Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.5% → 98.2%

Time: 3.5s

Alternatives: 9

Speedup: 0.9×

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: 89.5% 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: 98.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{\frac{x + x}{z\_m}}{y - t}\\ t_2 := y \cdot z\_m - t \cdot z\_m\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+226}:\\ \;\;\;\;\frac{x + x}{\left(y - t\right) \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (let* ((t_1 (/ (/ (+ x x) z_m) (- y t))) (t_2 (- (* y z_m) (* t z_m))))
   (*
    z_s
    (if (<= t_2 (- INFINITY))
      t_1
      (if (<= t_2 5e+226) (/ (+ x x) (* (- y t) z_m)) t_1)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = ((x + x) / z_m) / (y - t);
	double t_2 = (y * z_m) - (t * z_m);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 5e+226) {
		tmp = (x + x) / ((y - t) * z_m);
	} else {
		tmp = t_1;
	}
	return z_s * tmp;
}

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = ((x + x) / z_m) / (y - t);
	double t_2 = (y * z_m) - (t * z_m);
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= 5e+226) {
		tmp = (x + x) / ((y - t) * z_m);
	} else {
		tmp = t_1;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	t_1 = ((x + x) / z_m) / (y - t)
	t_2 = (y * z_m) - (t * z_m)
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= 5e+226:
		tmp = (x + x) / ((y - t) * z_m)
	else:
		tmp = t_1
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	t_1 = Float64(Float64(Float64(x + x) / z_m) / Float64(y - t))
	t_2 = Float64(Float64(y * z_m) - Float64(t * z_m))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 5e+226)
		tmp = Float64(Float64(x + x) / Float64(Float64(y - t) * z_m));
	else
		tmp = t_1;
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	t_1 = ((x + x) / z_m) / (y - t);
	t_2 = (y * z_m) - (t * z_m);
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= 5e+226)
		tmp = (x + x) / ((y - t) * z_m);
	else
		tmp = t_1;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

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_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(N[(x + x), $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 * If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 5e+226], N[(N[(x + x), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < -inf.0 or 5.0000000000000005e226 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 69.0%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 -inf.0 < (-.f64 (*.f64 y z) (*.f64 t z)) < 5.0000000000000005e226

   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 97.7

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

   3. Applied rewrites 97.7%

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

Alternative 2: 94.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-55}:\\ \;\;\;\;\frac{x + x}{\left(y - t\right) \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= x 6e-55) (/ (+ x x) (* (- y t) z_m)) (* (/ x (- y t)) (/ 2.0 z_m)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (x <= 6e-55) {
		tmp = (x + x) / ((y - t) * z_m);
	} else {
		tmp = (x / (y - t)) * (2.0 / z_m);
	}
	return z_s * tmp;
}

Fortran

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, y, z_m, t)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= 6d-55) then
        tmp = (x + x) / ((y - t) * z_m)
    else
        tmp = (x / (y - t)) * (2.0d0 / z_m)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (x <= 6e-55) {
		tmp = (x + x) / ((y - t) * z_m);
	} else {
		tmp = (x / (y - t)) * (2.0 / z_m);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if x <= 6e-55:
		tmp = (x + x) / ((y - t) * z_m)
	else:
		tmp = (x / (y - t)) * (2.0 / z_m)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (x <= 6e-55)
		tmp = Float64(Float64(x + x) / Float64(Float64(y - t) * z_m));
	else
		tmp = Float64(Float64(x / Float64(y - t)) * Float64(2.0 / z_m));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (x <= 6e-55)
		tmp = (x + x) / ((y - t) * z_m);
	else
		tmp = (x / (y - t)) * (2.0 / z_m);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

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_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[x, 6e-55], N[(N[(x + x), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(2.0 / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 6.00000000000000033e-55

