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

Percentage Accurate: 96.1% → 99.0%

Time: 8.3s

Alternatives: 5

Speedup: 0.4×

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: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := \left(\left(-x\right) \cdot y\right) \cdot z\\ \mathbf{if}\;y \cdot z \leq -1 \cdot 10^{+202}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \cdot z \leq 2 \cdot 10^{+83}:\\ \;\;\;\;\left(1 - y \cdot z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* (- x) y) z)))
   (if (<= (* y z) -1e+202)
     t_0
     (if (<= (* y z) 2e+83) (* (- 1.0 (* y z)) x) t_0))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = (-x * y) * z;
	double tmp;
	if ((y * z) <= -1e+202) {
		tmp = t_0;
	} else if ((y * z) <= 2e+83) {
		tmp = (1.0 - (y * z)) * x;
	} else {
		tmp = t_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 = (-x * y) * z
    if ((y * z) <= (-1d+202)) then
        tmp = t_0
    else if ((y * z) <= 2d+83) then
        tmp = (1.0d0 - (y * z)) * x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double t_0 = (-x * y) * z;
	double tmp;
	if ((y * z) <= -1e+202) {
		tmp = t_0;
	} else if ((y * z) <= 2e+83) {
		tmp = (1.0 - (y * z)) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	t_0 = (-x * y) * z
	tmp = 0
	if (y * z) <= -1e+202:
		tmp = t_0
	elif (y * z) <= 2e+83:
		tmp = (1.0 - (y * z)) * x
	else:
		tmp = t_0
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(Float64(Float64(-x) * y) * z)
	tmp = 0.0
	if (Float64(y * z) <= -1e+202)
		tmp = t_0;
	elseif (Float64(y * z) <= 2e+83)
		tmp = Float64(Float64(1.0 - Float64(y * z)) * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	t_0 = (-x * y) * z;
	tmp = 0.0;
	if ((y * z) <= -1e+202)
		tmp = t_0;
	elseif ((y * z) <= 2e+83)
		tmp = (1.0 - (y * z)) * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(N[((-x) * y), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[N[(y * z), $MachinePrecision], -1e+202], t$95$0, If[LessEqual[N[(y * z), $MachinePrecision], 2e+83], N[(N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -9.999999999999999e201 or 2.00000000000000006e83 < (*.f64 y z)

   1. Initial program 85.1%

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

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-*r* N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 98.7

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

   4. Applied rewrites 98.7%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-neg.f64 N/A

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

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

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

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

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

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

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

      9. lift-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. flip-+ N/A

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

      15. clear-num N/A

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

      16. lower-/.f64 N/A

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

      17. clear-num N/A

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

   6. Applied rewrites 99.2%

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

   7. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. associate-/r* N/A

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

      3. metadata-eval N/A

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

      4. distribute-neg-frac N/A

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

      5. lower-/.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 98.7

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

   9. Applied rewrites 98.7%

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

   10. Taylor expanded in z around inf

       \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right)} \]
   11. 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. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. mul-1-neg N/A

