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

Percentage Accurate: 84.0% → 97.3%

Time: 3.5s

Alternatives: 13

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: 84.0% 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: 97.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 10000:\\ \;\;\;\;\frac{x\_m \cdot \left(y - z\right)}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z - t} \cdot \left(z - y\right)\\ \end{array} \end{array} \]

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 10000.0)
    (/ (* x_m (- y z)) (- t z))
    (* (/ x_m (- z t)) (- z y)))))

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 <= 10000.0) {
		tmp = (x_m * (y - z)) / (t - z);
	} else {
		tmp = (x_m / (z - t)) * (z - y);
	}
	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 <= 10000.0d0) then
        tmp = (x_m * (y - z)) / (t - z)
    else
        tmp = (x_m / (z - t)) * (z - y)
    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 <= 10000.0) {
		tmp = (x_m * (y - z)) / (t - z);
	} else {
		tmp = (x_m / (z - t)) * (z - y);
	}
	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 <= 10000.0:
		tmp = (x_m * (y - z)) / (t - z)
	else:
		tmp = (x_m / (z - t)) * (z - y)
	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 <= 10000.0)
		tmp = Float64(Float64(x_m * Float64(y - z)) / Float64(t - z));
	else
		tmp = Float64(Float64(x_m / Float64(z - t)) * Float64(z - y));
	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 <= 10000.0)
		tmp = (x_m * (y - z)) / (t - z);
	else
		tmp = (x_m / (z - t)) * (z - y);
	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, 10000.0], N[(N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(z - t), $MachinePrecision]), $MachinePrecision] * N[(z - y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1e4

   1. Initial program 96.8%

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

   if 1e4 < x

   1. Initial program 70.5%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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. sub-negate N/A

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

      8. sub-negate N/A

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

      9. frac-2neg-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 97.5

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

   3. Applied rewrites 97.5%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. sub-negate N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 97.3

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

   5. Applied rewrites 97.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. sub-negate N/A

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

      8. frac-2neg N/A

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

      9. mult-flip N/A

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

      10. division-flip N/A

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

      11. mult-flip-rev N/A

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

      12. lower-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 97.0

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

   7. Applied rewrites 97.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift--.f64 96.7

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

   9. Applied rewrites 96.7%

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

Alternative 2: 96.8% 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 84.0%

   \[\frac{x \cdot \left(y - z\right)}{t - z} \]
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. sub-negate N/A

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

   8. sub-negate N/A

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

   9. frac-2neg-rev N/A

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

   10. lower-/.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower--.f64 97.3

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

3. Applied rewrites 97.3%

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

Alternative 3: 75.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m - x\_m \cdot \frac{y}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.14 \cdot 10^{+68}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -3.4 \cdot 10^{-177}:\\ \;\;\;\;x\_m \cdot \frac{y}{t - z}\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+26}:\\ \;\;\;\;\frac{x\_m \cdot \left(y - z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 -1.14e+68)
      t_1
      (if (<= z -3.4e-177)
        (* x_m (/ y (- t z)))
        (if (<= z 1.15e+26) (/ (* 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 <= -1.14e+68) {
		tmp = t_1;
	} else if (z <= -3.4e-177) {
		tmp = x_m * (y / (t - z));
	} else if (z <= 1.15e+26) {
		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 <= (-1.14d+68)) then
        tmp = t_1
    else if (z <= (-3.4d-177)) then
        tmp = x_m * (y / (t - z))
    else if (z <= 1.15d+26) 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 <= -1.14e+68) {
		tmp = t_1;
	} else if (z <= -3.4e-177) {
		tmp = x_m * (y / (t - z));
	} else if (z <= 1.15e+26) {
		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 <= -1.14e+68:
		tmp = t_1
	elif z <= -3.4e-177:
		tmp = x_m * (y / (t - z))
	elif z <= 1.15e+26:
		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 <= -1.14e+68)
		tmp = t_1;
	elseif (z <= -3.4e-177)
		tmp = Float64(x_m * Float64(y / Float64(t - z)));
	elseif (z <= 1.15e+26)
		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 <= -1.14e+68)
		tmp = t_1;
	elseif (z <= -3.4e-177)
		tmp = x_m * (y / (t - z));
	elseif (z <= 1.15e+26)
		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, -1.14e+68], t$95$1, If[LessEqual[z, -3.4e-177], N[(x$95$m * N[(y / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.15e+26], N[(N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < -1.13999999999999988e68 or 1.15e26 < z

