Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Percentage Accurate: 96.3% → 96.9%

Time: 4.5s

Alternatives: 8

Speedup: 1.3×

• 24.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return ((x * y) - (z * y)) * t;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x * y) - (z * y)) * t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return ((x * y) - (z * y)) * t;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x * y) - (z * y)) * t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.9% accurate, 1.3× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (t_s y_s x y_m z t_m)
 :precision binary64
 (* t_s (* y_s (* (* (- x z) y_m) t_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
assert(x < y_m && y_m < z && z < t_m);
double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	return t_s * (y_s * (((x - z) * y_m) * t_m));
}

Fortran

y\_m =     private
y\_s =     private
t\_m =     private
t\_s =     private
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
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(t_s, y_s, x, y_m, z, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    code = t_s * (y_s * (((x - z) * y_m) * t_m))
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
assert x < y_m && y_m < z && z < t_m;
public static double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	return t_s * (y_s * (((x - z) * y_m) * t_m));
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(t_s, y_s, x, y_m, z, t_m):
	return t_s * (y_s * (((x - z) * y_m) * t_m))

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
t\_m = abs(t)
t\_s = copysign(1.0, t)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(t_s, y_s, x, y_m, z, t_m)
	return Float64(t_s * Float64(y_s * Float64(Float64(Float64(x - z) * y_m) * t_m)))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp = code(t_s, y_s, x, y_m, z, t_m)
	tmp = t_s * (y_s * (((x - z) * y_m) * t_m));
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[t$95$s_, y$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(t$95$s * N[(y$95$s * N[(N[(N[(x - z), $MachinePrecision] * y$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

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

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

   5. sub-flip N/A

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

   6. mul-1-neg N/A

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

   7. add-flip N/A

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

   8. mul-1-neg N/A

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

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

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

   10. *-commutative N/A

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

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

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

   12. mul-1-neg N/A

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

   13. lower-*.f64 N/A

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

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

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

   15. mul-1-neg N/A

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

   16. sub-negate N/A

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

   17. lower--.f64 N/A

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

   18. mul-1-neg N/A

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

   19. lower-neg.f64 96.9

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

3. Applied rewrites 96.9%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. distribute-rgt-neg-out N/A

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

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

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

   6. sub-negate N/A

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

   7. lower-*.f64 N/A

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

   8. lower--.f64 96.9

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

5. Applied rewrites 96.9%

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

Alternative 2: 88.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ \begin{array}{l} t_2 := \left(y\_m \cdot x\right) \cdot t\_m\\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+99}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+92}:\\ \;\;\;\;\left(\left(x - z\right) \cdot t\_m\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (t_s y_s x y_m z t_m)
 :precision binary64
 (let* ((t_2 (* (* y_m x) t_m)))
   (*
    t_s
    (*
     y_s
     (if (<= x -2.6e+99)
       t_2
       (if (<= x 4.2e+92) (* (* (- x z) t_m) y_m) t_2))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
assert(x < y_m && y_m < z && z < t_m);
double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double t_2 = (y_m * x) * t_m;
	double tmp;
	if (x <= -2.6e+99) {
		tmp = t_2;
	} else if (x <= 4.2e+92) {
		tmp = ((x - z) * t_m) * y_m;
	} else {
		tmp = t_2;
	}
	return t_s * (y_s * tmp);
}

