Data.Colour.Matrix:inverse from colour-2.3.3, B

Percentage Accurate: 91.9% → 95.7%

Time: 3.4s

Alternatives: 8

Speedup: 0.6×

• 28.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 1.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a\_m} \cdot x - \frac{t}{a\_m} \cdot z\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= a_m 1.5e-66)
    (/ (- (* x y) (* z t)) a_m)
    (- (* (/ y a_m) x) (* (/ t a_m) z)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (a_m <= 1.5e-66) {
		tmp = ((x * y) - (z * t)) / a_m;
	} else {
		tmp = ((y / a_m) * x) - ((t / a_m) * z);
	}
	return a_s * tmp;
}

Fortran

a\_m =     private
a\_s =     private
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_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(a_s, x, y, z, t, a_m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    real(8) :: tmp
    if (a_m <= 1.5d-66) then
        tmp = ((x * y) - (z * t)) / a_m
    else
        tmp = ((y / a_m) * x) - ((t / a_m) * z)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (a_m <= 1.5e-66) {
		tmp = ((x * y) - (z * t)) / a_m;
	} else {
		tmp = ((y / a_m) * x) - ((t / a_m) * z);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if a_m <= 1.5e-66:
		tmp = ((x * y) - (z * t)) / a_m
	else:
		tmp = ((y / a_m) * x) - ((t / a_m) * z)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (a_m <= 1.5e-66)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a_m);
	else
		tmp = Float64(Float64(Float64(y / a_m) * x) - Float64(Float64(t / a_m) * z));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if (a_m <= 1.5e-66)
		tmp = ((x * y) - (z * t)) / a_m;
	else
		tmp = ((y / a_m) * x) - ((t / a_m) * z);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[a$95$m, 1.5e-66], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], N[(N[(N[(y / a$95$m), $MachinePrecision] * x), $MachinePrecision] - N[(N[(t / a$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 1.5000000000000001e-66

   1. Initial program 91.9%

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

   if 1.5000000000000001e-66 < a

   1. Initial program 91.9%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

         \[\leadsto \frac{\color{blue}{\frac{{\left(x \cdot y\right)}^{3} - {\left(z \cdot t\right)}^{3}}{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + \left(x \cdot y\right) \cdot \left(z \cdot t\right)\right)}}}{a} \]

      3. frac-2neg N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(\left({\left(x \cdot y\right)}^{3} - {\left(z \cdot t\right)}^{3}\right)\right)}{\mathsf{neg}\left(\left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + \left(x \cdot y\right) \cdot \left(z \cdot t\right)\right)\right)\right)}}}{a} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left({\left(x \cdot y\right)}^{3} - {\left(z \cdot t\right)}^{3}\right)\right)}{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + \left(x \cdot y\right) \cdot \left(z \cdot t\right)\right)}\right)}}{a} \]

      5. distribute-neg-frac N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left({\left(x \cdot y\right)}^{3} - {\left(z \cdot t\right)}^{3}\right)\right)\right)\right)}{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + \left(x \cdot y\right) \cdot \left(z \cdot t\right)\right)}}}{a} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left({\left(x \cdot y\right)}^{3} - {\left(z \cdot t\right)}^{3}\right)\right)\right)\right)}{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \color{blue}{\left(\left(x \cdot y\right) \cdot \left(z \cdot t\right) + \left(z \cdot t\right) \cdot \left(z \cdot t\right)\right)}}}{a} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left({\left(x \cdot y\right)}^{3} - {\left(z \cdot t\right)}^{3}\right)\right)\right)\right)}{\color{blue}{\left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(x \cdot y\right) \cdot \left(z \cdot t\right)\right) + \left(z \cdot t\right) \cdot \left(z \cdot t\right)}}}{a} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left({\left(x \cdot y\right)}^{3} - {\left(z \cdot t\right)}^{3}\right)\right)\right)\right)}{\left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \color{blue}{\left(z \cdot t\right) \cdot \left(x \cdot y\right)}\right) + \left(z \cdot t\right) \cdot \left(z \cdot t\right)}}{a} \]

      9. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left({\left(x \cdot y\right)}^{3} - {\left(z \cdot t\right)}^{3}\right)\right)\right)\right)}{\color{blue}{\left(z \cdot t\right) \cdot \left(z \cdot t\right) + \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(z \cdot t\right) \cdot \left(x \cdot y\right)\right)}}}{a} \]

   3. Applied rewrites 92.1%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. add-flip N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-neg.f64 N/A

