
	if (z <= -1.5e+22)
		tmp = t_1;
	elseif (z <= 1.45e-22)
		tmp = (x_m * (y - z)) / t;
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m - N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -1.5e+22], t$95$1, If[LessEqual[z, 1.45e-22], N[(N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.5e22 or 1.4500000000000001e-22 < z

   1. Initial program 83.7%

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

   2. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. mul-1-neg N/A

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

      5. sub-flip-reverse N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 47.6

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

   4. Applied rewrites 47.6%

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

   5. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 52.0

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

   7. Applied rewrites 52.0%

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

   if -1.5e22 < z < 1.4500000000000001e-22

   1. Initial program 83.7%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 47.9%

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

Alternative 4: 74.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m - x\_m \cdot \frac{y}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{-67}:\\ \;\;\;\;x\_m \cdot \frac{y - z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (- x_m (* x_m (/ y z)))))
   (*
    x_s
    (if (<= z -1.5e+22) t_1 (if (<= z 1.95e-67) (* x_m (/ (- y z) t)) 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 = x_m - (x_m * (y / z));
	double tmp;
	if (z <= -1.5e+22) {
		tmp = t_1;
	} else if (z <= 1.95e-67) {
		tmp = x_m * ((y - z) / t);
	} 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 = x_m - (x_m * (y / z))
    if (z <= (-1.5d+22)) then
        tmp = t_1
    else if (z <= 1.95d-67) then
        tmp = x_m * ((y - z) / t)
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m - (x_m * (y / z));
	double tmp;
	if (z <= -1.5e+22) {
		tmp = t_1;
	} else if (z <= 1.95e-67) {
		tmp = x_m * ((y - z) / t);
	} 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 = x_m - (x_m * (y / z))
	tmp = 0
	if z <= -1.5e+22:
		tmp = t_1
	elif z <= 1.95e-67:
		tmp = x_m * ((y - z) / t)
	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(x_m - Float64(x_m * Float64(y / z)))
	tmp = 0.0
	if (z <= -1.5e+22)
		tmp = t_1;
	elseif (z <= 1.95e-67)
		tmp = Float64(x_m * Float64(Float64(y - z) / t));
	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 = x_m - (x_m * (y / z));
	tmp = 0.0;
	if (z <= -1.5e+22)
		tmp = t_1;
	elseif (z <= 1.95e-67)
		tmp = x_m * ((y - z) / t);
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m - N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -1.5e+22], t$95$1, If[LessEqual[z, 1.95e-67], N[(x$95$m * N[(N[(y - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.5e22 or 1.9499999999999999e-67 < z

   1. Initial program 83.7%

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

   2. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. mul-1-neg N/A

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

      5. sub-flip-reverse N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 47.6

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

   4. Applied rewrites 47.6%

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

   5. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 52.0

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

   7. Applied rewrites 52.0%

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

   if -1.5e22 < z < 1.9499999999999999e-67

   1. Initial program 83.7%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. lower--.f64 97.2

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

   3. Applied rewrites 97.2%

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

   4. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-negate-rev N/A

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

      5. lower--.f64 51.0

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

   6. Applied rewrites 51.0%

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

Alternative 5: 74.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m - x\_m \cdot \frac{y}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 105000:\\ \;\;\;\;y \cdot \frac{x\_m}{t - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (- x_m (* x_m (/ y z)))))
   (*
    x_s
    (if (<= z -1.45e+22) t_1 (if (<= z 105000.0) (* y (/ x_m (- t z))) 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 = x_m - (x_m * (y / z));
	double tmp;
	if (z <= -1.45e+22) {
		tmp = t_1;
	} else if (z <= 105000.0) {
		tmp = y * (x_m / (t - z));
	} 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 = x_m - (x_m * (y / z))
    if (z <= (-1.45d+22)) then
        tmp = t_1
    else if (z <= 105000.0d0) then
        tmp = y * (x_m / (t - z))
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m - (x_m * (y / z));
	double tmp;
	if (z <= -1.45e+22) {
		tmp = t_1;
	} else if (z <= 105000.0) {
		tmp = y * (x_m / (t - z));
	} 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 = x_m - (x_m * (y / z))
	tmp = 0
	if z <= -1.45e+22:
		tmp = t_1
	elif z <= 105000.0:
		tmp = y * (x_m / (t - z))
	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(x_m - Float64(x_m * Float64(y / z)))
	tmp = 0.0
	if (z <= -1.45e+22)
		tmp = t_1;
	elseif (z <= 105000.0)
		tmp = Float64(y * Float64(x_m / Float64(t - z)));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m - (x_m * (y / z));
	tmp = 0.0;
	if (z <= -1.45e+22)
		tmp = t_1;
	elseif (z <= 105000.0)
		tmp = y * (x_m / (t - z));
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m - N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -1.45e+22], t$95$1, If[LessEqual[z, 105000.0], N[(y * N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.45e22 or 105000 < z

