Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 90.0% → 98.3%

Time: 6.4s

Alternatives: 9

Speedup: 1.3×

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot 2}{y \cdot z - t \cdot z} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ (* x 2.0) (- (* y z) (* t z))))

C

double code(double x, double y, double z, double t) {
	return (x * 2.0) / ((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 * 2.0d0) / ((y * z) - (t * z))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * 2.0) / ((y * z) - (t * z));
}

Python

def code(x, y, z, t):
	return (x * 2.0) / ((y * z) - (t * z))

Julia

function code(x, y, z, t)
	return Float64(Float64(x * 2.0) / Float64(Float64(y * z) - Float64(t * z)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * 2.0) / ((y * z) - (t * z));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * 2.0), $MachinePrecision] / N[(N[(y * z), $MachinePrecision] - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot 2}{y \cdot z - t \cdot z} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ (* x 2.0) (- (* y z) (* t z))))

C

double code(double x, double y, double z, double t) {
	return (x * 2.0) / ((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 * 2.0d0) / ((y * z) - (t * z))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * 2.0) / ((y * z) - (t * z));
}

Python

def code(x, y, z, t):
	return (x * 2.0) / ((y * z) - (t * z))

Julia

function code(x, y, z, t)
	return Float64(Float64(x * 2.0) / Float64(Float64(y * z) - Float64(t * z)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * 2.0) / ((y * z) - (t * z));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * 2.0), $MachinePrecision] / N[(N[(y * z), $MachinePrecision] - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := y \cdot z\_m - t \cdot z\_m\\ t_2 := \frac{x\_m \cdot 2}{t\_1}\\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-302}:\\ \;\;\;\;\frac{x\_m + x\_m}{t\_1}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{x\_m}{z\_m} \cdot \frac{2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m + x\_m}{\left(y - t\right) \cdot z\_m}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (let* ((t_1 (- (* y z_m) (* t z_m))) (t_2 (/ (* x_m 2.0) t_1)))
   (*
    z_s
    (*
     x_s
     (if (<= t_2 -5e-302)
       (/ (+ x_m x_m) t_1)
       (if (<= t_2 0.0)
         (* (/ x_m z_m) (/ 2.0 (- y t)))
         (/ (+ x_m x_m) (* (- y t) z_m))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double t_1 = (y * z_m) - (t * z_m);
	double t_2 = (x_m * 2.0) / t_1;
	double tmp;
	if (t_2 <= -5e-302) {
		tmp = (x_m + x_m) / t_1;
	} else if (t_2 <= 0.0) {
		tmp = (x_m / z_m) * (2.0 / (y - t));
	} else {
		tmp = (x_m + x_m) / ((y - t) * z_m);
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m =     private
x\_s =     private
z\_m =     private
z\_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(z_s, x_s, x_m, y, z_m, t)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (y * z_m) - (t * z_m)
    t_2 = (x_m * 2.0d0) / t_1
    if (t_2 <= (-5d-302)) then
        tmp = (x_m + x_m) / t_1
    else if (t_2 <= 0.0d0) then
        tmp = (x_m / z_m) * (2.0d0 / (y - t))
    else
        tmp = (x_m + x_m) / ((y - t) * z_m)
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double t_1 = (y * z_m) - (t * z_m);
	double t_2 = (x_m * 2.0) / t_1;
	double tmp;
	if (t_2 <= -5e-302) {
		tmp = (x_m + x_m) / t_1;
	} else if (t_2 <= 0.0) {
		tmp = (x_m / z_m) * (2.0 / (y - t));
	} else {
		tmp = (x_m + x_m) / ((y - t) * z_m);
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	t_1 = (y * z_m) - (t * z_m)
	t_2 = (x_m * 2.0) / t_1
	tmp = 0
	if t_2 <= -5e-302:
		tmp = (x_m + x_m) / t_1
	elif t_2 <= 0.0:
		tmp = (x_m / z_m) * (2.0 / (y - t))
	else:
		tmp = (x_m + x_m) / ((y - t) * z_m)
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	t_1 = Float64(Float64(y * z_m) - Float64(t * z_m))
	t_2 = Float64(Float64(x_m * 2.0) / t_1)
	tmp = 0.0
	if (t_2 <= -5e-302)
		tmp = Float64(Float64(x_m + x_m) / t_1);
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(x_m / z_m) * Float64(2.0 / Float64(y - t)));
	else
		tmp = Float64(Float64(x_m + x_m) / Float64(Float64(y - t) * z_m));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	t_1 = (y * z_m) - (t * z_m);
	t_2 = (x_m * 2.0) / t_1;
	tmp = 0.0;
	if (t_2 <= -5e-302)
		tmp = (x_m + x_m) / t_1;
	elseif (t_2 <= 0.0)
		tmp = (x_m / z_m) * (2.0 / (y - t));
	else
		tmp = (x_m + x_m) / ((y - t) * z_m);
	end
	tmp_2 = z_s * (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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(y * z$95$m), $MachinePrecision] - N[(t * z$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x$95$m * 2.0), $MachinePrecision] / t$95$1), $MachinePrecision]}, N[(z$95$s * N[(x$95$s * If[LessEqual[t$95$2, -5e-302], N[(N[(x$95$m + x$95$m), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m + x$95$m), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x #s(literal 2 binary64)) (-.f64 (*.f64 y z) (*.f64 t z))) < -5.00000000000000033e-302