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

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

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

      18. remove-double-neg N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 87.6

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

   5. Applied rewrites 87.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip-rev N/A

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

      7. remove-double-neg N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/r/ N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-neg.f64 N/A

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

      14. remove-double-neg 87.4

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

   7. Applied rewrites 87.4%

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

Alternative 2: 94.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \left(x - t \cdot \frac{z}{y}\right) \cdot \frac{y}{a\_m}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+160}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+193}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(-t\right) \cdot z\right)}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* (- x (* t (/ z y))) (/ y a_m))))
   (*
    a_s
    (if (<= (* x y) -5e+160)
      t_1
      (if (<= (* x y) 2e+193) (/ (fma y x (* (- t) z)) a_m) t_1)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (x - (t * (z / y))) * (y / a_m);
	double tmp;
	if ((x * y) <= -5e+160) {
		tmp = t_1;
	} else if ((x * y) <= 2e+193) {
		tmp = fma(y, x, (-t * z)) / a_m;
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(x - Float64(t * Float64(z / y))) * Float64(y / a_m))
	tmp = 0.0
	if (Float64(x * y) <= -5e+160)
		tmp = t_1;
	elseif (Float64(x * y) <= 2e+193)
		tmp = Float64(fma(y, x, Float64(Float64(-t) * z)) / a_m);
	else
		tmp = t_1;
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(x - N[(t * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], -5e+160], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 2e+193], N[(N[(y * x + N[((-t) * z), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -5.0000000000000002e160 or 2.00000000000000013e193 < (*.f64 x y)

   1. Initial program 91.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-to-mult N/A

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

      4. associate-/l* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. sub-to-mult-rev N/A

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

      13. lower--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 80.8

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

   3. Applied rewrites 80.8%

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

   if -5.0000000000000002e160 < (*.f64 x y) < 2.00000000000000013e193

   1. Initial program 91.9%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

         \[\leadsto \frac{\color{blue}{\frac{{\left(x \cdot y\right)}^{3} - {\left(z \cdot t\right)}^{3}}{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + \left(x \cdot y\right) \cdot \left(z \cdot t\right)\right)}}}{a} \]

      3. frac-2neg N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(\left({\left(x \cdot y\right)}^{3} - {\left(z \cdot t\right)}^{3}\right)\right)}{\mathsf{neg}\left(\left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + \left(x \cdot y\right) \cdot \left(z \cdot t\right)\right)\right)\right)}}}{a} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left({\left(x \cdot y\right)}^{3} - {\left(z \cdot t\right)}^{3}\right)\right)}{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + \left(x \cdot y\right) \cdot \left(z \cdot t\right)\right)}\right)}}{a} \]

      5. distribute-neg-frac N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left({\left(x \cdot y\right)}^{3} - {\left(z \cdot t\right)}^{3}\right)\right)\right)\right)}{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + \left(x \cdot y\right) \cdot \left(z \cdot t\right)\right)}}}{a} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left({\left(x \cdot y\right)}^{3} - {\left(z \cdot t\right)}^{3}\right)\right)\right)\right)}{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \color{blue}{\left(\left(x \cdot y\right) \cdot \left(z \cdot t\right) + \left(z \cdot t\right) \cdot \left(z \cdot t\right)\right)}}}{a} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left({\left(x \cdot y\right)}^{3} - {\left(z \cdot t\right)}^{3}\right)\right)\right)\right)}{\color{blue}{\left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(x \cdot y\right) \cdot \left(z \cdot t\right)\right) + \left(z \cdot t\right) \cdot \left(z \cdot t\right)}}}{a} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left({\left(x \cdot y\right)}^{3} - {\left(z \cdot t\right)}^{3}\right)\right)\right)\right)}{\left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \color{blue}{\left(z \cdot t\right) \cdot \left(x \cdot y\right)}\right) + \left(z \cdot t\right) \cdot \left(z \cdot t\right)}}{a} \]

      9. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left({\left(x \cdot y\right)}^{3} - {\left(z \cdot t\right)}^{3}\right)\right)\right)\right)}{\color{blue}{\left(z \cdot t\right) \cdot \left(z \cdot t\right) + \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(z \cdot t\right) \cdot \left(x \cdot y\right)\right)}}}{a} \]

