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

Percentage Accurate: 95.8% → 99.8%

Time: 7.0s

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: 95.8% 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: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \cdot z \leq -\infty \lor \neg \left(y \cdot z \leq 2 \cdot 10^{+193}\right):\\ \;\;\;\;\left(-y\right) \cdot \left(z \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= (* y z) (- INFINITY)) (not (<= (* y z) 2e+193)))
   (* (- 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 (((y * z) <= -((double) INFINITY)) || !((y * z) <= 2e+193)) {
		tmp = -y * (z * x);
	} else {
		tmp = x * (1.0 - (y * z));
	}
	return tmp;
}

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (((y * z) <= -Double.POSITIVE_INFINITY) || !((y * z) <= 2e+193)) {
		tmp = -y * (z * x);
	} else {
		tmp = x * (1.0 - (y * z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if ((y * z) <= -math.inf) or not ((y * z) <= 2e+193):
		tmp = -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 ((Float64(y * z) <= Float64(-Inf)) || !(Float64(y * z) <= 2e+193))
		tmp = Float64(Float64(-y) * Float64(z * x));
	else
		tmp = Float64(x * Float64(1.0 - Float64(y * z)));
	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) <= -Inf) || ~(((y * z) <= 2e+193)))
		tmp = -y * (z * x);
	else
		tmp = x * (1.0 - (y * 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_] := If[Or[LessEqual[N[(y * z), $MachinePrecision], (-Infinity)], N[Not[LessEqual[N[(y * z), $MachinePrecision], 2e+193]], $MachinePrecision]], N[((-y) * N[(z * x), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -inf.0 or 2.00000000000000013e193 < (*.f64 y z)

   1. Initial program 76.2%

      \[x \cdot \left(1 - y \cdot z\right) \]
   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. +-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 99.9

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

   4. Applied rewrites 99.9%

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

   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. mul-1-neg N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if -inf.0 < (*.f64 y z) < 2.00000000000000013e193

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -\infty \lor \neg \left(y \cdot z \leq 2 \cdot 10^{+193}\right):\\ \;\;\;\;\left(-y\right) \cdot \left(z \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \cdot z \leq -500000 \lor \neg \left(y \cdot z \leq 1\right):\\ \;\;\;\;\left(-y\right) \cdot \left(z \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \end{array} \]

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) -500000.0) (not (<= (* y z) 1.0)))
   (* (- y) (* z x))
   (* x 1.0)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (((y * z) <= -500000.0) || !((y * z) <= 1.0)) {
		tmp = -y * (z * x);
	} 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) <= (-500000.0d0)) .or. (.not. ((y * z) <= 1.0d0))) then
        tmp = -y * (z * x)
    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) <= -500000.0) || !((y * z) <= 1.0)) {
		tmp = -y * (z * x);
	} 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) <= -500000.0) or not ((y * z) <= 1.0):
		tmp = -y * (z * x)
	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) <= -500000.0) || !(Float64(y * z) <= 1.0))
		tmp = Float64(Float64(-y) * Float64(z * x));
	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) <= -500000.0) || ~(((y * z) <= 1.0)))
		tmp = -y * (z * x);
	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], -500000.0], N[Not[LessEqual[N[(y * z), $MachinePrecision], 1.0]], $MachinePrecision]], N[((-y) * N[(z * x), $MachinePrecision]), $MachinePrecision], N[(x * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -5e5 or 1 < (*.f64 y z)

   1. Initial program 91.3%

      \[x \cdot \left(1 - y \cdot z\right) \]
   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 92.2

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

   4. Applied rewrites 92.2%

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

   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. mul-1-neg N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 89.8

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

   7. Applied rewrites 89.8%

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

   if -5e5 < (*.f64 y z) < 1

   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 96.3%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -500000 \lor \neg \left(y \cdot z \leq 1\right):\\ \;\;\;\;\left(-y\right) \cdot \left(z \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \cdot z \leq -500000:\\ \;\;\;\;\left(x \cdot \left(-y\right)\right) \cdot z\\ \mathbf{elif}\;y \cdot z \leq 1:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \left(z \cdot x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* y z) -500000.0)
   (* (* x (- y)) z)
   (if (<= (* y z) 1.0) (* x 1.0) (* (- y) (* z x)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((y * z) <= -500000.0) {
		tmp = (x * -y) * z;
	} else if ((y * z) <= 1.0) {
		tmp = x * 1.0;
	} else {
		tmp = -y * (z * x);
	}
	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) <= (-500000.0d0)) then
        tmp = (x * -y) * z
    else if ((y * z) <= 1.0d0) then
        tmp = x * 1.0d0
    else
        tmp = -y * (z * x)
    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) <= -500000.0) {
		tmp = (x * -y) * z;
	} else if ((y * z) <= 1.0) {
		tmp = x * 1.0;
	} else {
		tmp = -y * (z * x);
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if (y * z) <= -500000.0:
		tmp = (x * -y) * z
	elif (y * z) <= 1.0:
		tmp = x * 1.0
	else:
		tmp = -y * (z * x)
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(y * z) <= -500000.0)
		tmp = Float64(Float64(x * Float64(-y)) * z);
	elseif (Float64(y * z) <= 1.0)
		tmp = Float64(x * 1.0);
	else
		tmp = Float64(Float64(-y) * Float64(z * x));
	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) <= -500000.0)
		tmp = (x * -y) * z;
	elseif ((y * z) <= 1.0)
		tmp = x * 1.0;
	else
		tmp = -y * (z * x);
	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[LessEqual[N[(y * z), $MachinePrecision], -500000.0], N[(N[(x * (-y)), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[N[(y * z), $MachinePrecision], 1.0], N[(x * 1.0), $MachinePrecision], N[((-y) * N[(z * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y z) < -5e5

   1. Initial program 91.5%

      \[x \cdot \left(1 - y \cdot z\right) \]
   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. 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 91.7

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

   4. Applied rewrites 91.7%

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

   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. mul-1-neg N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 88.9

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

   7. Applied rewrites 88.9%

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

   9. Applied rewrites 90.7%

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

   if -5e5 < (*.f64 y z) < 1

   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 96.3%

      \[\leadsto x \cdot \color{blue}{1} \]

   if 1 < (*.f64 y z)

   1. Initial program 91.1%

      \[x \cdot \left(1 - y \cdot z\right) \]
   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. 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 92.6

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

   4. Applied rewrites 92.6%

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

   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. mul-1-neg N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 90.5

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

   7. Applied rewrites 90.5%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -500000:\\ \;\;\;\;\left(x \cdot \left(-y\right)\right) \cdot z\\ \mathbf{elif}\;y \cdot z \leq 1:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \left(z \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(-y\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x 2e+21) (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 <= 2e+21) {
		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 <= 2e+21)
		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, 2e+21], 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 < 2e21

   1. Initial program 94.6%

      \[x \cdot \left(1 - y \cdot z\right) \]
   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. 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 94.2

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

   4. Applied rewrites 94.2%

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

   if 2e21 < x

   1. Initial program 100.0%

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

Alternative 5: 50.2% 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 95.8%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 51.7%

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

Reproduce

herbie shell --seed 2024332 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))