   1. Initial program 72.3%

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

   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. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 68.6

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

   4. Applied rewrites 68.6%

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

   5. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 78.7

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

   7. Applied rewrites 78.7%

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

   if -1.13999999999999988e68 < z < -3.4000000000000001e-177

   1. Initial program 93.2%

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

   2. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 63.3

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

   4. Applied rewrites 63.3%

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

   if -3.4000000000000001e-177 < z < 1.15e26

   1. Initial program 92.5%

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

   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 73.5%

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

Alternative 4: 75.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m - x\_m \cdot \frac{y}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.14 \cdot 10^{+68}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{-81}:\\ \;\;\;\;x\_m \cdot \frac{y}{t - z}\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{+26}:\\ \;\;\;\;\frac{x\_m \cdot y}{t - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 -1.14e+68)
      t_1
      (if (<= z -1.65e-81)
        (* x_m (/ y (- t z)))
        (if (<= z 4.3e+26) (/ (* x_m y) (- 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 <= -1.14e+68) {
		tmp = t_1;
	} else if (z <= -1.65e-81) {
		tmp = x_m * (y / (t - z));
	} else if (z <= 4.3e+26) {
		tmp = (x_m * y) / (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 <= (-1.14d+68)) then
        tmp = t_1
    else if (z <= (-1.65d-81)) then
        tmp = x_m * (y / (t - z))
    else if (z <= 4.3d+26) then
        tmp = (x_m * y) / (t - z)
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
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 <= -1.14e+68) {
		tmp = t_1;
	} else if (z <= -1.65e-81) {
		tmp = x_m * (y / (t - z));
	} else if (z <= 4.3e+26) {
		tmp = (x_m * y) / (t - z);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = x_m - (x_m * (y / z))
	tmp = 0
	if z <= -1.14e+68:
		tmp = t_1
	elif z <= -1.65e-81:
		tmp = x_m * (y / (t - z))
	elif z <= 4.3e+26:
		tmp = (x_m * y) / (t - z)
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
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 <= -1.14e+68)
		tmp = t_1;
	elseif (z <= -1.65e-81)
		tmp = Float64(x_m * Float64(y / Float64(t - z)));
	elseif (z <= 4.3e+26)
		tmp = Float64(Float64(x_m * y) / Float64(t - z));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m - (x_m * (y / z));
	tmp = 0.0;
	if (z <= -1.14e+68)
		tmp = t_1;
	elseif (z <= -1.65e-81)
		tmp = x_m * (y / (t - z));
	elseif (z <= 4.3e+26)
		tmp = (x_m * y) / (t - z);
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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, -1.14e+68], t$95$1, If[LessEqual[z, -1.65e-81], N[(x$95$m * N[(y / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.3e+26], N[(N[(x$95$m * y), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], t$95$1]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < -1.13999999999999988e68 or 4.2999999999999998e26 < z

   1. Initial program 72.2%

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

   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. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 68.6

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

   4. Applied rewrites 68.6%

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

   5. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 78.8

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

   7. Applied rewrites 78.8%

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

   if -1.13999999999999988e68 < z < -1.64999999999999994e-81

   1. Initial program 93.6%

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

   2. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 56.1

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

   4. Applied rewrites 56.1%

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

   if -1.64999999999999994e-81 < z < 4.2999999999999998e26

   1. Initial program 92.6%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 76.9%

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

Alternative 5: 73.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m - x\_m \cdot \frac{y}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.14 \cdot 10^{+68}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+26}:\\ \;\;\;\;x\_m \cdot \frac{y}{t - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 -1.14e+68) t_1 (if (<= z 6e+26) (* x_m (/ y (- 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 <= -1.14e+68) {
		tmp = t_1;
	} else if (z <= 6e+26) {
		tmp = x_m * (y / (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 <= (-1.14d+68)) then
        tmp = t_1
    else if (z <= 6d+26) then
        tmp = x_m * (y / (t - z))
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
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 <= -1.14e+68) {
		tmp = t_1;
	} else if (z <= 6e+26) {
		tmp = x_m * (y / (t - z));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = x_m - (x_m * (y / z))
	tmp = 0
	if z <= -1.14e+68:
		tmp = t_1
	elif z <= 6e+26:
		tmp = x_m * (y / (t - z))
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
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 <= -1.14e+68)
		tmp = t_1;
	elseif (z <= 6e+26)
		tmp = Float64(x_m * Float64(y / Float64(t - z)));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m - (x_m * (y / z));
	tmp = 0.0;
	if (z <= -1.14e+68)
		tmp = t_1;
	elseif (z <= 6e+26)
		tmp = x_m * (y / (t - z));
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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, -1.14e+68], t$95$1, If[LessEqual[z, 6e+26], N[(x$95$m * N[(y / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.13999999999999988e68 or 5.99999999999999994e26 < z