   3. Applied rewrites 92.1%

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

Alternative 3: 94.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;\frac{y}{a\_m} \cdot x\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+165}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(-t\right) \cdot z\right)}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a\_m}{y}}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* x y) (- INFINITY))
    (* (/ y a_m) x)
    (if (<= (* x y) 5e+165) (/ (fma y x (* (- t) z)) a_m) (/ x (/ a_m y))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -((double) INFINITY)) {
		tmp = (y / a_m) * x;
	} else if ((x * y) <= 5e+165) {
		tmp = fma(y, x, (-t * z)) / a_m;
	} else {
		tmp = x / (a_m / y);
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(x * y) <= Float64(-Inf))
		tmp = Float64(Float64(y / a_m) * x);
	elseif (Float64(x * y) <= 5e+165)
		tmp = Float64(fma(y, x, Float64(Float64(-t) * z)) / a_m);
	else
		tmp = Float64(x / Float64(a_m / y));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], (-Infinity)], N[(N[(y / a$95$m), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+165], N[(N[(y * x + N[((-t) * z), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], N[(x / N[(a$95$m / y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 91.9%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 51.1

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

   4. Applied rewrites 51.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 51.5

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

   6. Applied rewrites 51.5%

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

   if -inf.0 < (*.f64 x y) < 4.9999999999999997e165

   1. Initial program 91.9%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

         \[\leadsto \frac{\color{blue}{\frac{{\left(x \cdot y\right)}^{3} - {\left(z \cdot t\right)}^{3}}{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + \left(x \cdot y\right) \cdot \left(z \cdot t\right)\right)}}}{a} \]

      3. frac-2neg N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(\left({\left(x \cdot y\right)}^{3} - {\left(z \cdot t\right)}^{3}\right)\right)}{\mathsf{neg}\left(\left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + \left(x \cdot y\right) \cdot \left(z \cdot t\right)\right)\right)\right)}}}{a} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left({\left(x \cdot y\right)}^{3} - {\left(z \cdot t\right)}^{3}\right)\right)}{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + \left(x \cdot y\right) \cdot \left(z \cdot t\right)\right)}\right)}}{a} \]

      5. distribute-neg-frac N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left({\left(x \cdot y\right)}^{3} - {\left(z \cdot t\right)}^{3}\right)\right)\right)\right)}{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + \left(x \cdot y\right) \cdot \left(z \cdot t\right)\right)}}}{a} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left({\left(x \cdot y\right)}^{3} - {\left(z \cdot t\right)}^{3}\right)\right)\right)\right)}{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \color{blue}{\left(\left(x \cdot y\right) \cdot \left(z \cdot t\right) + \left(z \cdot t\right) \cdot \left(z \cdot t\right)\right)}}}{a} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left({\left(x \cdot y\right)}^{3} - {\left(z \cdot t\right)}^{3}\right)\right)\right)\right)}{\color{blue}{\left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(x \cdot y\right) \cdot \left(z \cdot t\right)\right) + \left(z \cdot t\right) \cdot \left(z \cdot t\right)}}}{a} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left({\left(x \cdot y\right)}^{3} - {\left(z \cdot t\right)}^{3}\right)\right)\right)\right)}{\left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \color{blue}{\left(z \cdot t\right) \cdot \left(x \cdot y\right)}\right) + \left(z \cdot t\right) \cdot \left(z \cdot t\right)}}{a} \]

      9. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left({\left(x \cdot y\right)}^{3} - {\left(z \cdot t\right)}^{3}\right)\right)\right)\right)}{\color{blue}{\left(z \cdot t\right) \cdot \left(z \cdot t\right) + \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(z \cdot t\right) \cdot \left(x \cdot y\right)\right)}}}{a} \]

   3. Applied rewrites 92.1%

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

   if 4.9999999999999997e165 < (*.f64 x y)

   1. Initial program 91.9%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 51.1

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

   4. Applied rewrites 51.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. div-flip N/A

         \[\leadsto x \cdot \frac{1}{\color{blue}{\frac{a}{y}}} \]

      5. mult-flip-rev N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{a}{y}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{a}{y}}} \]

      7. lower-/.f64 51.5

         \[\leadsto \frac{x}{\frac{a}{\color{blue}{y}}} \]

   6. Applied rewrites 51.5%

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

Alternative 4: 94.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;\frac{y}{a\_m} \cdot x\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+165}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a\_m}{y}}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* x y) (- INFINITY))
    (* (/ y a_m) x)
    (if (<= (* x y) 5e+165) (/ (- (* x y) (* z t)) a_m) (/ x (/ a_m y))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -((double) INFINITY)) {
		tmp = (y / a_m) * x;
	} else if ((x * y) <= 5e+165) {
		tmp = ((x * y) - (z * t)) / a_m;
	} else {
		tmp = x / (a_m / y);
	}
	return a_s * tmp;
}

