Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I

Percentage Accurate: 95.9% → 99.8%

Time: 4.8s

Alternatives: 4

Speedup: 0.3×

Specification

Math

\[\begin{array}{l} \\ x \cdot \left(1 - y \cdot z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* x (- 1.0 (* y z))))

C

double code(double x, double y, double z) {
	return x * (1.0 - (y * z));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x * (1.0d0 - (y * z))
end function

Java

public static double code(double x, double y, double z) {
	return x * (1.0 - (y * z));
}

Python

def code(x, y, z):
	return x * (1.0 - (y * z))

Julia

function code(x, y, z)
	return Float64(x * Float64(1.0 - Float64(y * z)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x * (1.0 - (y * z));
end

Wolfram

code[x_, y_, z_] := N[(x * N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 95.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 - y \cdot z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* x (- 1.0 (* y z))))

C

double code(double x, double y, double z) {
	return x * (1.0 - (y * z));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x * (1.0d0 - (y * z))
end function

Java

public static double code(double x, double y, double z) {
	return x * (1.0 - (y * z));
}

Python

def code(x, y, z):
	return x * (1.0 - (y * z))

Julia

function code(x, y, z)
	return Float64(x * Float64(1.0 - Float64(y * z)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x * (1.0 - (y * z));
end

Wolfram

code[x_, y_, z_] := N[(x * N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := 1 - y \cdot z\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+280} \lor \neg \left(t\_0 \leq 4 \cdot 10^{+223}\right):\\ \;\;\;\;\left(\left(-z\right) \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (* y z))))
   (if (or (<= t_0 -2e+280) (not (<= t_0 4e+223)))
     (* (* (- z) x) y)
     (* x t_0))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = 1.0 - (y * z);
	double tmp;
	if ((t_0 <= -2e+280) || !(t_0 <= 4e+223)) {
		tmp = (-z * x) * y;
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z 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(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (y * z)
    if ((t_0 <= (-2d+280)) .or. (.not. (t_0 <= 4d+223))) then
        tmp = (-z * x) * y
    else
        tmp = x * t_0
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double t_0 = 1.0 - (y * z);
	double tmp;
	if ((t_0 <= -2e+280) || !(t_0 <= 4e+223)) {
		tmp = (-z * x) * y;
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	t_0 = 1.0 - (y * z)
	tmp = 0
	if (t_0 <= -2e+280) or not (t_0 <= 4e+223):
		tmp = (-z * x) * y
	else:
		tmp = x * t_0
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y * z))
	tmp = 0.0
	if ((t_0 <= -2e+280) || !(t_0 <= 4e+223))
		tmp = Float64(Float64(Float64(-z) * x) * y);
	else
		tmp = Float64(x * t_0);
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (y * z);
	tmp = 0.0;
	if ((t_0 <= -2e+280) || ~((t_0 <= 4e+223)))
		tmp = (-z * x) * y;
	else
		tmp = x * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e+280], N[Not[LessEqual[t$95$0, 4e+223]], $MachinePrecision]], N[(N[((-z) * x), $MachinePrecision] * y), $MachinePrecision], N[(x * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 y z)) < -2.0000000000000001e280 or 4.00000000000000019e223 < (-.f64 #s(literal 1 binary64) (*.f64 y z))

   1. Initial program 69.6%

      \[x \cdot \left(1 - y \cdot z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 99.8

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 27.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{{\left(z \cdot x\right)}^{2} \cdot \left(-y\right)}, \sqrt{-y}, x\right)} \]

   7. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot \left(z \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot \left(z \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot x} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{y \cdot \left(\left(z \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot x\right)} \]

      3. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(x \cdot \left(z \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(y \cdot x\right) \cdot \left(z \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y\right)} \cdot \left(z \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot z\right)} \]

      7. unpow2 N/A

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

      8. rem-square-sqrt N/A

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

      9. mul-1-neg N/A

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

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

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

      11. associate-*r* N/A

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

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

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

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

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

      16. lower-*.f64 N/A

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

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

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-neg.f64 99.8

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

   9. Applied rewrites 99.8%

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

   if -2.0000000000000001e280 < (-.f64 #s(literal 1 binary64) (*.f64 y z)) < 4.00000000000000019e223

   1. Initial program 99.9%

      \[x \cdot \left(1 - y \cdot z\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \cdot z \leq -2 \cdot 10^{+280} \lor \neg \left(1 - y \cdot z \leq 4 \cdot 10^{+223}\right):\\ \;\;\;\;\left(\left(-z\right) \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := 1 - y \cdot z\\ \mathbf{if}\;t\_0 \leq -20000 \lor \neg \left(t\_0 \leq 2\right):\\ \;\;\;\;\left(\left(-z\right) \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (* y z))))
   (if (or (<= t_0 -20000.0) (not (<= t_0 2.0))) (* (* (- z) x) y) (* x 1.0))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = 1.0 - (y * z);
	double tmp;
	if ((t_0 <= -20000.0) || !(t_0 <= 2.0)) {
		tmp = (-z * x) * y;
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z 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(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (y * z)
    if ((t_0 <= (-20000.0d0)) .or. (.not. (t_0 <= 2.0d0))) then
        tmp = (-z * x) * y
    else
        tmp = x * 1.0d0
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double t_0 = 1.0 - (y * z);
	double tmp;
	if ((t_0 <= -20000.0) || !(t_0 <= 2.0)) {
		tmp = (-z * x) * y;
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	t_0 = 1.0 - (y * z)
	tmp = 0
	if (t_0 <= -20000.0) or not (t_0 <= 2.0):
		tmp = (-z * x) * y
	else:
		tmp = x * 1.0
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y * z))
	tmp = 0.0
	if ((t_0 <= -20000.0) || !(t_0 <= 2.0))
		tmp = Float64(Float64(Float64(-z) * x) * y);
	else
		tmp = Float64(x * 1.0);
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (y * z);
	tmp = 0.0;
	if ((t_0 <= -20000.0) || ~((t_0 <= 2.0)))
		tmp = (-z * x) * y;
	else
		tmp = x * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -20000.0], N[Not[LessEqual[t$95$0, 2.0]], $MachinePrecision]], N[(N[((-z) * x), $MachinePrecision] * y), $MachinePrecision], N[(x * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 y z)) < -2e4 or 2 < (-.f64 #s(literal 1 binary64) (*.f64 y z))