   1. Initial program 83.7%

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

   2. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. mul-1-neg N/A

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

      5. sub-flip-reverse N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 47.6

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

   4. Applied rewrites 47.6%

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

   5. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 52.0

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

   7. Applied rewrites 52.0%

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

   if -1.45e22 < z < 105000

   1. Initial program 83.7%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 50.2

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

   4. Applied rewrites 50.2%

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

Alternative 6: 70.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x\_m}{z}, x\_m\right)\\ \mathbf{elif}\;z \leq 500:\\ \;\;\;\;y \cdot \frac{x\_m}{t - z}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+153}:\\ \;\;\;\;z \cdot \frac{x\_m}{z - t}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot 1\\ \end{array} \end{array} \]

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 -1.45e+22)
    (fma t (/ x_m z) x_m)
    (if (<= z 500.0)
      (* y (/ x_m (- t z)))
      (if (<= z 6e+153) (* z (/ x_m (- z t))) (* x_m 1.0))))))

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 <= -1.45e+22) {
		tmp = fma(t, (x_m / z), x_m);
	} else if (z <= 500.0) {
		tmp = y * (x_m / (t - z));
	} else if (z <= 6e+153) {
		tmp = z * (x_m / (z - t));
	} else {
		tmp = x_m * 1.0;
	}
	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 <= -1.45e+22)
		tmp = fma(t, Float64(x_m / z), x_m);
	elseif (z <= 500.0)
		tmp = Float64(y * Float64(x_m / Float64(t - z)));
	elseif (z <= 6e+153)
		tmp = Float64(z * Float64(x_m / Float64(z - t)));
	else
		tmp = Float64(x_m * 1.0);
	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, -1.45e+22], N[(t * N[(x$95$m / z), $MachinePrecision] + x$95$m), $MachinePrecision], If[LessEqual[z, 500.0], N[(y * N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6e+153], N[(z * N[(x$95$m / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * 1.0), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if z < -1.45e22

   1. Initial program 83.7%

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

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z}{t - z}} \]
   3. 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. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. sub-negate-rev N/A

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

      7. lower--.f64 44.4

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

   4. Applied rewrites 44.4%

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

   5. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{\frac{t \cdot x}{z}} \]
   6. 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 36.3

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

   7. Applied rewrites 36.3%

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

   if -1.45e22 < z < 500

   1. Initial program 83.7%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 50.2

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

   4. Applied rewrites 50.2%

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

   if 500 < z < 6.00000000000000037e153

   1. Initial program 83.7%

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

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z}{t - z}} \]
   3. 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. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. sub-negate-rev N/A

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

      7. lower--.f64 44.4

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

   4. Applied rewrites 44.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 46.2

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

   6. Applied rewrites 46.2%

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

   if 6.00000000000000037e153 < z

   1. Initial program 83.7%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. lower--.f64 97.2

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

   3. Applied rewrites 97.2%

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

   4. Taylor expanded in z around inf

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

   6. Applied rewrites 35.1%

      \[\leadsto x \cdot \color{blue}{1} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 68.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x\_m}{z}, x\_m\right)\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+118}:\\ \;\;\;\;y \cdot \frac{x\_m}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot 1\\ \end{array} \end{array} \]

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 -1.45e+22)
    (fma t (/ x_m z) x_m)
    (if (<= z 5.5e+118) (* y (/ x_m (- t z))) (* x_m 1.0)))))

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 <= -1.45e+22) {
		tmp = fma(t, (x_m / z), x_m);
	} else if (z <= 5.5e+118) {
		tmp = y * (x_m / (t - z));
	} else {
		tmp = x_m * 1.0;
	}
	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 <= -1.45e+22)
		tmp = fma(t, Float64(x_m / z), x_m);
	elseif (z <= 5.5e+118)
		tmp = Float64(y * Float64(x_m / Float64(t - z)));
	else
		tmp = Float64(x_m * 1.0);
	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, -1.45e+22], N[(t * N[(x$95$m / z), $MachinePrecision] + x$95$m), $MachinePrecision], If[LessEqual[z, 5.5e+118], N[(y * N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * 1.0), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1.45e22