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -Double.POSITIVE_INFINITY) {
		tmp = (y / a_m) * x;
	} else if ((x * y) <= 5e+165) {
		tmp = ((x * y) - (z * t)) / a_m;
	} else {
		tmp = x / (a_m / y);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if (x * y) <= -math.inf:
		tmp = (y / a_m) * x
	elif (x * y) <= 5e+165:
		tmp = ((x * y) - (z * t)) / a_m
	else:
		tmp = x / (a_m / y)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(x * y) <= Float64(-Inf))
		tmp = Float64(Float64(y / a_m) * x);
	elseif (Float64(x * y) <= 5e+165)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a_m);
	else
		tmp = Float64(x / Float64(a_m / y));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if ((x * y) <= -Inf)
		tmp = (y / a_m) * x;
	elseif ((x * y) <= 5e+165)
		tmp = ((x * y) - (z * t)) / a_m;
	else
		tmp = x / (a_m / y);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], (-Infinity)], N[(N[(y / a$95$m), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+165], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], N[(x / N[(a$95$m / y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 91.9%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 51.1

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

   4. Applied rewrites 51.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 51.5

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

   6. Applied rewrites 51.5%

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

   if -inf.0 < (*.f64 x y) < 4.9999999999999997e165

   1. Initial program 91.9%

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

   if 4.9999999999999997e165 < (*.f64 x y)

   1. Initial program 91.9%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 51.1

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

   4. Applied rewrites 51.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. div-flip N/A

         \[\leadsto x \cdot \frac{1}{\color{blue}{\frac{a}{y}}} \]

      5. mult-flip-rev N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{a}{y}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{a}{y}}} \]

      7. lower-/.f64 51.5

         \[\leadsto \frac{x}{\frac{a}{\color{blue}{y}}} \]

   6. Applied rewrites 51.5%

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

Alternative 5: 72.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \frac{\left(-z\right) \cdot t}{a\_m}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+52}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 10^{+50}:\\ \;\;\;\;\frac{x \cdot y}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (/ (* (- z) t) a_m)))
   (*
    a_s
    (if (<= (* z t) -1e+52) t_1 (if (<= (* z t) 1e+50) (/ (* x y) a_m) t_1)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (-z * t) / a_m;
	double tmp;
	if ((z * t) <= -1e+52) {
		tmp = t_1;
	} else if ((z * t) <= 1e+50) {
		tmp = (x * y) / a_m;
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Fortran

a\_m =     private
a\_s =     private
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_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(a_s, x, y, z, t, a_m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-z * t) / a_m
    if ((z * t) <= (-1d+52)) then
        tmp = t_1
    else if ((z * t) <= 1d+50) then
        tmp = (x * y) / a_m
    else
        tmp = t_1
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (-z * t) / a_m;
	double tmp;
	if ((z * t) <= -1e+52) {
		tmp = t_1;
	} else if ((z * t) <= 1e+50) {
		tmp = (x * y) / a_m;
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	t_1 = (-z * t) / a_m
	tmp = 0
	if (z * t) <= -1e+52:
		tmp = t_1
	elif (z * t) <= 1e+50:
		tmp = (x * y) / a_m
	else:
		tmp = t_1
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(Float64(-z) * t) / a_m)
	tmp = 0.0
	if (Float64(z * t) <= -1e+52)
		tmp = t_1;
	elseif (Float64(z * t) <= 1e+50)
		tmp = Float64(Float64(x * y) / a_m);
	else
		tmp = t_1;
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = (-z * t) / a_m;
	tmp = 0.0;
	if ((z * t) <= -1e+52)
		tmp = t_1;
	elseif ((z * t) <= 1e+50)
		tmp = (x * y) / a_m;
	else
		tmp = t_1;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[((-z) * t), $MachinePrecision] / a$95$m), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(z * t), $MachinePrecision], -1e+52], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 1e+50], N[(N[(x * y), $MachinePrecision] / a$95$m), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -9.9999999999999999e51 or 1.0000000000000001e50 < (*.f64 z t)

   1. Initial program 91.9%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 50.2

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

   4. Applied rewrites 50.2%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

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

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

      7. remove-double-neg N/A

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

      8. lift-neg.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. remove-double-neg 50.2