   1. Initial program 88.1%

      \[x \cdot \left(1 - y \cdot z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 91.3

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

   4. Applied rewrites 91.3%

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

   6. Applied rewrites 21.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{{\left(z \cdot x\right)}^{2} \cdot \left(-y\right)}, \sqrt{-y}, x\right)} \]

   7. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot \left(z \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot \left(z \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot x} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{y \cdot \left(\left(z \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot x\right)} \]

      3. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(x \cdot \left(z \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(y \cdot x\right) \cdot \left(z \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y\right)} \cdot \left(z \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot z\right)} \]

      7. unpow2 N/A

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

      8. rem-square-sqrt N/A

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

      9. mul-1-neg N/A

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

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

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

      11. associate-*r* N/A

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

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

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

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

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

      16. lower-*.f64 N/A

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

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

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-neg.f64 90.1

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

   9. Applied rewrites 90.1%

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

   if -2e4 < (-.f64 #s(literal 1 binary64) (*.f64 y z)) < 2

   1. Initial program 100.0%

      \[x \cdot \left(1 - y \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.3%

      \[\leadsto x \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \cdot z \leq -20000 \lor \neg \left(1 - y \cdot z \leq 2\right):\\ \;\;\;\;\left(\left(-z\right) \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := 1 - y \cdot z\\ \mathbf{if}\;t\_0 \leq -20000 \lor \neg \left(t\_0 \leq 2\right):\\ \;\;\;\;\left(\left(-y\right) \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (* y z))))
   (if (or (<= t_0 -20000.0) (not (<= t_0 2.0))) (* (* (- y) x) z) (* x 1.0))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = 1.0 - (y * z);
	double tmp;
	if ((t_0 <= -20000.0) || !(t_0 <= 2.0)) {
		tmp = (-y * x) * z;
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z 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(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (y * z)
    if ((t_0 <= (-20000.0d0)) .or. (.not. (t_0 <= 2.0d0))) then
        tmp = (-y * x) * z
    else
        tmp = x * 1.0d0
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double t_0 = 1.0 - (y * z);
	double tmp;
	if ((t_0 <= -20000.0) || !(t_0 <= 2.0)) {
		tmp = (-y * x) * z;
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	t_0 = 1.0 - (y * z)
	tmp = 0
	if (t_0 <= -20000.0) or not (t_0 <= 2.0):
		tmp = (-y * x) * z
	else:
		tmp = x * 1.0
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y * z))
	tmp = 0.0
	if ((t_0 <= -20000.0) || !(t_0 <= 2.0))
		tmp = Float64(Float64(Float64(-y) * x) * z);
	else
		tmp = Float64(x * 1.0);
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (y * z);
	tmp = 0.0;
	if ((t_0 <= -20000.0) || ~((t_0 <= 2.0)))
		tmp = (-y * x) * z;
	else
		tmp = x * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -20000.0], N[Not[LessEqual[t$95$0, 2.0]], $MachinePrecision]], N[(N[((-y) * x), $MachinePrecision] * z), $MachinePrecision], N[(x * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 y z)) < -2e4 or 2 < (-.f64 #s(literal 1 binary64) (*.f64 y z))

   1. Initial program 88.1%

      \[x \cdot \left(1 - y \cdot z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 91.3

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

   4. Applied rewrites 91.3%

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

   6. Applied rewrites 21.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{{\left(z \cdot x\right)}^{2} \cdot \left(-y\right)}, \sqrt{-y}, x\right)} \]

   7. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(y \cdot \left(z \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(y \cdot \left(z \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(x \cdot y\right) \cdot \left(z \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right) \]

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

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \left(\mathsf{neg}\left(z \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(x \cdot y\right) \cdot \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot z}\right)\right) \]

      5. unpow2 N/A

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

      6. rem-square-sqrt N/A

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

      7. mul-1-neg N/A

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

      8. remove-double-neg N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 0.7

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

   9. Applied rewrites 0.7%

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

   11. Applied rewrites 94.3%

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

   if -2e4 < (-.f64 #s(literal 1 binary64) (*.f64 y z)) < 2

   1. Initial program 100.0%

      \[x \cdot \left(1 - y \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.3%

      \[\leadsto x \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \cdot z \leq -20000 \lor \neg \left(1 - y \cdot z \leq 2\right):\\ \;\;\;\;\left(\left(-y\right) \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ x \cdot 1 \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (* x 1.0))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return x * 1.0;
}

Fortran

NOTE: x, y, and z 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(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x * 1.0d0
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	return x * 1.0;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return x * 1.0

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(x * 1.0)
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = x * 1.0;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(x * 1.0), $MachinePrecision]

Derivation

1. Initial program 93.8%

   \[x \cdot \left(1 - y \cdot z\right) \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto x \cdot \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 49.1%

   \[\leadsto x \cdot \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024359 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))