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

Percentage Accurate: 96.0% → 97.9%

Time: 3.9s

Alternatives: 5

Speedup: 0.7×

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: 96.0% 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: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(\left(-x\_m\right) \cdot y, z, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m - \left(z \cdot y\right) \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (<= x_m 1e+27) (fma (* (- x_m) y) z x_m) (- x_m (* (* z y) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 1e+27) {
		tmp = fma((-x_m * y), z, x_m);
	} else {
		tmp = x_m - ((z * y) * x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 1e+27)
		tmp = fma(Float64(Float64(-x_m) * y), z, x_m);
	else
		tmp = Float64(x_m - Float64(Float64(z * y) * x_m));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 1e+27], N[(N[((-x$95$m) * y), $MachinePrecision] * z + x$95$m), $MachinePrecision], N[(x$95$m - N[(N[(z * y), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1e27

   1. Initial program 93.2%

      \[x \cdot \left(1 - y \cdot z\right) \]
   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. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. mul-1-neg N/A

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

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

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

      11. associate-*r* N/A

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

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

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-*.f64 94.6

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

   4. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-x\right) \cdot y, z, x \cdot 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 94.6

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

   6. Applied rewrites 94.6%

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

   if 1e27 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(\left(-x\right) \cdot y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ \begin{array}{l} t_0 := 1 - y \cdot z\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -50000000 \lor \neg \left(t\_0 \leq 2\right):\\ \;\;\;\;\left(\left(-x\_m\right) \cdot y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (* y z))))
   (*
    x_s
    (if (or (<= t_0 -50000000.0) (not (<= t_0 2.0)))
      (* (* (- x_m) y) z)
      x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = 1.0 - (y * z);
	double tmp;
	if ((t_0 <= -50000000.0) || !(t_0 <= 2.0)) {
		tmp = (-x_m * y) * z;
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
NOTE: x_m, 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_s, x_m, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    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 <= (-50000000.0d0)) .or. (.not. (t_0 <= 2.0d0))) then
        tmp = (-x_m * y) * z
    else
        tmp = x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z;
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = 1.0 - (y * z);
	double tmp;
	if ((t_0 <= -50000000.0) || !(t_0 <= 2.0)) {
		tmp = (-x_m * y) * z;
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z] = sort([x_m, y, z])
def code(x_s, x_m, y, z):
	t_0 = 1.0 - (y * z)
	tmp = 0
	if (t_0 <= -50000000.0) or not (t_0 <= 2.0):
		tmp = (-x_m * y) * z
	else:
		tmp = x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	t_0 = Float64(1.0 - Float64(y * z))
	tmp = 0.0
	if ((t_0 <= -50000000.0) || !(t_0 <= 2.0))
		tmp = Float64(Float64(Float64(-x_m) * y) * z);
	else
		tmp = x_m;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z = num2cell(sort([x_m, y, z])){:}
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = 1.0 - (y * z);
	tmp = 0.0;
	if ((t_0 <= -50000000.0) || ~((t_0 <= 2.0)))
		tmp = (-x_m * y) * z;
	else
		tmp = x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[Or[LessEqual[t$95$0, -50000000.0], N[Not[LessEqual[t$95$0, 2.0]], $MachinePrecision]], N[(N[((-x$95$m) * y), $MachinePrecision] * z), $MachinePrecision], x$95$m]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 y z)) < -5e7 or 2 < (-.f64 #s(literal 1 binary64) (*.f64 y z))

   1. Initial program 89.1%

      \[x \cdot \left(1 - y \cdot z\right) \]
   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. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. mul-1-neg N/A

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

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

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

      11. associate-*r* N/A

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

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

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-*.f64 93.6

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

   4. Applied rewrites 93.6%

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

   6. Applied rewrites 33.6%

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

   7. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. associate-*l* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. lift-neg.f64 92.1

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

   9. Applied rewrites 92.1%

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

   if -5e7 < (-.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

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

   5. Applied rewrites 98.4%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \cdot z \leq -50000000 \lor \neg \left(1 - y \cdot z \leq 2\right):\\ \;\;\;\;\left(\left(-x\right) \cdot y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ \begin{array}{l} t_0 := 1 - y \cdot z\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -50000000 \lor \neg \left(t\_0 \leq 2\right):\\ \;\;\;\;\left(\left(-x\_m\right) \cdot z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (* y z))))
   (*
    x_s
    (if (or (<= t_0 -50000000.0) (not (<= t_0 2.0)))
      (* (* (- x_m) z) y)
      x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = 1.0 - (y * z);
	double tmp;
	if ((t_0 <= -50000000.0) || !(t_0 <= 2.0)) {
		tmp = (-x_m * z) * y;
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
NOTE: x_m, 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_s, x_m, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    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 <= (-50000000.0d0)) .or. (.not. (t_0 <= 2.0d0))) then
        tmp = (-x_m * z) * y
    else
        tmp = x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z;
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = 1.0 - (y * z);
	double tmp;
	if ((t_0 <= -50000000.0) || !(t_0 <= 2.0)) {
		tmp = (-x_m * z) * y;
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z] = sort([x_m, y, z])
def code(x_s, x_m, y, z):
	t_0 = 1.0 - (y * z)
	tmp = 0
	if (t_0 <= -50000000.0) or not (t_0 <= 2.0):
		tmp = (-x_m * z) * y
	else:
		tmp = x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	t_0 = Float64(1.0 - Float64(y * z))
	tmp = 0.0
	if ((t_0 <= -50000000.0) || !(t_0 <= 2.0))
		tmp = Float64(Float64(Float64(-x_m) * z) * y);
	else
		tmp = x_m;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z = num2cell(sort([x_m, y, z])){:}
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = 1.0 - (y * z);
	tmp = 0.0;
	if ((t_0 <= -50000000.0) || ~((t_0 <= 2.0)))
		tmp = (-x_m * z) * y;
	else
		tmp = x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[Or[LessEqual[t$95$0, -50000000.0], N[Not[LessEqual[t$95$0, 2.0]], $MachinePrecision]], N[(N[((-x$95$m) * z), $MachinePrecision] * y), $MachinePrecision], x$95$m]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 y z)) < -5e7 or 2 < (-.f64 #s(literal 1 binary64) (*.f64 y z))