   1. Initial program 90.8%

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

      1. lift-*.f64 N/A

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

      2. *-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 90.8

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

      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 93.1

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

   3. Applied rewrites 93.1%

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

   if 6.00000000000000033e-55 < x

   1. Initial program 86.4%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 96.3

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

   3. Applied rewrites 96.3%

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

Alternative 3: 94.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-55}:\\ \;\;\;\;\frac{x + x}{\left(y - t\right) \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + x}{y - t}}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= x 6e-55) (/ (+ x x) (* (- y t) z_m)) (/ (/ (+ x x) (- y t)) z_m))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (x <= 6e-55) {
		tmp = (x + x) / ((y - t) * z_m);
	} else {
		tmp = ((x + x) / (y - t)) / z_m;
	}
	return z_s * tmp;
}

Fortran

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, y, z_m, t)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= 6d-55) then
        tmp = (x + x) / ((y - t) * z_m)
    else
        tmp = ((x + x) / (y - t)) / z_m
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (x <= 6e-55) {
		tmp = (x + x) / ((y - t) * z_m);
	} else {
		tmp = ((x + x) / (y - t)) / z_m;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if x <= 6e-55:
		tmp = (x + x) / ((y - t) * z_m)
	else:
		tmp = ((x + x) / (y - t)) / z_m
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (x <= 6e-55)
		tmp = Float64(Float64(x + x) / Float64(Float64(y - t) * z_m));
	else
		tmp = Float64(Float64(Float64(x + x) / Float64(y - t)) / z_m);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (x <= 6e-55)
		tmp = (x + x) / ((y - t) * z_m);
	else
		tmp = ((x + x) / (y - t)) / z_m;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

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_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[x, 6e-55], N[(N[(x + x), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x + x), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 6.00000000000000033e-55

   1. Initial program 90.8%

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

      1. lift-*.f64 N/A

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

      2. *-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 90.8

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

      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 93.1

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

   3. Applied rewrites 93.1%

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

   if 6.00000000000000033e-55 < x

   1. Initial program 86.4%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 96.3

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

   3. Applied rewrites 96.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. count-2-rev N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift--.f64 96.4

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

   5. Applied rewrites 96.4%

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

Alternative 4: 91.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (((y * z_m) - (t * z_m)) <= -((double) INFINITY)) {
		tmp = ((x + x) / z_m) / y;
	} else {
		tmp = (x + x) / ((y - t) * z_m);
	}
	return z_s * tmp;
}

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (((y * z_m) - (t * z_m)) <= -Double.POSITIVE_INFINITY) {
		tmp = ((x + x) / z_m) / y;
	} else {
		tmp = (x + x) / ((y - t) * z_m);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if ((y * z_m) - (t * z_m)) <= -math.inf:
		tmp = ((x + x) / z_m) / y
	else:
		tmp = (x + x) / ((y - t) * z_m)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (Float64(Float64(y * z_m) - Float64(t * z_m)) <= Float64(-Inf))
		tmp = Float64(Float64(Float64(x + x) / z_m) / y);
	else
		tmp = Float64(Float64(x + x) / Float64(Float64(y - t) * z_m));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (((y * z_m) - (t * z_m)) <= -Inf)
		tmp = ((x + x) / z_m) / y;
	else
		tmp = (x + x) / ((y - t) * z_m);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

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_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[N[(N[(y * z$95$m), $MachinePrecision] - N[(t * z$95$m), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(N[(x + x), $MachinePrecision] / z$95$m), $MachinePrecision] / y), $MachinePrecision], N[(N[(x + x), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < -inf.0

   1. Initial program 69.3%

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

      1. lift-*.f64 N/A

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

      2. 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 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in y around inf

