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

Percentage Accurate: 96.1% → 99.1%

Time: 7.7s

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

real(8) function code(x, y, z)
    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: 96.1% 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

real(8) function code(x, y, z)
    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: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := 1 - y \cdot z\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+120}:\\ \;\;\;\;\left(x \cdot z\right) \cdot \left(-y\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+100}:\\ \;\;\;\;x \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-y\right) \cdot x\right) \cdot z\\ \end{array} \end{array} \]

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 (<= t_0 -2e+120)
     (* (* x z) (- y))
     (if (<= t_0 2e+100) (* x t_0) (* (* (- y) x) z)))))

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+120) {
		tmp = (x * z) * -y;
	} else if (t_0 <= 2e+100) {
		tmp = x * t_0;
	} else {
		tmp = (-y * x) * z;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    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+120)) then
        tmp = (x * z) * -y
    else if (t_0 <= 2d+100) then
        tmp = x * t_0
    else
        tmp = (-y * x) * z
    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+120) {
		tmp = (x * z) * -y;
	} else if (t_0 <= 2e+100) {
		tmp = x * t_0;
	} else {
		tmp = (-y * x) * z;
	}
	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+120:
		tmp = (x * z) * -y
	elif t_0 <= 2e+100:
		tmp = x * t_0
	else:
		tmp = (-y * x) * z
	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+120)
		tmp = Float64(Float64(x * z) * Float64(-y));
	elseif (t_0 <= 2e+100)
		tmp = Float64(x * t_0);
	else
		tmp = Float64(Float64(Float64(-y) * x) * z);
	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+120)
		tmp = (x * z) * -y;
	elseif (t_0 <= 2e+100)
		tmp = x * t_0;
	else
		tmp = (-y * x) * z;
	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[LessEqual[t$95$0, -2e+120], N[(N[(x * z), $MachinePrecision] * (-y)), $MachinePrecision], If[LessEqual[t$95$0, 2e+100], N[(x * t$95$0), $MachinePrecision], N[(N[((-y) * x), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 y z)) < -2e120

   1. Initial program 88.5%

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

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower--.f64 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 37.1

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

   4. Applied rewrites 37.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 37.1

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

   6. Applied rewrites 37.1%

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

   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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 99.8

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

   9. Applied rewrites 99.8%

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

   11. Applied rewrites 96.1%

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

   if -2e120 < (-.f64 #s(literal 1 binary64) (*.f64 y z)) < 2.00000000000000003e100

   1. Initial program 99.9%

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

   if 2.00000000000000003e100 < (-.f64 #s(literal 1 binary64) (*.f64 y z))

   1. Initial program 84.6%

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

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower--.f64 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 40.7

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

   4. Applied rewrites 40.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 40.7

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

   6. Applied rewrites 40.7%

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

   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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 99.9

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

   9. Applied rewrites 99.9%

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

Alternative 2: 93.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := 1 - y \cdot z\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+20} \lor \neg \left(t\_0 \leq 20000\right):\\ \;\;\;\;\left(\left(-y\right) \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \end{array} \]

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 -2e+20) (not (<= t_0 20000.0)))
     (* (* (- 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 <= -2e+20) || !(t_0 <= 20000.0)) {
		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.
real(8) function code(x, y, z)
    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+20)) .or. (.not. (t_0 <= 20000.0d0))) 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 <= -2e+20) || !(t_0 <= 20000.0)) {
		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 <= -2e+20) or not (t_0 <= 20000.0):
		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 <= -2e+20) || !(t_0 <= 20000.0))
		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 <= -2e+20) || ~((t_0 <= 20000.0)))
		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, -2e+20], N[Not[LessEqual[t$95$0, 20000.0]], $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)) < -2e20 or 2e4 < (-.f64 #s(literal 1 binary64) (*.f64 y z))

   1. Initial program 90.0%

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

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower--.f64 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 52.7

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

   4. Applied rewrites 52.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 52.7

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

   6. Applied rewrites 52.7%

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

   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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 94.7

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

   9. Applied rewrites 94.7%

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

   if -2e20 < (-.f64 #s(literal 1 binary64) (*.f64 y z)) < 2e4

   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 98.0%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \cdot z \leq -2 \cdot 10^{+20} \lor \neg \left(1 - y \cdot z \leq 20000\right):\\ \;\;\;\;\left(\left(-y\right) \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \cdot z \leq -10000 \lor \neg \left(y \cdot z \leq 2 \cdot 10^{-7}\right):\\ \;\;\;\;\left(x \cdot z\right) \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= (* y z) -10000.0) (not (<= (* y z) 2e-7)))
   (* (* x z) (- y))
   (* x 1.0)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (((y * z) <= -10000.0) || !((y * z) <= 2e-7)) {
		tmp = (x * z) * -y;
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((y * z) <= (-10000.0d0)) .or. (.not. ((y * z) <= 2d-7))) then
        tmp = (x * z) * -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 tmp;
	if (((y * z) <= -10000.0) || !((y * z) <= 2e-7)) {
		tmp = (x * z) * -y;
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if ((y * z) <= -10000.0) or not ((y * z) <= 2e-7):
		tmp = (x * z) * -y
	else:
		tmp = x * 1.0
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if ((Float64(y * z) <= -10000.0) || !(Float64(y * z) <= 2e-7))
		tmp = Float64(Float64(x * z) * Float64(-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)
	tmp = 0.0;
	if (((y * z) <= -10000.0) || ~(((y * z) <= 2e-7)))
		tmp = (x * z) * -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_] := If[Or[LessEqual[N[(y * z), $MachinePrecision], -10000.0], N[Not[LessEqual[N[(y * z), $MachinePrecision], 2e-7]], $MachinePrecision]], N[(N[(x * z), $MachinePrecision] * (-y)), $MachinePrecision], N[(x * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -1e4 or 1.9999999999999999e-7 < (*.f64 y z)

   1. Initial program 90.1%

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

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower--.f64 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 53.1

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

   4. Applied rewrites 53.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 53.1

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

   6. Applied rewrites 53.1%

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

   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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 94.3

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

   9. Applied rewrites 94.3%

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

   11. Applied rewrites 91.0%

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

   if -1e4 < (*.f64 y z) < 1.9999999999999999e-7

   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 98.7%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -10000 \lor \neg \left(y \cdot z \leq 2 \cdot 10^{-7}\right):\\ \;\;\;\;\left(x \cdot z\right) \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(-y\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x 1.5e-23) (fma (* x (- y)) z x) (* x (- 1.0 (* y z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.5e-23) {
		tmp = fma((x * -y), z, x);
	} else {
		tmp = x * (1.0 - (y * z));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (x <= 1.5e-23)
		tmp = fma(Float64(x * Float64(-y)), z, x);
	else
		tmp = Float64(x * Float64(1.0 - Float64(y * z)));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[x, 1.5e-23], N[(N[(x * (-y)), $MachinePrecision] * z + x), $MachinePrecision], N[(x * N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.50000000000000001e-23

   1. Initial program 93.3%

      \[x \cdot \left(1 - y \cdot z\right) \]
   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. associate-*r* N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 94.9

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

   4. Applied rewrites 94.9%

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

   if 1.50000000000000001e-23 < x

   1. Initial program 99.9%

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

Alternative 5: 51.9% 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.
real(8) function code(x, y, z)
    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 95.1%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 51.4%

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

Reproduce

herbie shell --seed 2024338 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))