   1. Initial program 72.2%

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

   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. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 68.6

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

   4. Applied rewrites 68.6%

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

   5. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 78.7

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

   7. Applied rewrites 78.7%

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

   if -1.13999999999999988e68 < z < 5.99999999999999994e26

   1. Initial program 92.8%

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

   2. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 73.5

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

   4. Applied rewrites 73.5%

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

Alternative 6: 67.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m - x\_m \cdot \frac{y}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-121}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-110}:\\ \;\;\;\;\frac{y \cdot x\_m}{t}\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-10}:\\ \;\;\;\;z \cdot \frac{x\_m}{-t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 -1.9e-121)
      t_1
      (if (<= z 1.2e-110)
        (/ (* y x_m) t)
        (if (<= z 2.4e-10) (* z (/ x_m (- 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 <= -1.9e-121) {
		tmp = t_1;
	} else if (z <= 1.2e-110) {
		tmp = (y * x_m) / t;
	} else if (z <= 2.4e-10) {
		tmp = z * (x_m / -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 <= (-1.9d-121)) then
        tmp = t_1
    else if (z <= 1.2d-110) then
        tmp = (y * x_m) / t
    else if (z <= 2.4d-10) then
        tmp = z * (x_m / -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 <= -1.9e-121) {
		tmp = t_1;
	} else if (z <= 1.2e-110) {
		tmp = (y * x_m) / t;
	} else if (z <= 2.4e-10) {
		tmp = z * (x_m / -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 <= -1.9e-121:
		tmp = t_1
	elif z <= 1.2e-110:
		tmp = (y * x_m) / t
	elif z <= 2.4e-10:
		tmp = z * (x_m / -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 <= -1.9e-121)
		tmp = t_1;
	elseif (z <= 1.2e-110)
		tmp = Float64(Float64(y * x_m) / t);
	elseif (z <= 2.4e-10)
		tmp = Float64(z * Float64(x_m / Float64(-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 <= -1.9e-121)
		tmp = t_1;
	elseif (z <= 1.2e-110)
		tmp = (y * x_m) / t;
	elseif (z <= 2.4e-10)
		tmp = z * (x_m / -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, -1.9e-121], t$95$1, If[LessEqual[z, 1.2e-110], N[(N[(y * x$95$m), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 2.4e-10], N[(z * N[(x$95$m / (-t)), $MachinePrecision]), $MachinePrecision], t$95$1]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < -1.9e-121 or 2.4e-10 < z

   1. Initial program 78.7%

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

   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. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 62.0