Fortran

y\_m =     private
y\_s =     private
t\_m =     private
t\_s =     private
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
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(t_s, y_s, x, y_m, z, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (y_m * x) * t_m
    if (x <= (-2.6d+99)) then
        tmp = t_2
    else if (x <= 4.2d+92) then
        tmp = ((x - z) * t_m) * y_m
    else
        tmp = t_2
    end if
    code = t_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
assert x < y_m && y_m < z && z < t_m;
public static double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double t_2 = (y_m * x) * t_m;
	double tmp;
	if (x <= -2.6e+99) {
		tmp = t_2;
	} else if (x <= 4.2e+92) {
		tmp = ((x - z) * t_m) * y_m;
	} else {
		tmp = t_2;
	}
	return t_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(t_s, y_s, x, y_m, z, t_m):
	t_2 = (y_m * x) * t_m
	tmp = 0
	if x <= -2.6e+99:
		tmp = t_2
	elif x <= 4.2e+92:
		tmp = ((x - z) * t_m) * y_m
	else:
		tmp = t_2
	return t_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
t\_m = abs(t)
t\_s = copysign(1.0, t)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(t_s, y_s, x, y_m, z, t_m)
	t_2 = Float64(Float64(y_m * x) * t_m)
	tmp = 0.0
	if (x <= -2.6e+99)
		tmp = t_2;
	elseif (x <= 4.2e+92)
		tmp = Float64(Float64(Float64(x - z) * t_m) * y_m);
	else
		tmp = t_2;
	end
	return Float64(t_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(t_s, y_s, x, y_m, z, t_m)
	t_2 = (y_m * x) * t_m;
	tmp = 0.0;
	if (x <= -2.6e+99)
		tmp = t_2;
	elseif (x <= 4.2e+92)
		tmp = ((x - z) * t_m) * y_m;
	else
		tmp = t_2;
	end
	tmp_2 = t_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[t$95$s_, y$95$s_, x_, y$95$m_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(y$95$m * x), $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * N[(y$95$s * If[LessEqual[x, -2.6e+99], t$95$2, If[LessEqual[x, 4.2e+92], N[(N[(N[(x - z), $MachinePrecision] * t$95$m), $MachinePrecision] * y$95$m), $MachinePrecision], t$95$2]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.6e99 or 4.19999999999999972e92 < x

   1. Initial program 98.2%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 84.5

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

   4. Applied rewrites 84.5%

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

   if -2.6e99 < x < 4.19999999999999972e92

   1. Initial program 95.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. sub-flip N/A

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

      8. mul-1-neg N/A

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

      9. add-flip N/A

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

      10. mul-1-neg N/A

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

      11. distribute-rgt-neg-out N/A

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

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

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

      13. mul-1-neg N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

   3. Applied rewrites 91.3%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift--.f64 N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. sub-negate N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 91.3

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

   5. Applied rewrites 91.3%

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

Alternative 3: 76.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ \begin{array}{l} t_2 := \left(y\_m \cdot x\right) \cdot t\_m\\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{+55}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(-z\right) \cdot y\_m\right) \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (t_s y_s x y_m z t_m)
 :precision binary64
 (let* ((t_2 (* (* y_m x) t_m)))
   (*
    t_s
    (*
     y_s
     (if (<= x -9.6e+55) t_2 (if (<= x 3.1e-69) (* (* (- z) y_m) t_m) t_2))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
assert(x < y_m && y_m < z && z < t_m);
double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double t_2 = (y_m * x) * t_m;
	double tmp;
	if (x <= -9.6e+55) {
		tmp = t_2;
	} else if (x <= 3.1e-69) {
		tmp = (-z * y_m) * t_m;
	} else {
		tmp = t_2;
	}
	return t_s * (y_s * tmp);
}

Fortran

y\_m =     private
y\_s =     private
t\_m =     private
t\_s =     private
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
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(t_s, y_s, x, y_m, z, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (y_m * x) * t_m
    if (x <= (-9.6d+55)) then
        tmp = t_2
    else if (x <= 3.1d-69) then
        tmp = (-z * y_m) * t_m
    else
        tmp = t_2
    end if
    code = t_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
assert x < y_m && y_m < z && z < t_m;
public static double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double t_2 = (y_m * x) * t_m;
	double tmp;
	if (x <= -9.6e+55) {
		tmp = t_2;
	} else if (x <= 3.1e-69) {
		tmp = (-z * y_m) * t_m;
	} else {
		tmp = t_2;
	}
	return t_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(t_s, y_s, x, y_m, z, t_m):
	t_2 = (y_m * x) * t_m
	tmp = 0
	if x <= -9.6e+55:
		tmp = t_2
	elif x <= 3.1e-69:
		tmp = (-z * y_m) * t_m
	else:
		tmp = t_2
	return t_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
t\_m = abs(t)
t\_s = copysign(1.0, t)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(t_s, y_s, x, y_m, z, t_m)
	t_2 = Float64(Float64(y_m * x) * t_m)
	tmp = 0.0
	if (x <= -9.6e+55)
		tmp = t_2;
	elseif (x <= 3.1e-69)
		tmp = Float64(Float64(Float64(-z) * y_m) * t_m);
	else
		tmp = t_2;
	end
	return Float64(t_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(t_s, y_s, x, y_m, z, t_m)
	t_2 = (y_m * x) * t_m;
	tmp = 0.0;
	if (x <= -9.6e+55)
		tmp = t_2;
	elseif (x <= 3.1e-69)
		tmp = (-z * y_m) * t_m;
	else
		tmp = t_2;
	end
	tmp_2 = t_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[t$95$s_, y$95$s_, x_, y$95$m_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(y$95$m * x), $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * N[(y$95$s * If[LessEqual[x, -9.6e+55], t$95$2, If[LessEqual[x, 3.1e-69], N[(N[((-z) * y$95$m), $MachinePrecision] * t$95$m), $MachinePrecision], t$95$2]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.5999999999999997e55 or 3.0999999999999999e-69 < x