   1. Initial program 89.1%

      \[x \cdot \left(1 - y \cdot z\right) \]
   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. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. mul-1-neg N/A

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

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

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

      11. associate-*r* N/A

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

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

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-*.f64 93.6

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

   4. Applied rewrites 93.6%

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

   5. Taylor expanded in y around inf

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

      1. associate-*l* N/A

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

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

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

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

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

      4. mul-1-neg N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-lft-identity N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

   7. Applied rewrites 95.2%

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

   if -5e7 < (-.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

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

   5. Applied rewrites 98.4%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \cdot z \leq -50000000 \lor \neg \left(1 - y \cdot z \leq 2\right):\\ \;\;\;\;\left(\left(-x\right) \cdot z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;1 - y \cdot z \leq 10^{+185}:\\ \;\;\;\;x\_m - \left(z \cdot y\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x\_m\right) \cdot z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (- 1.0 (* y z)) 1e+185)
    (- x_m (* (* z y) x_m))
    (* (* (- x_m) z) y))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((1.0 - (y * z)) <= 1e+185) {
		tmp = x_m - ((z * y) * x_m);
	} else {
		tmp = (-x_m * z) * y;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
NOTE: x_m, 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_s, x_m, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((1.0d0 - (y * z)) <= 1d+185) then
        tmp = x_m - ((z * y) * x_m)
    else
        tmp = (-x_m * z) * y
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z;
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((1.0 - (y * z)) <= 1e+185) {
		tmp = x_m - ((z * y) * x_m);
	} else {
		tmp = (-x_m * z) * y;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z] = sort([x_m, y, z])
def code(x_s, x_m, y, z):
	tmp = 0
	if (1.0 - (y * z)) <= 1e+185:
		tmp = x_m - ((z * y) * x_m)
	else:
		tmp = (-x_m * z) * y
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(1.0 - Float64(y * z)) <= 1e+185)
		tmp = Float64(x_m - Float64(Float64(z * y) * x_m));
	else
		tmp = Float64(Float64(Float64(-x_m) * z) * y);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z = num2cell(sort([x_m, y, z])){:}
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((1.0 - (y * z)) <= 1e+185)
		tmp = x_m - ((z * y) * x_m);
	else
		tmp = (-x_m * z) * y;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision], 1e+185], N[(x$95$m - N[(N[(z * y), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[((-x$95$m) * z), $MachinePrecision] * y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 y z)) < 9.9999999999999998e184

   1. Initial program 98.6%

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

   3. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 98.6

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

   5. Applied rewrites 98.6%

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

   if 9.9999999999999998e184 < (-.f64 #s(literal 1 binary64) (*.f64 y z))

   1. Initial program 68.7%

      \[x \cdot \left(1 - y \cdot z\right) \]
   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. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. mul-1-neg N/A

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

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

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

      11. associate-*r* N/A

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

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

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around inf

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

      1. associate-*l* N/A

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

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

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

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

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

      4. mul-1-neg N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-lft-identity N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

   7. Applied rewrites 99.8%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \cdot z \leq 10^{+185}:\\ \;\;\;\;x - \left(z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x\right) \cdot z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.2% accurate, 14.0× speedup

Math

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

FPCore

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

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	return x_s * x_m;
}

Fortran

x\_m =     private
x\_s =     private
NOTE: x_m, 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_s, x_m, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x_s * x_m
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z;
public static double code(double x_s, double x_m, double y, double z) {
	return x_s * x_m;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z] = sort([x_m, y, z])
def code(x_s, x_m, y, z):
	return x_s * x_m

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	return Float64(x_s * x_m)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z = num2cell(sort([x_m, y, z])){:}
function tmp = code(x_s, x_m, y, z)
	tmp = x_s * x_m;
end

Wolfram

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

Derivation

1. Initial program 94.8%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 52.7%

   \[\leadsto \color{blue}{x} \]

6. Final simplification 52.7%

   \[\leadsto x \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025028 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))