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

   6. Applied rewrites 68.8%

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

   if -inf.0 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 91.6%

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

      1. lift-*.f64 N/A

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

      2. *-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 91.6

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

      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 94.1

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

   3. Applied rewrites 94.1%

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

Alternative 5: 73.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{\frac{x}{t} \cdot -2}{z\_m}\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.68:\\ \;\;\;\;\frac{\frac{x + x}{y}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (let* ((t_1 (/ (* (/ x t) -2.0) z_m)))
   (*
    z_s
    (if (<= t -1.15e-37) t_1 (if (<= t 1.68) (/ (/ (+ x x) y) z_m) t_1)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = ((x / t) * -2.0) / z_m;
	double tmp;
	if (t <= -1.15e-37) {
		tmp = t_1;
	} else if (t <= 1.68) {
		tmp = ((x + x) / y) / z_m;
	} else {
		tmp = t_1;
	}
	return z_s * tmp;
}

Fortran

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, y, z_m, t)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    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 / t) * (-2.0d0)) / z_m
    if (t <= (-1.15d-37)) then
        tmp = t_1
    else if (t <= 1.68d0) then
        tmp = ((x + x) / y) / z_m
    else
        tmp = t_1
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = ((x / t) * -2.0) / z_m;
	double tmp;
	if (t <= -1.15e-37) {
		tmp = t_1;
	} else if (t <= 1.68) {
		tmp = ((x + x) / y) / z_m;
	} else {
		tmp = t_1;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	t_1 = ((x / t) * -2.0) / z_m
	tmp = 0
	if t <= -1.15e-37:
		tmp = t_1
	elif t <= 1.68:
		tmp = ((x + x) / y) / z_m
	else:
		tmp = t_1
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	t_1 = Float64(Float64(Float64(x / t) * -2.0) / z_m)
	tmp = 0.0
	if (t <= -1.15e-37)
		tmp = t_1;
	elseif (t <= 1.68)
		tmp = Float64(Float64(Float64(x + x) / y) / z_m);
	else
		tmp = t_1;
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	t_1 = ((x / t) * -2.0) / z_m;
	tmp = 0.0;
	if (t <= -1.15e-37)
		tmp = t_1;
	elseif (t <= 1.68)
		tmp = ((x + x) / y) / z_m;
	else
		tmp = t_1;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

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_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(N[(x / t), $MachinePrecision] * -2.0), $MachinePrecision] / z$95$m), $MachinePrecision]}, N[(z$95$s * If[LessEqual[t, -1.15e-37], t$95$1, If[LessEqual[t, 1.68], N[(N[(N[(x + x), $MachinePrecision] / y), $MachinePrecision] / z$95$m), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.15e-37 or 1.67999999999999994 < t

   1. Initial program 88.1%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 91.4

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

   3. Applied rewrites 91.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. count-2-rev N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift--.f64 91.5

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

   5. Applied rewrites 91.5%

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

   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 74.0

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

   8. Applied rewrites 74.0%

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

   if -1.15e-37 < t < 1.67999999999999994

   1. Initial program 91.2%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 92.4

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

   3. Applied rewrites 92.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. count-2-rev N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift--.f64 92.5

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

   5. Applied rewrites 92.5%

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

   6. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 73.4

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

   8. Applied rewrites 73.4%

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

Alternative 6: 73.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-37}:\\ \;\;\;\;\frac{x}{t \cdot z\_m} \cdot -2\\ \mathbf{elif}\;t \leq 1.68:\\ \;\;\;\;\frac{\frac{x + x}{y}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z\_m}}{t} \cdot -2\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= t -1.15e-37)
    (* (/ x (* t z_m)) -2.0)
    (if (<= t 1.68) (/ (/ (+ x x) y) z_m) (* (/ (/ x z_m) t) -2.0)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (t <= -1.15e-37) {
		tmp = (x / (t * z_m)) * -2.0;
	} else if (t <= 1.68) {
		tmp = ((x + x) / y) / z_m;
	} else {
		tmp = ((x / z_m) / t) * -2.0;
	}
	return z_s * tmp;
}