   1. Initial program 98.4%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 76.7

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

   4. Applied rewrites 76.7%

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

   if -9.5999999999999997e55 < x < 3.0999999999999999e-69

   1. Initial program 94.1%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 77.0

         \[\leadsto \left(\left(-z\right) \cdot y\right) \cdot t \]

   4. Applied rewrites 77.0%

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

Alternative 4: 76.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ \begin{array}{l} t_2 := \left(y\_m \cdot x\right) \cdot t\_m\\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+55}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-70}:\\ \;\;\;\;\left(-z\right) \cdot \left(t\_m \cdot y\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (t_s y_s x y_m z t_m)
 :precision binary64
 (let* ((t_2 (* (* y_m x) t_m)))
   (*
    t_s
    (*
     y_s
     (if (<= x -2.8e+55) t_2 (if (<= x 3.5e-70) (* (- z) (* t_m y_m)) t_2))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
assert(x < y_m && y_m < z && z < t_m);
double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double t_2 = (y_m * x) * t_m;
	double tmp;
	if (x <= -2.8e+55) {
		tmp = t_2;
	} else if (x <= 3.5e-70) {
		tmp = -z * (t_m * y_m);
	} else {
		tmp = t_2;
	}
	return t_s * (y_s * tmp);
}

Fortran

y\_m =     private
y\_s =     private
t\_m =     private
t\_s =     private
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
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(t_s, y_s, x, y_m, z, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (y_m * x) * t_m
    if (x <= (-2.8d+55)) then
        tmp = t_2
    else if (x <= 3.5d-70) then
        tmp = -z * (t_m * y_m)
    else
        tmp = t_2
    end if
    code = t_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
assert x < y_m && y_m < z && z < t_m;
public static double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double t_2 = (y_m * x) * t_m;
	double tmp;
	if (x <= -2.8e+55) {
		tmp = t_2;
	} else if (x <= 3.5e-70) {
		tmp = -z * (t_m * y_m);
	} else {
		tmp = t_2;
	}
	return t_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(t_s, y_s, x, y_m, z, t_m):
	t_2 = (y_m * x) * t_m
	tmp = 0
	if x <= -2.8e+55:
		tmp = t_2
	elif x <= 3.5e-70:
		tmp = -z * (t_m * y_m)
	else:
		tmp = t_2
	return t_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
t\_m = abs(t)
t\_s = copysign(1.0, t)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(t_s, y_s, x, y_m, z, t_m)
	t_2 = Float64(Float64(y_m * x) * t_m)
	tmp = 0.0
	if (x <= -2.8e+55)
		tmp = t_2;
	elseif (x <= 3.5e-70)
		tmp = Float64(Float64(-z) * Float64(t_m * y_m));
	else
		tmp = t_2;
	end
	return Float64(t_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(t_s, y_s, x, y_m, z, t_m)
	t_2 = (y_m * x) * t_m;
	tmp = 0.0;
	if (x <= -2.8e+55)
		tmp = t_2;
	elseif (x <= 3.5e-70)
		tmp = -z * (t_m * y_m);
	else
		tmp = t_2;
	end
	tmp_2 = t_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[t$95$s_, y$95$s_, x_, y$95$m_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(y$95$m * x), $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * N[(y$95$s * If[LessEqual[x, -2.8e+55], t$95$2, If[LessEqual[x, 3.5e-70], N[((-z) * N[(t$95$m * y$95$m), $MachinePrecision]), $MachinePrecision], t$95$2]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.8000000000000001e55 or 3.49999999999999974e-70 < x