   1. Initial program 98.5%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot 2}}{y \cdot z - t \cdot z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{2 \cdot x}}{y \cdot z - t \cdot z} \]

      3. count-2-rev N/A

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

      4. lower-+.f64 98.5

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

   3. Applied rewrites 98.5%

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

   if -5.00000000000000033e-302 < (/.f64 (*.f64 x #s(literal 2 binary64)) (-.f64 (*.f64 y z) (*.f64 t z))) < -0.0

   1. Initial program 79.6%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot 2}}{y \cdot z - t \cdot z} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot 2}{y \cdot z - t \cdot z}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot z} - t \cdot z} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot 2}{y \cdot z - \color{blue}{t \cdot z}} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot z - t \cdot z}} \]

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

         \[\leadsto \frac{x \cdot 2}{\color{blue}{z \cdot \left(y - t\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{2}{y - t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{2}{y - t}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{z}} \cdot \frac{2}{y - t} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{2}{y - t}} \]

      11. lower--.f64 99.6

          \[\leadsto \frac{x}{z} \cdot \frac{2}{\color{blue}{y - t}} \]

   3. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{2}{y - t}} \]

   if -0.0 < (/.f64 (*.f64 x #s(literal 2 binary64)) (-.f64 (*.f64 y z) (*.f64 t z)))

   1. Initial program 90.5%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot 2}}{y \cdot z - t \cdot z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{2 \cdot x}}{y \cdot z - t \cdot z} \]

      3. count-2-rev N/A

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

      4. lower-+.f64 90.5

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

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

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 96.3

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

   3. Applied rewrites 96.3%

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

Alternative 2: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := y \cdot z\_m - t \cdot z\_m\\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{x\_m + x\_m}{z\_m}}{y - t}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\frac{x\_m + x\_m}{\left(y - t\right) \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z\_m} \cdot \frac{2}{y - t}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (let* ((t_1 (- (* y z_m) (* t z_m))))
   (*
    z_s
    (*
     x_s
     (if (<= t_1 -5e+231)
       (/ (/ (+ x_m x_m) z_m) (- y t))
       (if (<= t_1 5e+153)
         (/ (+ x_m x_m) (* (- y t) z_m))
         (* (/ x_m z_m) (/ 2.0 (- y t)))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double t_1 = (y * z_m) - (t * z_m);
	double tmp;
	if (t_1 <= -5e+231) {
		tmp = ((x_m + x_m) / z_m) / (y - t);
	} else if (t_1 <= 5e+153) {
		tmp = (x_m + x_m) / ((y - t) * z_m);
	} else {
		tmp = (x_m / z_m) * (2.0 / (y - t));
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m =     private
x\_s =     private
z\_m =     private
z\_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(z_s, x_s, x_m, y, z_m, t)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y * z_m) - (t * z_m)
    if (t_1 <= (-5d+231)) then
        tmp = ((x_m + x_m) / z_m) / (y - t)
    else if (t_1 <= 5d+153) then
        tmp = (x_m + x_m) / ((y - t) * z_m)
    else
        tmp = (x_m / z_m) * (2.0d0 / (y - t))
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double t_1 = (y * z_m) - (t * z_m);
	double tmp;
	if (t_1 <= -5e+231) {
		tmp = ((x_m + x_m) / z_m) / (y - t);
	} else if (t_1 <= 5e+153) {
		tmp = (x_m + x_m) / ((y - t) * z_m);
	} else {
		tmp = (x_m / z_m) * (2.0 / (y - t));
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	t_1 = (y * z_m) - (t * z_m)
	tmp = 0
	if t_1 <= -5e+231:
		tmp = ((x_m + x_m) / z_m) / (y - t)
	elif t_1 <= 5e+153:
		tmp = (x_m + x_m) / ((y - t) * z_m)
	else:
		tmp = (x_m / z_m) * (2.0 / (y - t))
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	t_1 = Float64(Float64(y * z_m) - Float64(t * z_m))
	tmp = 0.0
	if (t_1 <= -5e+231)
		tmp = Float64(Float64(Float64(x_m + x_m) / z_m) / Float64(y - t));
	elseif (t_1 <= 5e+153)
		tmp = Float64(Float64(x_m + x_m) / Float64(Float64(y - t) * z_m));
	else
		tmp = Float64(Float64(x_m / z_m) * Float64(2.0 / Float64(y - t)));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	t_1 = (y * z_m) - (t * z_m);
	tmp = 0.0;
	if (t_1 <= -5e+231)
		tmp = ((x_m + x_m) / z_m) / (y - t);
	elseif (t_1 <= 5e+153)
		tmp = (x_m + x_m) / ((y - t) * z_m);
	else
		tmp = (x_m / z_m) * (2.0 / (y - t));
	end
	tmp_2 = z_s * (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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(y * z$95$m), $MachinePrecision] - N[(t * z$95$m), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * N[(x$95$s * If[LessEqual[t$95$1, -5e+231], N[(N[(N[(x$95$m + x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+153], N[(N[(x$95$m + x$95$m), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < -5.00000000000000028e231