Fortran

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, y, z_m, t)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.15d-37)) then
        tmp = (x / (t * z_m)) * (-2.0d0)
    else if (t <= 1.68d0) then
        tmp = ((x + x) / y) / z_m
    else
        tmp = ((x / z_m) / t) * (-2.0d0)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (t <= -1.15e-37) {
		tmp = (x / (t * z_m)) * -2.0;
	} else if (t <= 1.68) {
		tmp = ((x + x) / y) / z_m;
	} else {
		tmp = ((x / z_m) / t) * -2.0;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if t <= -1.15e-37:
		tmp = (x / (t * z_m)) * -2.0
	elif t <= 1.68:
		tmp = ((x + x) / y) / z_m
	else:
		tmp = ((x / z_m) / t) * -2.0
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (t <= -1.15e-37)
		tmp = Float64(Float64(x / Float64(t * z_m)) * -2.0);
	elseif (t <= 1.68)
		tmp = Float64(Float64(Float64(x + x) / y) / z_m);
	else
		tmp = Float64(Float64(Float64(x / z_m) / t) * -2.0);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (t <= -1.15e-37)
		tmp = (x / (t * z_m)) * -2.0;
	elseif (t <= 1.68)
		tmp = ((x + x) / y) / z_m;
	else
		tmp = ((x / z_m) / t) * -2.0;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

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_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[t, -1.15e-37], N[(N[(x / N[(t * z$95$m), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t, 1.68], N[(N[(N[(x + x), $MachinePrecision] / y), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(x / z$95$m), $MachinePrecision] / t), $MachinePrecision] * -2.0), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -1.15e-37

   1. Initial program 87.4%

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

   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 71.3

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

   4. Applied rewrites 71.3%

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

   if -1.15e-37 < t < 1.67999999999999994

   1. Initial program 91.2%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 92.4

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

   3. Applied rewrites 92.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. count-2-rev N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift--.f64 92.5

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

   5. Applied rewrites 92.5%

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

   6. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 73.4

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

   8. Applied rewrites 73.4%

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

   if 1.67999999999999994 < t

   1. Initial program 88.8%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 91.1

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

   3. Applied rewrites 91.1%

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

   4. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. frac-times N/A

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

      3. count-2-rev N/A

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

      4. associate-/l/ N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 75.7

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

   6. Applied rewrites 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 74.5

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

   8. Applied rewrites 74.5%

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

Alternative 7: 73.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{x}{t \cdot z\_m} \cdot -2\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.68:\\ \;\;\;\;\frac{\frac{x + x}{y}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (let* ((t_1 (* (/ x (* t z_m)) -2.0)))
   (*
    z_s
    (if (<= t -1.15e-37) t_1 (if (<= t 1.68) (/ (/ (+ x x) y) z_m) t_1)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = (x / (t * z_m)) * -2.0;
	double tmp;
	if (t <= -1.15e-37) {
		tmp = t_1;
	} else if (t <= 1.68) {
		tmp = ((x + x) / y) / z_m;
	} else {
		tmp = t_1;
	}
	return z_s * tmp;
}

Fortran

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, y, z_m, t)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    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 / (t * z_m)) * (-2.0d0)
    if (t <= (-1.15d-37)) then
        tmp = t_1
    else if (t <= 1.68d0) then
        tmp = ((x + x) / y) / z_m
    else
        tmp = t_1
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = (x / (t * z_m)) * -2.0;
	double tmp;
	if (t <= -1.15e-37) {
		tmp = t_1;
	} else if (t <= 1.68) {
		tmp = ((x + x) / y) / z_m;
	} else {
		tmp = t_1;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	t_1 = (x / (t * z_m)) * -2.0
	tmp = 0
	if t <= -1.15e-37:
		tmp = t_1
	elif t <= 1.68:
		tmp = ((x + x) / y) / z_m
	else:
		tmp = t_1
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	t_1 = Float64(Float64(x / Float64(t * z_m)) * -2.0)
	tmp = 0.0
	if (t <= -1.15e-37)
		tmp = t_1;
	elseif (t <= 1.68)
		tmp = Float64(Float64(Float64(x + x) / y) / z_m);
	else
		tmp = t_1;
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	t_1 = (x / (t * z_m)) * -2.0;
	tmp = 0.0;
	if (t <= -1.15e-37)
		tmp = t_1;
	elseif (t <= 1.68)
		tmp = ((x + x) / y) / z_m;
	else
		tmp = t_1;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