   1. Initial program 83.7%

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

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z}{t - z}} \]
   3. 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. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. sub-negate-rev N/A

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

      7. lower--.f64 44.4

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

   4. Applied rewrites 44.4%

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

   5. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{\frac{t \cdot x}{z}} \]
   6. 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 36.3

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

   7. Applied rewrites 36.3%

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

   if -1.45e22 < z < 5.5000000000000003e118

   1. Initial program 83.7%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 50.2

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

   4. Applied rewrites 50.2%

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

   if 5.5000000000000003e118 < z

   1. Initial program 83.7%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. lower--.f64 97.2

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

   3. Applied rewrites 97.2%

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

   4. Taylor expanded in z around inf

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

   6. Applied rewrites 35.1%

      \[\leadsto x \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 62.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x\_m}{z}, x\_m\right)\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+80}:\\ \;\;\;\;x\_m \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot 1\\ \end{array} \end{array} \]

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 -1.65e+22)
    (fma t (/ x_m z) x_m)
    (if (<= z 3e+80) (* x_m (/ y t)) (* x_m 1.0)))))

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 <= -1.65e+22) {
		tmp = fma(t, (x_m / z), x_m);
	} else if (z <= 3e+80) {
		tmp = x_m * (y / t);
	} else {
		tmp = x_m * 1.0;
	}
	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 <= -1.65e+22)
		tmp = fma(t, Float64(x_m / z), x_m);
	elseif (z <= 3e+80)
		tmp = Float64(x_m * Float64(y / t));
	else
		tmp = Float64(x_m * 1.0);
	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, -1.65e+22], N[(t * N[(x$95$m / z), $MachinePrecision] + x$95$m), $MachinePrecision], If[LessEqual[z, 3e+80], N[(x$95$m * N[(y / t), $MachinePrecision]), $MachinePrecision], N[(x$95$m * 1.0), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1.6499999999999999e22

   1. Initial program 83.7%

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

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z}{t - z}} \]
   3. 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. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. sub-negate-rev N/A

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

      7. lower--.f64 44.4

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

   4. Applied rewrites 44.4%

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

   5. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{\frac{t \cdot x}{z}} \]
   6. 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 36.3

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

   7. Applied rewrites 36.3%

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

   if -1.6499999999999999e22 < z < 2.99999999999999987e80

   1. Initial program 83.7%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. lower--.f64 97.2

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

   3. Applied rewrites 97.2%

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

   4. Taylor expanded in z around 0

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

      1. lower-/.f64 40.4

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

   6. Applied rewrites 40.4%

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

   if 2.99999999999999987e80 < z

   1. Initial program 83.7%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. lower--.f64 97.2

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

   3. Applied rewrites 97.2%

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

   4. Taylor expanded in z around inf

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

   6. Applied rewrites 35.1%

      \[\leadsto x \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 62.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{+22}:\\ \;\;\;\;x\_m \cdot 1\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+80}:\\ \;\;\;\;x\_m \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot 1\\ \end{array} \end{array} \]

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 -1.65e+22)
    (* x_m 1.0)
    (if (<= z 3e+80) (* x_m (/ y t)) (* x_m 1.0)))))

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 <= -1.65e+22) {
		tmp = x_m * 1.0;
	} else if (z <= 3e+80) {
		tmp = x_m * (y / t);
	} else {
		tmp = x_m * 1.0;
	}
	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 <= (-1.65d+22)) then
        tmp = x_m * 1.0d0
    else if (z <= 3d+80) then
        tmp = x_m * (y / t)
    else
        tmp = x_m * 1.0d0
    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 <= -1.65e+22) {
		tmp = x_m * 1.0;
	} else if (z <= 3e+80) {
		tmp = x_m * (y / t);
	} else {
		tmp = x_m * 1.0;
	}
	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 <= -1.65e+22:
		tmp = x_m * 1.0
	elif z <= 3e+80:
		tmp = x_m * (y / t)
	else:
		tmp = x_m * 1.0
	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 <= -1.65e+22)
		tmp = Float64(x_m * 1.0);
	elseif (z <= 3e+80)
		tmp = Float64(x_m * Float64(y / t));
	else
		tmp = Float64(x_m * 1.0);
	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 <= -1.65e+22)
		tmp = x_m * 1.0;
	elseif (z <= 3e+80)
		tmp = x_m * (y / t);
	else
		tmp = x_m * 1.0;
	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, -1.65e+22], N[(x$95$m * 1.0), $MachinePrecision], If[LessEqual[z, 3e+80], N[(x$95$m * N[(y / t), $MachinePrecision]), $MachinePrecision], N[(x$95$m * 1.0), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.6499999999999999e22 or 2.99999999999999987e80 < z