   1. Initial program 78.0%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot 2}}{y \cdot z - t \cdot z} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot 2}{y \cdot z - t \cdot z}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot z} - t \cdot z} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot 2}{y \cdot z - \color{blue}{t \cdot z}} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot z - t \cdot z}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{2 \cdot x}}{y \cdot z - t \cdot z} \]

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{2 \cdot \frac{x}{y \cdot z - t \cdot z}} \]

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

         \[\leadsto 2 \cdot \frac{x}{\color{blue}{z \cdot \left(y - t\right)}} \]

      9. count-2-rev N/A

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

      10. associate-/r* N/A

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

      11. associate-/r* N/A

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

      12. div-add-rev N/A

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

      13. count-2-rev N/A

          \[\leadsto \frac{\color{blue}{2 \cdot \frac{x}{z}}}{y - t} \]

      14. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{2 \cdot \frac{x}{z}}{y - t}} \]

      15. associate-*r/ N/A

          \[\leadsto \frac{\color{blue}{\frac{2 \cdot x}{z}}}{y - t} \]

      16. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{x \cdot 2}}{z}}{y - t} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{x \cdot 2}{z}}}{y - t} \]

      18. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{2 \cdot x}}{z}}{y - t} \]

      19. count-2-rev N/A

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

      20. lower-+.f64 N/A

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

      21. lower--.f64 99.6

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

   3. Applied rewrites 99.6%

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

   if -5.00000000000000028e231 < (-.f64 (*.f64 y z) (*.f64 t z)) < 5.00000000000000018e153

   1. Initial program 96.9%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot 2}}{y \cdot z - t \cdot z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{2 \cdot x}}{y \cdot z - t \cdot z} \]

      3. count-2-rev N/A

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

      4. lower-+.f64 96.9

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

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

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 97.9

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

   3. Applied rewrites 97.9%

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

   if 5.00000000000000018e153 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 78.1%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot 2}}{y \cdot z - t \cdot z} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot 2}{y \cdot z - t \cdot z}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot z} - t \cdot z} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot 2}{y \cdot z - \color{blue}{t \cdot z}} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot z - t \cdot z}} \]

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

         \[\leadsto \frac{x \cdot 2}{\color{blue}{z \cdot \left(y - t\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{2}{y - t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{2}{y - t}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{z}} \cdot \frac{2}{y - t} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{2}{y - t}} \]

      11. lower--.f64 98.4

          \[\leadsto \frac{x}{z} \cdot \frac{2}{\color{blue}{y - t}} \]