   1. Initial program 98.4%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 76.7

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

   4. Applied rewrites 76.7%

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

   if -2.8000000000000001e55 < x < 3.49999999999999974e-70

   1. Initial program 94.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

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

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

      5. sub-flip N/A

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

      6. mul-1-neg N/A

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

      7. add-flip N/A

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

      8. mul-1-neg N/A

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

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

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

      10. *-commutative N/A

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

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

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

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

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

      15. mul-1-neg N/A

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

      16. sub-negate N/A

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

      17. lower--.f64 N/A

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

      18. mul-1-neg N/A

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

      19. lower-neg.f64 94.1

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

   3. Applied rewrites 94.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. sub-negate N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 93.1

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

   5. Applied rewrites 93.1%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 75.7

         \[\leadsto \left(-z\right) \cdot \left(t \cdot y\right) \]

   8. Applied rewrites 75.7%

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

Alternative 5: 74.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ \begin{array}{l} t_2 := \left(y\_m \cdot x\right) \cdot t\_m\\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -1.18 \cdot 10^{+55}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-69}:\\ \;\;\;\;\left(z \cdot \left(-t\_m\right)\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (t_s y_s x y_m z t_m)
 :precision binary64
 (let* ((t_2 (* (* y_m x) t_m)))
   (*
    t_s
    (*
     y_s
     (if (<= x -1.18e+55)
       t_2
       (if (<= x 2.6e-69) (* (* z (- t_m)) y_m) t_2))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
assert(x < y_m && y_m < z && z < t_m);
double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double t_2 = (y_m * x) * t_m;
	double tmp;
	if (x <= -1.18e+55) {
		tmp = t_2;
	} else if (x <= 2.6e-69) {
		tmp = (z * -t_m) * y_m;
	} else {
		tmp = t_2;
	}
	return t_s * (y_s * tmp);
}

Fortran

y\_m =     private
y\_s =     private
t\_m =     private
t\_s =     private
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
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(t_s, y_s, x, y_m, z, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (y_m * x) * t_m
    if (x <= (-1.18d+55)) then
        tmp = t_2
    else if (x <= 2.6d-69) then
        tmp = (z * -t_m) * y_m
    else
        tmp = t_2
    end if
    code = t_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
assert x < y_m && y_m < z && z < t_m;
public static double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double t_2 = (y_m * x) * t_m;
	double tmp;
	if (x <= -1.18e+55) {
		tmp = t_2;
	} else if (x <= 2.6e-69) {
		tmp = (z * -t_m) * y_m;
	} else {
		tmp = t_2;
	}
	return t_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(t_s, y_s, x, y_m, z, t_m):
	t_2 = (y_m * x) * t_m
	tmp = 0
	if x <= -1.18e+55:
		tmp = t_2
	elif x <= 2.6e-69:
		tmp = (z * -t_m) * y_m
	else:
		tmp = t_2
	return t_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
t\_m = abs(t)
t\_s = copysign(1.0, t)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(t_s, y_s, x, y_m, z, t_m)
	t_2 = Float64(Float64(y_m * x) * t_m)
	tmp = 0.0
	if (x <= -1.18e+55)
		tmp = t_2;
	elseif (x <= 2.6e-69)
		tmp = Float64(Float64(z * Float64(-t_m)) * y_m);
	else
		tmp = t_2;
	end
	return Float64(t_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(t_s, y_s, x, y_m, z, t_m)
	t_2 = (y_m * x) * t_m;
	tmp = 0.0;
	if (x <= -1.18e+55)
		tmp = t_2;
	elseif (x <= 2.6e-69)
		tmp = (z * -t_m) * y_m;
	else
		tmp = t_2;
	end
	tmp_2 = t_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[t$95$s_, y$95$s_, x_, y$95$m_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(y$95$m * x), $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * N[(y$95$s * If[LessEqual[x, -1.18e+55], t$95$2, If[LessEqual[x, 2.6e-69], N[(N[(z * (-t$95$m)), $MachinePrecision] * y$95$m), $MachinePrecision], t$95$2]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1799999999999999e55 or 2.6000000000000002e-69 < x