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_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(x / N[(t * z$95$m), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]}, N[(z$95$s * If[LessEqual[t, -1.15e-37], t$95$1, If[LessEqual[t, 1.68], N[(N[(N[(x + x), $MachinePrecision] / y), $MachinePrecision] / z$95$m), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.15e-37 or 1.67999999999999994 < t

   1. Initial program 88.1%

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

   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 73.4

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

   4. Applied rewrites 73.4%

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

   if -1.15e-37 < t < 1.67999999999999994

   1. Initial program 91.2%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 92.4

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

   3. Applied rewrites 92.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. count-2-rev N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift--.f64 92.5

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

   5. Applied rewrites 92.5%

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

   6. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 73.4

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

   8. Applied rewrites 73.4%

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

Alternative 8: 73.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{x}{t \cdot z\_m} \cdot -2\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.35:\\ \;\;\;\;\frac{x + x}{z\_m \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (let* ((t_1 (* (/ x (* t z_m)) -2.0)))
   (*
    z_s
    (if (<= t -1.15e-37) t_1 (if (<= t 1.35) (/ (+ x x) (* z_m y)) t_1)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = (x / (t * z_m)) * -2.0;
	double tmp;
	if (t <= -1.15e-37) {
		tmp = t_1;
	} else if (t <= 1.35) {
		tmp = (x + x) / (z_m * y);
	} else {
		tmp = t_1;
	}
	return z_s * tmp;
}

Fortran

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, y, z_m, t)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    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 / (t * z_m)) * (-2.0d0)
    if (t <= (-1.15d-37)) then
        tmp = t_1
    else if (t <= 1.35d0) then
        tmp = (x + x) / (z_m * y)
    else
        tmp = t_1
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double t_1 = (x / (t * z_m)) * -2.0;
	double tmp;
	if (t <= -1.15e-37) {
		tmp = t_1;
	} else if (t <= 1.35) {
		tmp = (x + x) / (z_m * y);
	} else {
		tmp = t_1;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	t_1 = (x / (t * z_m)) * -2.0
	tmp = 0
	if t <= -1.15e-37:
		tmp = t_1
	elif t <= 1.35:
		tmp = (x + x) / (z_m * y)
	else:
		tmp = t_1
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	t_1 = Float64(Float64(x / Float64(t * z_m)) * -2.0)
	tmp = 0.0
	if (t <= -1.15e-37)
		tmp = t_1;
	elseif (t <= 1.35)
		tmp = Float64(Float64(x + x) / Float64(z_m * y));
	else
		tmp = t_1;
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	t_1 = (x / (t * z_m)) * -2.0;
	tmp = 0.0;
	if (t <= -1.15e-37)
		tmp = t_1;
	elseif (t <= 1.35)
		tmp = (x + x) / (z_m * y);
	else
		tmp = t_1;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

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_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(x / N[(t * z$95$m), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]}, N[(z$95$s * If[LessEqual[t, -1.15e-37], t$95$1, If[LessEqual[t, 1.35], N[(N[(x + x), $MachinePrecision] / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.15e-37 or 1.3500000000000001 < t

   1. Initial program 88.1%

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

   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 73.4

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

   4. Applied rewrites 73.4%

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

   if -1.15e-37 < t < 1.3500000000000001

   1. Initial program 91.2%

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

   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 73.4

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

   4. Applied rewrites 73.4%

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

Alternative 9: 53.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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, y, z_m, t)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    code = z_s * ((x + x) / (z_m * y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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_, y_, z$95$m_, t_] := N[(z$95$s * N[(N[(x + x), $MachinePrecision] / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.5%

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

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 53.3

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

4. Applied rewrites 53.3%

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

Reproduce

herbie shell --seed 2025119 
(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))))