   1. Initial program 83.7%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. lower--.f64 97.2

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

   3. Applied rewrites 97.2%

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

   4. Taylor expanded in z around inf

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

   6. Applied rewrites 35.1%

      \[\leadsto x \cdot \color{blue}{1} \]

   if -1.6499999999999999e22 < z < 2.99999999999999987e80

   1. Initial program 83.7%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. lower--.f64 97.2

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

   3. Applied rewrites 97.2%

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

   4. Taylor expanded in z around 0

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

      1. lower-/.f64 40.4

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

   6. Applied rewrites 40.4%

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

Alternative 10: 61.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+22}:\\ \;\;\;\;x\_m \cdot 1\\ \mathbf{elif}\;z \leq 105000:\\ \;\;\;\;y \cdot \frac{x\_m}{t}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot 1\\ \end{array} \end{array} \]

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 -1.3e+22)
    (* x_m 1.0)
    (if (<= z 105000.0) (* y (/ x_m t)) (* x_m 1.0)))))

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 <= -1.3e+22) {
		tmp = x_m * 1.0;
	} else if (z <= 105000.0) {
		tmp = y * (x_m / t);
	} else {
		tmp = x_m * 1.0;
	}
	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 <= (-1.3d+22)) then
        tmp = x_m * 1.0d0
    else if (z <= 105000.0d0) then
        tmp = y * (x_m / t)
    else
        tmp = x_m * 1.0d0
    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 <= -1.3e+22) {
		tmp = x_m * 1.0;
	} else if (z <= 105000.0) {
		tmp = y * (x_m / t);
	} else {
		tmp = x_m * 1.0;
	}
	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 <= -1.3e+22:
		tmp = x_m * 1.0
	elif z <= 105000.0:
		tmp = y * (x_m / t)
	else:
		tmp = x_m * 1.0
	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 <= -1.3e+22)
		tmp = Float64(x_m * 1.0);
	elseif (z <= 105000.0)
		tmp = Float64(y * Float64(x_m / t));
	else
		tmp = Float64(x_m * 1.0);
	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 <= -1.3e+22)
		tmp = x_m * 1.0;
	elseif (z <= 105000.0)
		tmp = y * (x_m / t);
	else
		tmp = x_m * 1.0;
	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, -1.3e+22], N[(x$95$m * 1.0), $MachinePrecision], If[LessEqual[z, 105000.0], N[(y * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision], N[(x$95$m * 1.0), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.3e22 or 105000 < z

   1. Initial program 83.7%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. lower--.f64 97.2

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

   3. Applied rewrites 97.2%

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

   4. Taylor expanded in z around inf

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

   6. Applied rewrites 35.1%

      \[\leadsto x \cdot \color{blue}{1} \]

   if -1.3e22 < z < 105000

   1. Initial program 83.7%

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

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{x \cdot y}{t}} \]
   3. 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 38.3

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

   4. Applied rewrites 38.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 37.9

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

   6. Applied rewrites 37.9%

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

Alternative 11: 35.1% accurate, 3.2× speedup

Math

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

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 1.0)))

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 * 1.0);
}

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 * 1.0d0)
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 * 1.0);
}

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 * 1.0)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	return Float64(x_s * Float64(x_m * 1.0))
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 * 1.0);
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 * N[(x$95$m * 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.7%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. frac-2neg N/A

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

   8. lower-/.f64 N/A

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

   9. sub-negate-rev N/A

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

   10. lower--.f64 N/A

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

   11. sub-negate-rev N/A

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

   12. lower--.f64 97.2

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

3. Applied rewrites 97.2%

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

4. Taylor expanded in z around inf

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

6. Applied rewrites 35.1%

   \[\leadsto x \cdot \color{blue}{1} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025136 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3"
  :precision binary64
  (/ (* x (- y z)) (- t z)))