   1. Initial program 98.4%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 76.7

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

   4. Applied rewrites 76.7%

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

   if -1.1799999999999999e55 < x < 2.6000000000000002e-69

   1. Initial program 94.1%

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

   2. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

         \[\leadsto \left(z \cdot \left(-1 \cdot t\right)\right) \cdot y \]

      8. mul-1-neg N/A

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

      9. lower-neg.f64 72.7

         \[\leadsto \left(z \cdot \left(-t\right)\right) \cdot y \]

   4. Applied rewrites 72.7%

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

Alternative 6: 57.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.7 \cdot 10^{+73}:\\ \;\;\;\;\left(y\_m \cdot x\right) \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t\_m \cdot y\_m\right)\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (t_s y_s x y_m z t_m)
 :precision binary64
 (* t_s (* y_s (if (<= t_m 2.7e+73) (* (* y_m x) t_m) (* x (* t_m y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
assert(x < y_m && y_m < z && z < t_m);
double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 2.7e+73) {
		tmp = (y_m * x) * t_m;
	} else {
		tmp = x * (t_m * y_m);
	}
	return t_s * (y_s * tmp);
}

Fortran

y\_m =     private
y\_s =     private
t\_m =     private
t\_s =     private
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
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(t_s, y_s, x, y_m, z, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 2.7d+73) then
        tmp = (y_m * x) * t_m
    else
        tmp = x * (t_m * y_m)
    end if
    code = t_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
assert x < y_m && y_m < z && z < t_m;
public static double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 2.7e+73) {
		tmp = (y_m * x) * t_m;
	} else {
		tmp = x * (t_m * y_m);
	}
	return t_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(t_s, y_s, x, y_m, z, t_m):
	tmp = 0
	if t_m <= 2.7e+73:
		tmp = (y_m * x) * t_m
	else:
		tmp = x * (t_m * y_m)
	return t_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
t\_m = abs(t)
t\_s = copysign(1.0, t)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(t_s, y_s, x, y_m, z, t_m)
	tmp = 0.0
	if (t_m <= 2.7e+73)
		tmp = Float64(Float64(y_m * x) * t_m);
	else
		tmp = Float64(x * Float64(t_m * y_m));
	end
	return Float64(t_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(t_s, y_s, x, y_m, z, t_m)
	tmp = 0.0;
	if (t_m <= 2.7e+73)
		tmp = (y_m * x) * t_m;
	else
		tmp = x * (t_m * y_m);
	end
	tmp_2 = t_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[t$95$s_, y$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(t$95$s * N[(y$95$s * If[LessEqual[t$95$m, 2.7e+73], N[(N[(y$95$m * x), $MachinePrecision] * t$95$m), $MachinePrecision], N[(x * N[(t$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.6999999999999999e73

   1. Initial program 98.8%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 61.0

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

   4. Applied rewrites 61.0%

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

   if 2.6999999999999999e73 < t

   1. Initial program 94.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

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

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

      5. sub-flip N/A

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

      6. mul-1-neg N/A

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

      7. add-flip N/A

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

      8. mul-1-neg N/A

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

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

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

      10. *-commutative N/A

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

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

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

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

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

      15. mul-1-neg N/A

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

      16. sub-negate N/A

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

      17. lower--.f64 N/A

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

      18. mul-1-neg N/A

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

      19. lower-neg.f64 95.7

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

   3. Applied rewrites 95.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. sub-negate N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 97.2

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 55.6%

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

Alternative 7: 57.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 100000000:\\ \;\;\;\;\left(t\_m \cdot x\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t\_m \cdot y\_m\right)\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (t_s y_s x y_m z t_m)
 :precision binary64
 (* t_s (* y_s (if (<= t_m 100000000.0) (* (* t_m x) y_m) (* x (* t_m y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
assert(x < y_m && y_m < z && z < t_m);
double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 100000000.0) {
		tmp = (t_m * x) * y_m;
	} else {
		tmp = x * (t_m * y_m);
	}
	return t_s * (y_s * tmp);
}