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

   6. Applied rewrites 50.2%

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

   if -9.9999999999999999e51 < (*.f64 z t) < 1.0000000000000001e50

   1. Initial program 91.9%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 51.1

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

   4. Applied rewrites 51.1%

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

Alternative 6: 52.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+103}:\\ \;\;\;\;y \cdot \frac{x}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a\_m}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (* a_s (if (<= x -5e+103) (* y (/ x a_m)) (/ (* x y) a_m))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (x <= -5e+103) {
		tmp = y * (x / a_m);
	} else {
		tmp = (x * y) / a_m;
	}
	return a_s * tmp;
}

Fortran

a\_m =     private
a\_s =     private
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_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(a_s, x, y, z, t, a_m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    real(8) :: tmp
    if (x <= (-5d+103)) then
        tmp = y * (x / a_m)
    else
        tmp = (x * y) / a_m
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (x <= -5e+103) {
		tmp = y * (x / a_m);
	} else {
		tmp = (x * y) / a_m;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if x <= -5e+103:
		tmp = y * (x / a_m)
	else:
		tmp = (x * y) / a_m
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (x <= -5e+103)
		tmp = Float64(y * Float64(x / a_m));
	else
		tmp = Float64(Float64(x * y) / a_m);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if (x <= -5e+103)
		tmp = y * (x / a_m);
	else
		tmp = (x * y) / a_m;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[x, -5e+103], N[(y * N[(x / a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / a$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < -5e103

   1. Initial program 91.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-to-mult N/A

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

      4. associate-/l* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. sub-to-mult-rev N/A

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

      13. lower--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 81.1

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

   3. Applied rewrites 81.1%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 51.5%

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

   if -5e103 < x

   1. Initial program 91.9%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 51.1

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

   4. Applied rewrites 51.1%

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

Alternative 7: 52.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 10^{-60}:\\ \;\;\;\;\frac{x \cdot y}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a\_m} \cdot x\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (* a_s (if (<= a_m 1e-60) (/ (* x y) a_m) (* (/ y a_m) x))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (a_m <= 1e-60) {
		tmp = (x * y) / a_m;
	} else {
		tmp = (y / a_m) * x;
	}
	return a_s * tmp;
}

Fortran

a\_m =     private
a\_s =     private
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_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(a_s, x, y, z, t, a_m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    real(8) :: tmp
    if (a_m <= 1d-60) then
        tmp = (x * y) / a_m
    else
        tmp = (y / a_m) * x
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (a_m <= 1e-60) {
		tmp = (x * y) / a_m;
	} else {
		tmp = (y / a_m) * x;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if a_m <= 1e-60:
		tmp = (x * y) / a_m
	else:
		tmp = (y / a_m) * x
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (a_m <= 1e-60)
		tmp = Float64(Float64(x * y) / a_m);
	else
		tmp = Float64(Float64(y / a_m) * x);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if (a_m <= 1e-60)
		tmp = (x * y) / a_m;
	else
		tmp = (y / a_m) * x;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[a$95$m, 1e-60], N[(N[(x * y), $MachinePrecision] / a$95$m), $MachinePrecision], N[(N[(y / a$95$m), $MachinePrecision] * x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 9.9999999999999997e-61

   1. Initial program 91.9%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 51.1

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

   4. Applied rewrites 51.1%

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

   if 9.9999999999999997e-61 < a

   1. Initial program 91.9%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 51.1

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

   4. Applied rewrites 51.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 51.5

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

   6. Applied rewrites 51.5%

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

Alternative 8: 51.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	return a_s * ((y / a_m) * x);
}

Fortran

a\_m =     private
a\_s =     private
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_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(a_s, x, y, z, t, a_m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    code = a_s * ((y / a_m) * x)
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	return a_s * ((y / a_m) * x);
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	return a_s * ((y / a_m) * x)

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	return Float64(a_s * Float64(Float64(y / a_m) * x))
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp = code(a_s, x, y, z, t, a_m)
	tmp = a_s * ((y / a_m) * x);
end

Wolfram

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

Derivation

1. Initial program 91.9%

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

2. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 51.1

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

4. Applied rewrites 51.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 51.5

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

6. Applied rewrites 51.5%

   \[\leadsto \frac{y}{a} \cdot \color{blue}{x} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025156 
(FPCore (x y z t a)
  :name "Data.Colour.Matrix:inverse from colour-2.3.3, B"
  :precision binary64
  (/ (- (* x y) (* z t)) a))