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

   4. Applied rewrites 62.0%

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

   5. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 69.3

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

   7. Applied rewrites 69.3%

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

   if -1.9e-121 < z < 1.20000000000000003e-110

   1. Initial program 92.2%

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

   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 73.0

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

   4. Applied rewrites 73.0%

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

   if 1.20000000000000003e-110 < z < 2.4e-10

   1. Initial program 93.0%

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

   2. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. sub-negate N/A

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

      4. frac-2neg-rev N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 43.9

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

   4. Applied rewrites 43.9%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 31.5

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

   7. Applied rewrites 31.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 31.6

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

   9. Applied rewrites 31.6%

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

Alternative 7: 61.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{\left(z - y\right) \cdot x\_m}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x\_m}{z}, x\_m\right)\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{-121}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-110}:\\ \;\;\;\;\frac{y \cdot x\_m}{t}\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-10}:\\ \;\;\;\;z \cdot \frac{x\_m}{-t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (/ (* (- z y) x_m) z)))
   (*
    x_s
    (if (<= z -3.6e+100)
      (fma t (/ x_m z) x_m)
      (if (<= z -1.9e-121)
        t_1
        (if (<= z 1.2e-110)
          (/ (* y x_m) t)
          (if (<= z 2.4e-10) (* z (/ x_m (- 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 = ((z - y) * x_m) / z;
	double tmp;
	if (z <= -3.6e+100) {
		tmp = fma(t, (x_m / z), x_m);
	} else if (z <= -1.9e-121) {
		tmp = t_1;
	} else if (z <= 1.2e-110) {
		tmp = (y * x_m) / t;
	} else if (z <= 2.4e-10) {
		tmp = z * (x_m / -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(Float64(Float64(z - y) * x_m) / z)
	tmp = 0.0
	if (z <= -3.6e+100)
		tmp = fma(t, Float64(x_m / z), x_m);
	elseif (z <= -1.9e-121)
		tmp = t_1;
	elseif (z <= 1.2e-110)
		tmp = Float64(Float64(y * x_m) / t);
	elseif (z <= 2.4e-10)
		tmp = Float64(z * Float64(x_m / Float64(-t)));
	else
		tmp = t_1;
	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_] := Block[{t$95$1 = N[(N[(N[(z - y), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -3.6e+100], N[(t * N[(x$95$m / z), $MachinePrecision] + x$95$m), $MachinePrecision], If[LessEqual[z, -1.9e-121], t$95$1, If[LessEqual[z, 1.2e-110], N[(N[(y * x$95$m), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 2.4e-10], N[(z * N[(x$95$m / (-t)), $MachinePrecision]), $MachinePrecision], t$95$1]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if z < -3.6e100

   1. Initial program 67.3%

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

   2. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. sub-negate N/A

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

      4. frac-2neg-rev N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 55.9

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

   4. Applied rewrites 55.9%

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

   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 68.9

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

   7. Applied rewrites 68.9%

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

   if -3.6e100 < z < -1.9e-121 or 2.4e-10 < z

   1. Initial program 83.1%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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. sub-negate N/A

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

      8. sub-negate N/A

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

      9. frac-2neg-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 99.2

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 53.2

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

   6. Applied rewrites 53.2%

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

   if -1.9e-121 < z < 1.20000000000000003e-110

   1. Initial program 92.2%

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

   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 73.0

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

   4. Applied rewrites 73.0%

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

   if 1.20000000000000003e-110 < z < 2.4e-10

   1. Initial program 93.0%

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

   2. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. sub-negate N/A

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

      4. frac-2neg-rev N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 43.9

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

   4. Applied rewrites 43.9%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 31.5

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

   7. Applied rewrites 31.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 31.6

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

   9. Applied rewrites 31.6%

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

Alternative 8: 60.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, \frac{x\_m}{z}, x\_m\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+70}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{-110}:\\ \;\;\;\;x\_m \cdot \frac{y}{t}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+22}:\\ \;\;\;\;z \cdot \frac{x\_m}{-t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (fma t (/ x_m z) x_m)))
   (*
    x_s
    (if (<= z -8.5e+70)
      t_1
      (if (<= z 1.95e-110)
        (* x_m (/ y t))
        (if (<= z 1.5e+22) (* z (/ x_m (- 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 = fma(t, (x_m / z), x_m);
	double tmp;
	if (z <= -8.5e+70) {
		tmp = t_1;
	} else if (z <= 1.95e-110) {
		tmp = x_m * (y / t);
	} else if (z <= 1.5e+22) {
		tmp = z * (x_m / -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 = fma(t, Float64(x_m / z), x_m)
	tmp = 0.0
	if (z <= -8.5e+70)
		tmp = t_1;
	elseif (z <= 1.95e-110)
		tmp = Float64(x_m * Float64(y / t));
	elseif (z <= 1.5e+22)
		tmp = Float64(z * Float64(x_m / Float64(-t)));
	else
		tmp = t_1;
	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_] := Block[{t$95$1 = N[(t * N[(x$95$m / z), $MachinePrecision] + x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -8.5e+70], t$95$1, If[LessEqual[z, 1.95e-110], N[(x$95$m * N[(y / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.5e+22], N[(z * N[(x$95$m / (-t)), $MachinePrecision]), $MachinePrecision], t$95$1]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < -8.4999999999999996e70 or 1.5e22 < z