   3. Applied rewrites 98.4%

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

Alternative 3: 98.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{\frac{x\_m + x\_m}{z\_m}}{y - t}\\ t_2 := y \cdot z\_m - t \cdot z\_m\\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+231}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\frac{x\_m + x\_m}{\left(y - t\right) \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (let* ((t_1 (/ (/ (+ x_m x_m) z_m) (- y t))) (t_2 (- (* y z_m) (* t z_m))))
   (*
    z_s
    (*
     x_s
     (if (<= t_2 -5e+231)
       t_1
       (if (<= t_2 5e+153) (/ (+ x_m x_m) (* (- y t) z_m)) t_1))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double t_1 = ((x_m + x_m) / z_m) / (y - t);
	double t_2 = (y * z_m) - (t * z_m);
	double tmp;
	if (t_2 <= -5e+231) {
		tmp = t_1;
	} else if (t_2 <= 5e+153) {
		tmp = (x_m + x_m) / ((y - t) * z_m);
	} else {
		tmp = t_1;
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m =     private
x\_s =     private
z\_m =     private
z\_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(z_s, x_s, x_m, y, z_m, t)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((x_m + x_m) / z_m) / (y - t)
    t_2 = (y * z_m) - (t * z_m)
    if (t_2 <= (-5d+231)) then
        tmp = t_1
    else if (t_2 <= 5d+153) then
        tmp = (x_m + x_m) / ((y - t) * z_m)
    else
        tmp = t_1
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double t_1 = ((x_m + x_m) / z_m) / (y - t);
	double t_2 = (y * z_m) - (t * z_m);
	double tmp;
	if (t_2 <= -5e+231) {
		tmp = t_1;
	} else if (t_2 <= 5e+153) {
		tmp = (x_m + x_m) / ((y - t) * z_m);
	} else {
		tmp = t_1;
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	t_1 = ((x_m + x_m) / z_m) / (y - t)
	t_2 = (y * z_m) - (t * z_m)
	tmp = 0
	if t_2 <= -5e+231:
		tmp = t_1
	elif t_2 <= 5e+153:
		tmp = (x_m + x_m) / ((y - t) * z_m)
	else:
		tmp = t_1
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	t_1 = Float64(Float64(Float64(x_m + x_m) / z_m) / Float64(y - t))
	t_2 = Float64(Float64(y * z_m) - Float64(t * z_m))
	tmp = 0.0
	if (t_2 <= -5e+231)
		tmp = t_1;
	elseif (t_2 <= 5e+153)
		tmp = Float64(Float64(x_m + x_m) / Float64(Float64(y - t) * z_m));
	else
		tmp = t_1;
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	t_1 = ((x_m + x_m) / z_m) / (y - t);
	t_2 = (y * z_m) - (t * z_m);
	tmp = 0.0;
	if (t_2 <= -5e+231)
		tmp = t_1;
	elseif (t_2 <= 5e+153)
		tmp = (x_m + x_m) / ((y - t) * z_m);
	else
		tmp = t_1;
	end
	tmp_2 = z_s * (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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(N[(x$95$m + x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * z$95$m), $MachinePrecision] - N[(t * z$95$m), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * N[(x$95$s * If[LessEqual[t$95$2, -5e+231], t$95$1, If[LessEqual[t$95$2, 5e+153], N[(N[(x$95$m + x$95$m), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < -5.00000000000000028e231 or 5.00000000000000018e153 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 78.1%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot 2}}{y \cdot z - t \cdot z} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot 2}{y \cdot z - t \cdot z}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot z} - t \cdot z} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot 2}{y \cdot z - \color{blue}{t \cdot z}} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot z - t \cdot z}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{2 \cdot x}}{y \cdot z - t \cdot z} \]

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{2 \cdot \frac{x}{y \cdot z - t \cdot z}} \]

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

         \[\leadsto 2 \cdot \frac{x}{\color{blue}{z \cdot \left(y - t\right)}} \]

      9. count-2-rev N/A

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

      10. associate-/r* N/A

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

      11. associate-/r* N/A

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

      12. div-add-rev N/A

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

      13. count-2-rev N/A

          \[\leadsto \frac{\color{blue}{2 \cdot \frac{x}{z}}}{y - t} \]

      14. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{2 \cdot \frac{x}{z}}{y - t}} \]

      15. associate-*r/ N/A

          \[\leadsto \frac{\color{blue}{\frac{2 \cdot x}{z}}}{y - t} \]

      16. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{x \cdot 2}}{z}}{y - t} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{x \cdot 2}{z}}}{y - t} \]

      18. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{2 \cdot x}}{z}}{y - t} \]

      19. count-2-rev N/A

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

      20. lower-+.f64 N/A

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

      21. lower--.f64 98.9

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

   3. Applied rewrites 98.9%

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

   if -5.00000000000000028e231 < (-.f64 (*.f64 y z) (*.f64 t z)) < 5.00000000000000018e153

   1. Initial program 96.9%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot 2}}{y \cdot z - t \cdot z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{2 \cdot x}}{y \cdot z - t \cdot z} \]

      3. count-2-rev N/A

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

      4. lower-+.f64 96.9

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

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

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 97.9

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

   3. Applied rewrites 97.9%

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

Alternative 4: 92.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

x\_m =     private
x\_s =     private
z\_m =     private
z\_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(z_s, x_s, x_m, y, z_m, t)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    code = z_s * (x_s * ((x_m + x_m) / ((y - t) * z_m)))
end function

Java

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

Python

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

Julia

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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp = code(z_s, x_s, x_m, y, z_m, t)
	tmp = z_s * (x_s * ((x_m + x_m) / ((y - t) * z_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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * N[(N[(x$95$m + x$95$m), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.0%

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

   1. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot 2}}{y \cdot z - t \cdot z} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{2 \cdot x}}{y \cdot z - t \cdot z} \]