Fortran

y\_m =     private
y\_s =     private
t\_m =     private
t\_s =     private
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
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(t_s, y_s, x, y_m, z, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 100000000.0d0) then
        tmp = (t_m * x) * y_m
    else
        tmp = x * (t_m * y_m)
    end if
    code = t_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
assert x < y_m && y_m < z && z < t_m;
public static double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 100000000.0) {
		tmp = (t_m * x) * y_m;
	} else {
		tmp = x * (t_m * y_m);
	}
	return t_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(t_s, y_s, x, y_m, z, t_m):
	tmp = 0
	if t_m <= 100000000.0:
		tmp = (t_m * x) * y_m
	else:
		tmp = x * (t_m * y_m)
	return t_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
t\_m = abs(t)
t\_s = copysign(1.0, t)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(t_s, y_s, x, y_m, z, t_m)
	tmp = 0.0
	if (t_m <= 100000000.0)
		tmp = Float64(Float64(t_m * x) * y_m);
	else
		tmp = Float64(x * Float64(t_m * y_m));
	end
	return Float64(t_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(t_s, y_s, x, y_m, z, t_m)
	tmp = 0.0;
	if (t_m <= 100000000.0)
		tmp = (t_m * x) * y_m;
	else
		tmp = x * (t_m * y_m);
	end
	tmp_2 = t_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[t$95$s_, y$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(t$95$s * N[(y$95$s * If[LessEqual[t$95$m, 100000000.0], N[(N[(t$95$m * x), $MachinePrecision] * y$95$m), $MachinePrecision], N[(x * N[(t$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1e8

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. sub-flip N/A

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

      8. mul-1-neg N/A

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

      9. add-flip N/A

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

      10. mul-1-neg N/A

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

      11. distribute-rgt-neg-out N/A

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

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

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

      13. mul-1-neg N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 64.9

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

   6. Applied rewrites 64.9%

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

   if 1e8 < t

   1. Initial program 95.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

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

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

      5. sub-flip N/A

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

      6. mul-1-neg N/A

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

      7. add-flip N/A

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

      8. mul-1-neg N/A

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

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

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

      10. *-commutative N/A

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

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

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

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

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

      15. mul-1-neg N/A

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

      16. sub-negate N/A

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

      17. lower--.f64 N/A

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

      18. mul-1-neg N/A

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

      19. lower-neg.f64 95.9

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

   3. Applied rewrites 95.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. sub-negate N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 97.6

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 55.2%

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

Alternative 8: 51.3% accurate, 1.8× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (t_s y_s x y_m z t_m)
 :precision binary64
 (* t_s (* y_s (* (* t_m x) y_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
assert(x < y_m && y_m < z && z < t_m);
double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	return t_s * (y_s * ((t_m * x) * y_m));
}

Fortran

y\_m =     private
y\_s =     private
t\_m =     private
t\_s =     private
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
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(t_s, y_s, x, y_m, z, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    code = t_s * (y_s * ((t_m * x) * y_m))
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
assert x < y_m && y_m < z && z < t_m;
public static double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	return t_s * (y_s * ((t_m * x) * y_m));
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(t_s, y_s, x, y_m, z, t_m):
	return t_s * (y_s * ((t_m * x) * y_m))

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
t\_m = abs(t)
t\_s = copysign(1.0, t)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(t_s, y_s, x, y_m, z, t_m)
	return Float64(t_s * Float64(y_s * Float64(Float64(t_m * x) * y_m)))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp = code(t_s, y_s, x, y_m, z, t_m)
	tmp = t_s * (y_s * ((t_m * x) * y_m));
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[t$95$s_, y$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(t$95$s * N[(y$95$s * N[(N[(t$95$m * x), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. *-commutative N/A

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

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

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

   7. sub-flip N/A

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

   8. mul-1-neg N/A

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

   9. add-flip N/A

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

   10. mul-1-neg N/A

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

   11. distribute-rgt-neg-out N/A

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

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

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

   13. mul-1-neg N/A

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

   14. associate-*r* N/A

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

   15. *-commutative N/A

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

   16. associate-*r* N/A

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

   17. lower-*.f64 N/A

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

3. Applied rewrites 87.2%

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

4. Taylor expanded in x around inf

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

   1. lower-*.f64 51.3

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

6. Applied rewrites 51.3%

   \[\leadsto \color{blue}{\left(t \cdot x\right)} \cdot y \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025122 
(FPCore (x y z t)
  :name "Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3"
  :precision binary64
  (* (- (* x y) (* z y)) t))