   1. Initial program 72.5%

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

   2. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. sub-negate N/A

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

      4. frac-2neg-rev N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 57.6

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

   4. Applied rewrites 57.6%

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

   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 63.3

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

   7. Applied rewrites 63.3%

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

   if -8.4999999999999996e70 < z < 1.95e-110

   1. Initial program 92.5%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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. sub-negate N/A

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

      8. sub-negate N/A

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

      9. frac-2neg-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 94.9

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

   3. Applied rewrites 94.9%

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

   4. Taylor expanded in z around 0

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

      1. lower-/.f64 63.5

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

   6. Applied rewrites 63.5%

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

   if 1.95e-110 < z < 1.5e22

   1. Initial program 93.3%

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

   2. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. sub-negate N/A

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

      4. frac-2neg-rev N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 46.4

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

   4. Applied rewrites 46.4%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 30.3

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

   7. Applied rewrites 30.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 30.5

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

   9. Applied rewrites 30.5%

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

Alternative 9: 60.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.14 \cdot 10^{+68}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{-110}:\\ \;\;\;\;x\_m \cdot \frac{y}{t}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+22}:\\ \;\;\;\;z \cdot \frac{x\_m}{-t}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

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 -1.14e+68)
    x_m
    (if (<= z 1.95e-110)
      (* x_m (/ y t))
      (if (<= z 1.5e+22) (* z (/ x_m (- 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 <= -1.14e+68) {
		tmp = x_m;
	} else if (z <= 1.95e-110) {
		tmp = x_m * (y / t);
	} else if (z <= 1.5e+22) {
		tmp = z * (x_m / -t);
	} 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 <= (-1.14d+68)) then
        tmp = x_m
    else if (z <= 1.95d-110) then
        tmp = x_m * (y / t)
    else if (z <= 1.5d+22) then
        tmp = z * (x_m / -t)
    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 <= -1.14e+68) {
		tmp = x_m;
	} else if (z <= 1.95e-110) {
		tmp = x_m * (y / t);
	} else if (z <= 1.5e+22) {
		tmp = z * (x_m / -t);
	} 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 <= -1.14e+68:
		tmp = x_m
	elif z <= 1.95e-110:
		tmp = x_m * (y / t)
	elif z <= 1.5e+22:
		tmp = z * (x_m / -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 <= -1.14e+68)
		tmp = x_m;
	elseif (z <= 1.95e-110)
		tmp = Float64(x_m * Float64(y / t));
	elseif (z <= 1.5e+22)
		tmp = Float64(z * Float64(x_m / Float64(-t)));
	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 <= -1.14e+68)
		tmp = x_m;
	elseif (z <= 1.95e-110)
		tmp = x_m * (y / t);
	elseif (z <= 1.5e+22)
		tmp = z * (x_m / -t);
	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, -1.14e+68], x$95$m, If[LessEqual[z, 1.95e-110], N[(x$95$m * N[(y / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.5e+22], N[(z * N[(x$95$m / (-t)), $MachinePrecision]), $MachinePrecision], x$95$m]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1.13999999999999988e68 or 1.5e22 < z

   1. Initial program 72.5%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 63.2%

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

   if -1.13999999999999988e68 < z < 1.95e-110

   1. Initial program 92.6%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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. sub-negate N/A

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

      8. sub-negate N/A

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

      9. frac-2neg-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 94.9

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

   3. Applied rewrites 94.9%

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

   4. Taylor expanded in z around 0

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

      1. lower-/.f64 63.8

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

   6. Applied rewrites 63.8%

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

   if 1.95e-110 < z < 1.5e22

   1. Initial program 93.3%

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

   2. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. sub-negate N/A

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

      4. frac-2neg-rev N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 46.4

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

   4. Applied rewrites 46.4%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 30.3

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

   7. Applied rewrites 30.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 30.5

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

   9. Applied rewrites 30.5%

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

Alternative 10: 60.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.14 \cdot 10^{+68}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+27}:\\ \;\;\;\;x\_m \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