   3. count-2-rev N/A

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

   4. lower-+.f64 90.0

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift--.f64 N/A

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

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

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower--.f64 92.2

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

3. Applied rewrites 92.2%

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

Alternative 5: 74.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -8.6 \cdot 10^{+54}:\\ \;\;\;\;\frac{x\_m}{z\_m} \cdot \frac{-2}{t}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-91}:\\ \;\;\;\;\frac{\frac{x\_m + x\_m}{z\_m}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{t} \cdot -2}{z\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= t -8.6e+54)
     (* (/ x_m z_m) (/ -2.0 t))
     (if (<= t 6.5e-91)
       (/ (/ (+ x_m x_m) z_m) y)
       (/ (* (/ x_m t) -2.0) z_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (t <= -8.6e+54) {
		tmp = (x_m / z_m) * (-2.0 / t);
	} else if (t <= 6.5e-91) {
		tmp = ((x_m + x_m) / z_m) / y;
	} else {
		tmp = ((x_m / t) * -2.0) / z_m;
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m =     private
x\_s =     private
z\_m =     private
z\_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(z_s, x_s, x_m, y, z_m, t)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-8.6d+54)) then
        tmp = (x_m / z_m) * ((-2.0d0) / t)
    else if (t <= 6.5d-91) then
        tmp = ((x_m + x_m) / z_m) / y
    else
        tmp = ((x_m / t) * (-2.0d0)) / z_m
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (t <= -8.6e+54) {
		tmp = (x_m / z_m) * (-2.0 / t);
	} else if (t <= 6.5e-91) {
		tmp = ((x_m + x_m) / z_m) / y;
	} else {
		tmp = ((x_m / t) * -2.0) / z_m;
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if t <= -8.6e+54:
		tmp = (x_m / z_m) * (-2.0 / t)
	elif t <= 6.5e-91:
		tmp = ((x_m + x_m) / z_m) / y
	else:
		tmp = ((x_m / t) * -2.0) / z_m
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if (t <= -8.6e+54)
		tmp = Float64(Float64(x_m / z_m) * Float64(-2.0 / t));
	elseif (t <= 6.5e-91)
		tmp = Float64(Float64(Float64(x_m + x_m) / z_m) / y);
	else
		tmp = Float64(Float64(Float64(x_m / t) * -2.0) / z_m);
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if (t <= -8.6e+54)
		tmp = (x_m / z_m) * (-2.0 / t);
	elseif (t <= 6.5e-91)
		tmp = ((x_m + x_m) / z_m) / y;
	else
		tmp = ((x_m / t) * -2.0) / z_m;
	end
	tmp_2 = z_s * (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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[LessEqual[t, -8.6e+54], N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(-2.0 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.5e-91], N[(N[(N[(x$95$m + x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[(x$95$m / t), $MachinePrecision] * -2.0), $MachinePrecision] / z$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -8.59999999999999952e54

   1. Initial program 87.8%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot 2}}{y \cdot z - t \cdot z} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot 2}{y \cdot z - t \cdot z}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot z} - t \cdot z} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot 2}{y \cdot z - \color{blue}{t \cdot z}} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot z - t \cdot z}} \]

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

         \[\leadsto \frac{x \cdot 2}{\color{blue}{z \cdot \left(y - t\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{2}{y - t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{2}{y - t}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{z}} \cdot \frac{2}{y - t} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{2}{y - t}} \]

      11. lower--.f64 90.8

          \[\leadsto \frac{x}{z} \cdot \frac{2}{\color{blue}{y - t}} \]

   3. Applied rewrites 90.8%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{2}{y - t}} \]

   4. Taylor expanded in y around 0

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{-2}{t}} \]
   5. Step-by-step derivation

      1. lower-/.f64 79.3

         \[\leadsto \frac{x}{z} \cdot \frac{-2}{\color{blue}{t}} \]

   6. Applied rewrites 79.3%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{-2}{t}} \]

   if -8.59999999999999952e54 < t < 6.5000000000000001e-91

   1. Initial program 92.2%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]

   2. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot x}{\color{blue}{y \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x \cdot 2}{\color{blue}{y} \cdot z} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot z}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2 \cdot x}{\color{blue}{y} \cdot z} \]

      5. count-2-rev N/A

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

      6. lower-+.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 73.1

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

   4. Applied rewrites 73.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. count-2-rev N/A

         \[\leadsto \frac{\frac{2 \cdot x}{z}}{y} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \frac{x}{z}}{y} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{2 \cdot \frac{x}{z}}{\color{blue}{y}} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{\frac{2 \cdot x}{z}}{y} \]

      9. count-2-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 74.2

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

   6. Applied rewrites 74.2%

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

   if 6.5000000000000001e-91 < t

   1. Initial program 88.3%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot 2}}{y \cdot z - t \cdot z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{2 \cdot x}}{y \cdot z - t \cdot z} \]

      3. count-2-rev N/A

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

      4. lower-+.f64 88.3

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

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

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 91.2

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

   3. Applied rewrites 91.2%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift--.f64 93.3

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

   5. Applied rewrites 93.3%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{x}{t}}}{z} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{x}{t} \cdot \color{blue}{-2}}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{x}{t} \cdot \color{blue}{-2}}{z} \]

      3. lower-/.f64 71.7

         \[\leadsto \frac{\frac{x}{t} \cdot -2}{z} \]

   8. Applied rewrites 71.7%

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

Alternative 6: 73.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -8.6 \cdot 10^{+54}:\\ \;\;\;\;\frac{x\_m}{z\_m} \cdot \frac{-2}{t}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-91}:\\ \;\;\;\;\frac{\frac{x\_m + x\_m}{z\_m}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{t \cdot z\_m} \cdot -2\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= t -8.6e+54)
     (* (/ x_m z_m) (/ -2.0 t))
     (if (<= t 6.5e-91)
       (/ (/ (+ x_m x_m) z_m) y)
       (* (/ x_m (* t z_m)) -2.0))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (t <= -8.6e+54) {
		tmp = (x_m / z_m) * (-2.0 / t);
	} else if (t <= 6.5e-91) {
		tmp = ((x_m + x_m) / z_m) / y;
	} else {
		tmp = (x_m / (t * z_m)) * -2.0;
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m =     private
x\_s =     private
z\_m =     private
z\_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(z_s, x_s, x_m, y, z_m, t)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-8.6d+54)) then
        tmp = (x_m / z_m) * ((-2.0d0) / t)
    else if (t <= 6.5d-91) then
        tmp = ((x_m + x_m) / z_m) / y
    else
        tmp = (x_m / (t * z_m)) * (-2.0d0)
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (t <= -8.6e+54) {
		tmp = (x_m / z_m) * (-2.0 / t);
	} else if (t <= 6.5e-91) {
		tmp = ((x_m + x_m) / z_m) / y;
	} else {
		tmp = (x_m / (t * z_m)) * -2.0;
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if t <= -8.6e+54:
		tmp = (x_m / z_m) * (-2.0 / t)
	elif t <= 6.5e-91:
		tmp = ((x_m + x_m) / z_m) / y
	else:
		tmp = (x_m / (t * z_m)) * -2.0
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if (t <= -8.6e+54)
		tmp = Float64(Float64(x_m / z_m) * Float64(-2.0 / t));
	elseif (t <= 6.5e-91)
		tmp = Float64(Float64(Float64(x_m + x_m) / z_m) / y);
	else
		tmp = Float64(Float64(x_m / Float64(t * z_m)) * -2.0);
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if (t <= -8.6e+54)
		tmp = (x_m / z_m) * (-2.0 / t);
	elseif (t <= 6.5e-91)
		tmp = ((x_m + x_m) / z_m) / y;
	else
		tmp = (x_m / (t * z_m)) * -2.0;
	end
	tmp_2 = z_s * (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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[LessEqual[t, -8.6e+54], N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(-2.0 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.5e-91], N[(N[(N[(x$95$m + x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision] / y), $MachinePrecision], N[(N[(x$95$m / N[(t * z$95$m), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -8.59999999999999952e54

   1. Initial program 87.8%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot 2}}{y \cdot z - t \cdot z} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot 2}{y \cdot z - t \cdot z}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot z} - t \cdot z} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot 2}{y \cdot z - \color{blue}{t \cdot z}} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot z - t \cdot z}} \]

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

         \[\leadsto \frac{x \cdot 2}{\color{blue}{z \cdot \left(y - t\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{2}{y - t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{2}{y - t}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{z}} \cdot \frac{2}{y - t} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{2}{y - t}} \]

      11. lower--.f64 90.8

          \[\leadsto \frac{x}{z} \cdot \frac{2}{\color{blue}{y - t}} \]

   3. Applied rewrites 90.8%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{2}{y - t}} \]

   4. Taylor expanded in y around 0

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{-2}{t}} \]
   5. Step-by-step derivation

      1. lower-/.f64 79.3

         \[\leadsto \frac{x}{z} \cdot \frac{-2}{\color{blue}{t}} \]

   6. Applied rewrites 79.3%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{-2}{t}} \]

   if -8.59999999999999952e54 < t < 6.5000000000000001e-91

   1. Initial program 92.2%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]

   2. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot x}{\color{blue}{y \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x \cdot 2}{\color{blue}{y} \cdot z} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot z}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2 \cdot x}{\color{blue}{y} \cdot z} \]