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 -1.14e+68) x_m (if (<= z 1.3e+27) (* x_m (/ y 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 <= -1.14e+68) {
		tmp = x_m;
	} else if (z <= 1.3e+27) {
		tmp = x_m * (y / t);
	} 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 <= (-1.14d+68)) then
        tmp = x_m
    else if (z <= 1.3d+27) then
        tmp = x_m * (y / t)
    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 <= -1.14e+68) {
		tmp = x_m;
	} else if (z <= 1.3e+27) {
		tmp = x_m * (y / t);
	} 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 <= -1.14e+68:
		tmp = x_m
	elif z <= 1.3e+27:
		tmp = x_m * (y / 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 <= -1.14e+68)
		tmp = x_m;
	elseif (z <= 1.3e+27)
		tmp = Float64(x_m * Float64(y / t));
	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 <= -1.14e+68)
		tmp = x_m;
	elseif (z <= 1.3e+27)
		tmp = x_m * (y / t);
	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, -1.14e+68], x$95$m, If[LessEqual[z, 1.3e+27], N[(x$95$m * N[(y / t), $MachinePrecision]), $MachinePrecision], x$95$m]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.13999999999999988e68 or 1.30000000000000004e27 < z

   1. Initial program 72.2%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 63.6%

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

   if -1.13999999999999988e68 < z < 1.30000000000000004e27

   1. Initial program 92.8%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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. sub-negate N/A

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

      8. sub-negate N/A

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

      9. frac-2neg-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 95.5

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

   3. Applied rewrites 95.5%

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

   4. Taylor expanded in z around 0

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

      1. lower-/.f64 60.2

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

   6. Applied rewrites 60.2%

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

Alternative 11: 60.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.14 \cdot 10^{+68}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+27}:\\ \;\;\;\;\frac{y \cdot x\_m}{t}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

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 -1.14e+68) x_m (if (<= z 1.3e+27) (/ (* y x_m) 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 <= -1.14e+68) {
		tmp = x_m;
	} else if (z <= 1.3e+27) {
		tmp = (y * x_m) / t;
	} 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 <= (-1.14d+68)) then
        tmp = x_m
    else if (z <= 1.3d+27) then
        tmp = (y * x_m) / t
    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 <= -1.14e+68) {
		tmp = x_m;
	} else if (z <= 1.3e+27) {
		tmp = (y * x_m) / t;
	} 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 <= -1.14e+68:
		tmp = x_m
	elif z <= 1.3e+27:
		tmp = (y * x_m) / 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 <= -1.14e+68)
		tmp = x_m;
	elseif (z <= 1.3e+27)
		tmp = Float64(Float64(y * x_m) / t);
	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 <= -1.14e+68)
		tmp = x_m;
	elseif (z <= 1.3e+27)
		tmp = (y * x_m) / t;
	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, -1.14e+68], x$95$m, If[LessEqual[z, 1.3e+27], N[(N[(y * x$95$m), $MachinePrecision] / t), $MachinePrecision], x$95$m]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.13999999999999988e68 or 1.30000000000000004e27 < z

   1. Initial program 72.2%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 63.6%

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

   if -1.13999999999999988e68 < z < 1.30000000000000004e27

   1. Initial program 92.8%

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

   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 57.7

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

   4. Applied rewrites 57.7%

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

Alternative 12: 36.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot \left(y - z\right)}{t - z} \leq 0:\\ \;\;\;\;t \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

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 (- y z)) (- t z)) 0.0) (* t (/ 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 * (y - z)) / (t - z)) <= 0.0) {
		tmp = t * (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 * (y - z)) / (t - z)) <= 0.0d0) then
        tmp = t * (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 * (y - z)) / (t - z)) <= 0.0) {
		tmp = t * (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 * (y - z)) / (t - z)) <= 0.0:
		tmp = t * (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 (Float64(Float64(x_m * Float64(y - z)) / Float64(t - z)) <= 0.0)
		tmp = Float64(t * Float64(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 * (y - z)) / (t - z)) <= 0.0)
		tmp = t * (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[N[(N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], 0.0], N[(t * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], x$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < -0.0

   1. Initial program 88.2%

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

   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. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 25.1

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

   4. Applied rewrites 25.1%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lift-*.f64 4.8

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

   7. Applied rewrites 4.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 6.8

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

   9. Applied rewrites 6.8%

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

   if -0.0 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z))

   1. Initial program 81.5%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 54.6%

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

Alternative 13: 35.5% accurate, 12.6× 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 84.0%

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

2. Taylor expanded in z around inf

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

4. Applied rewrites 35.5%

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

Reproduce

herbie shell --seed 2025127 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3"
  :precision binary64
  (/ (* x (- y z)) (- t z)))