      5. count-2-rev N/A

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

      6. lower-+.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 73.1

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

   4. Applied rewrites 73.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. count-2-rev N/A

         \[\leadsto \frac{\frac{2 \cdot x}{z}}{y} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \frac{x}{z}}{y} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{2 \cdot \frac{x}{z}}{\color{blue}{y}} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{\frac{2 \cdot x}{z}}{y} \]

      9. count-2-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 74.2

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

   6. Applied rewrites 74.2%

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

   if 6.5000000000000001e-91 < t

   1. Initial program 88.3%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{x}{t \cdot z} \cdot \color{blue}{-2} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{t \cdot z} \cdot \color{blue}{-2} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{x}{t \cdot z} \cdot -2 \]

      4. lift-*.f64 69.9

         \[\leadsto \frac{x}{t \cdot z} \cdot -2 \]

   4. Applied rewrites 69.9%

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

Alternative 7: 73.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{x\_m}{t \cdot z\_m} \cdot -2\\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -4.9 \cdot 10^{+72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-91}:\\ \;\;\;\;\frac{\frac{x\_m + x\_m}{z\_m}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (let* ((t_1 (* (/ x_m (* t z_m)) -2.0)))
   (*
    z_s
    (*
     x_s
     (if (<= t -4.9e+72)
       t_1
       (if (<= t 6.5e-91) (/ (/ (+ x_m x_m) z_m) y) t_1))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double t_1 = (x_m / (t * z_m)) * -2.0;
	double tmp;
	if (t <= -4.9e+72) {
		tmp = t_1;
	} else if (t <= 6.5e-91) {
		tmp = ((x_m + x_m) / z_m) / y;
	} else {
		tmp = t_1;
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m =     private
x\_s =     private
z\_m =     private
z\_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(z_s, x_s, x_m, y, z_m, t)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x_m / (t * z_m)) * (-2.0d0)
    if (t <= (-4.9d+72)) then
        tmp = t_1
    else if (t <= 6.5d-91) then
        tmp = ((x_m + x_m) / z_m) / y
    else
        tmp = t_1
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double t_1 = (x_m / (t * z_m)) * -2.0;
	double tmp;
	if (t <= -4.9e+72) {
		tmp = t_1;
	} else if (t <= 6.5e-91) {
		tmp = ((x_m + x_m) / z_m) / y;
	} else {
		tmp = t_1;
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	t_1 = (x_m / (t * z_m)) * -2.0
	tmp = 0
	if t <= -4.9e+72:
		tmp = t_1
	elif t <= 6.5e-91:
		tmp = ((x_m + x_m) / z_m) / y
	else:
		tmp = t_1
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	t_1 = Float64(Float64(x_m / Float64(t * z_m)) * -2.0)
	tmp = 0.0
	if (t <= -4.9e+72)
		tmp = t_1;
	elseif (t <= 6.5e-91)
		tmp = Float64(Float64(Float64(x_m + x_m) / z_m) / y);
	else
		tmp = t_1;
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	t_1 = (x_m / (t * z_m)) * -2.0;
	tmp = 0.0;
	if (t <= -4.9e+72)
		tmp = t_1;
	elseif (t <= 6.5e-91)
		tmp = ((x_m + x_m) / z_m) / y;
	else
		tmp = t_1;
	end
	tmp_2 = z_s * (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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(x$95$m / N[(t * z$95$m), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]}, N[(z$95$s * N[(x$95$s * If[LessEqual[t, -4.9e+72], t$95$1, If[LessEqual[t, 6.5e-91], N[(N[(N[(x$95$m + x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision] / y), $MachinePrecision], t$95$1]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.90000000000000006e72 or 6.5000000000000001e-91 < t

   1. Initial program 87.9%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{x}{t \cdot z} \cdot \color{blue}{-2} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{t \cdot z} \cdot \color{blue}{-2} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{x}{t \cdot z} \cdot -2 \]

      4. lift-*.f64 74.4

         \[\leadsto \frac{x}{t \cdot z} \cdot -2 \]

   4. Applied rewrites 74.4%

      \[\leadsto \color{blue}{\frac{x}{t \cdot z} \cdot -2} \]

   if -4.90000000000000006e72 < t < 6.5000000000000001e-91

   1. Initial program 92.2%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]

   2. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot x}{\color{blue}{y \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x \cdot 2}{\color{blue}{y} \cdot z} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot z}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2 \cdot x}{\color{blue}{y} \cdot z} \]

      5. count-2-rev N/A

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

      6. lower-+.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 72.0

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

   4. Applied rewrites 72.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. count-2-rev N/A

         \[\leadsto \frac{\frac{2 \cdot x}{z}}{y} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \frac{x}{z}}{y} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{2 \cdot \frac{x}{z}}{\color{blue}{y}} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{\frac{2 \cdot x}{z}}{y} \]

      9. count-2-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 73.3

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

   6. Applied rewrites 73.3%

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

Alternative 8: 73.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{x\_m}{t \cdot z\_m} \cdot -2\\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -4.9 \cdot 10^{+72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{-91}:\\ \;\;\;\;\frac{x\_m + x\_m}{z\_m \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (let* ((t_1 (* (/ x_m (* t z_m)) -2.0)))
   (*
    z_s
    (*
     x_s
     (if (<= t -4.9e+72)
       t_1
       (if (<= t 3.9e-91) (/ (+ x_m x_m) (* z_m y)) t_1))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double t_1 = (x_m / (t * z_m)) * -2.0;
	double tmp;
	if (t <= -4.9e+72) {
		tmp = t_1;
	} else if (t <= 3.9e-91) {
		tmp = (x_m + x_m) / (z_m * y);
	} else {
		tmp = t_1;
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m =     private
x\_s =     private
z\_m =     private
z\_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(z_s, x_s, x_m, y, z_m, t)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x_m / (t * z_m)) * (-2.0d0)
    if (t <= (-4.9d+72)) then
        tmp = t_1
    else if (t <= 3.9d-91) then
        tmp = (x_m + x_m) / (z_m * y)
    else
        tmp = t_1
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double t_1 = (x_m / (t * z_m)) * -2.0;
	double tmp;
	if (t <= -4.9e+72) {
		tmp = t_1;
	} else if (t <= 3.9e-91) {
		tmp = (x_m + x_m) / (z_m * y);
	} else {
		tmp = t_1;
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	t_1 = (x_m / (t * z_m)) * -2.0
	tmp = 0
	if t <= -4.9e+72:
		tmp = t_1
	elif t <= 3.9e-91:
		tmp = (x_m + x_m) / (z_m * y)
	else:
		tmp = t_1
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	t_1 = Float64(Float64(x_m / Float64(t * z_m)) * -2.0)
	tmp = 0.0
	if (t <= -4.9e+72)
		tmp = t_1;
	elseif (t <= 3.9e-91)
		tmp = Float64(Float64(x_m + x_m) / Float64(z_m * y));
	else
		tmp = t_1;
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	t_1 = (x_m / (t * z_m)) * -2.0;
	tmp = 0.0;
	if (t <= -4.9e+72)
		tmp = t_1;
	elseif (t <= 3.9e-91)
		tmp = (x_m + x_m) / (z_m * y);
	else
		tmp = t_1;
	end
	tmp_2 = z_s * (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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(x$95$m / N[(t * z$95$m), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]}, N[(z$95$s * N[(x$95$s * If[LessEqual[t, -4.9e+72], t$95$1, If[LessEqual[t, 3.9e-91], N[(N[(x$95$m + x$95$m), $MachinePrecision] / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.90000000000000006e72 or 3.89999999999999994e-91 < t

   1. Initial program 87.9%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{x}{t \cdot z} \cdot \color{blue}{-2} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{t \cdot z} \cdot \color{blue}{-2} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{x}{t \cdot z} \cdot -2 \]

      4. lift-*.f64 74.4

         \[\leadsto \frac{x}{t \cdot z} \cdot -2 \]

   4. Applied rewrites 74.4%

      \[\leadsto \color{blue}{\frac{x}{t \cdot z} \cdot -2} \]

   if -4.90000000000000006e72 < t < 3.89999999999999994e-91

   1. Initial program 92.2%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]

   2. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot x}{\color{blue}{y \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x \cdot 2}{\color{blue}{y} \cdot z} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot z}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2 \cdot x}{\color{blue}{y} \cdot z} \]

      5. count-2-rev N/A

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

      6. lower-+.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 72.0

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

   4. Applied rewrites 72.0%

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

Alternative 9: 52.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

x\_m =     private
x\_s =     private
z\_m =     private
z\_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(z_s, x_s, x_m, y, z_m, t)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    code = z_s * (x_s * ((x_m + x_m) / (z_m * y)))
end function

Java

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

Python

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

Julia

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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp = code(z_s, x_s, x_m, y, z_m, t)
	tmp = z_s * (x_s * ((x_m + x_m) / (z_m * y)));
end

Wolfram

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

Derivation

1. Initial program 90.0%

   \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]

2. Taylor expanded in y around inf

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

   1. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot x}{\color{blue}{y \cdot z}} \]

   2. *-commutative N/A

      \[\leadsto \frac{x \cdot 2}{\color{blue}{y} \cdot z} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot z}} \]

   4. *-commutative N/A

      \[\leadsto \frac{2 \cdot x}{\color{blue}{y} \cdot z} \]

   5. count-2-rev N/A

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

   6. lower-+.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 52.1

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

4. Applied rewrites 52.1%

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

Reproduce

herbie shell --seed 2025114 
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"
  :precision binary64
  (/ (* x 2.0) (- (* y z) (* t z))))