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

       8. lower-neg.f64 N/A

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

       9. lower-*.f64 99.3

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

   12. Applied rewrites 99.3%

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

   if -9.999999999999999e201 < (*.f64 y z) < 2.00000000000000006e83

   1. Initial program 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -1 \cdot 10^{+202}:\\ \;\;\;\;\left(\left(-x\right) \cdot y\right) \cdot z\\ \mathbf{elif}\;y \cdot z \leq 2 \cdot 10^{+83}:\\ \;\;\;\;\left(1 - y \cdot z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x\right) \cdot y\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := \left(\left(-x\right) \cdot y\right) \cdot z\\ \mathbf{if}\;y \cdot z \leq -1 \cdot 10^{+202}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \cdot z \leq -5000000000:\\ \;\;\;\;\left(y \cdot \left(-z\right)\right) \cdot x\\ \mathbf{elif}\;y \cdot z \leq 0.02:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* (- x) y) z)))
   (if (<= (* y z) -1e+202)
     t_0
     (if (<= (* y z) -5000000000.0)
       (* (* y (- z)) x)
       (if (<= (* y z) 0.02) (* 1.0 x) t_0)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = (-x * y) * z;
	double tmp;
	if ((y * z) <= -1e+202) {
		tmp = t_0;
	} else if ((y * z) <= -5000000000.0) {
		tmp = (y * -z) * x;
	} else if ((y * z) <= 0.02) {
		tmp = 1.0 * x;
	} else {
		tmp = t_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 = (-x * y) * z
    if ((y * z) <= (-1d+202)) then
        tmp = t_0
    else if ((y * z) <= (-5000000000.0d0)) then
        tmp = (y * -z) * x
    else if ((y * z) <= 0.02d0) then
        tmp = 1.0d0 * x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double t_0 = (-x * y) * z;
	double tmp;
	if ((y * z) <= -1e+202) {
		tmp = t_0;
	} else if ((y * z) <= -5000000000.0) {
		tmp = (y * -z) * x;
	} else if ((y * z) <= 0.02) {
		tmp = 1.0 * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	t_0 = (-x * y) * z
	tmp = 0
	if (y * z) <= -1e+202:
		tmp = t_0
	elif (y * z) <= -5000000000.0:
		tmp = (y * -z) * x
	elif (y * z) <= 0.02:
		tmp = 1.0 * x
	else:
		tmp = t_0
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(Float64(Float64(-x) * y) * z)
	tmp = 0.0
	if (Float64(y * z) <= -1e+202)
		tmp = t_0;
	elseif (Float64(y * z) <= -5000000000.0)
		tmp = Float64(Float64(y * Float64(-z)) * x);
	elseif (Float64(y * z) <= 0.02)
		tmp = Float64(1.0 * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	t_0 = (-x * y) * z;
	tmp = 0.0;
	if ((y * z) <= -1e+202)
		tmp = t_0;
	elseif ((y * z) <= -5000000000.0)
		tmp = (y * -z) * x;
	elseif ((y * z) <= 0.02)
		tmp = 1.0 * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(N[((-x) * y), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[N[(y * z), $MachinePrecision], -1e+202], t$95$0, If[LessEqual[N[(y * z), $MachinePrecision], -5000000000.0], N[(N[(y * (-z)), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[N[(y * z), $MachinePrecision], 0.02], N[(1.0 * x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y z) < -9.999999999999999e201 or 0.0200000000000000004 < (*.f64 y z)

   1. Initial program 87.9%

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

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-*r* N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 94.1

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

   4. Applied rewrites 94.1%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-neg.f64 N/A

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

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

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

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

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

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

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

      9. lift-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. flip-+ N/A

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

      15. clear-num N/A

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

      16. lower-/.f64 N/A

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

      17. clear-num N/A

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

   6. Applied rewrites 96.6%

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

   7. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. associate-/r* N/A

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

      3. metadata-eval N/A

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

      4. distribute-neg-frac N/A

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

      5. lower-/.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 93.9

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

   9. Applied rewrites 93.9%

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

   10. Taylor expanded in z around inf

       \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right)} \]
   11. 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. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. mul-1-neg N/A

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

       8. lower-neg.f64 N/A

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

       9. lower-*.f64 95.1

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

   12. Applied rewrites 95.1%

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

   if -9.999999999999999e201 < (*.f64 y z) < -5e9

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 97.9

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

   5. Applied rewrites 97.9%

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

   if -5e9 < (*.f64 y z) < 0.0200000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 96.0%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -1 \cdot 10^{+202}:\\ \;\;\;\;\left(\left(-x\right) \cdot y\right) \cdot z\\ \mathbf{elif}\;y \cdot z \leq -5000000000:\\ \;\;\;\;\left(y \cdot \left(-z\right)\right) \cdot x\\ \mathbf{elif}\;y \cdot z \leq 0.02:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x\right) \cdot y\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := \left(\left(-x\right) \cdot y\right) \cdot z\\ \mathbf{if}\;y \cdot z \leq -5000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \cdot z \leq 0.02:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* (- x) y) z)))
   (if (<= (* y z) -5000000000.0) t_0 (if (<= (* y z) 0.02) (* 1.0 x) t_0))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = (-x * y) * z;
	double tmp;
	if ((y * z) <= -5000000000.0) {
		tmp = t_0;
	} else if ((y * z) <= 0.02) {
		tmp = 1.0 * x;
	} else {
		tmp = t_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 = (-x * y) * z
    if ((y * z) <= (-5000000000.0d0)) then
        tmp = t_0
    else if ((y * z) <= 0.02d0) then
        tmp = 1.0d0 * x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double t_0 = (-x * y) * z;
	double tmp;
	if ((y * z) <= -5000000000.0) {
		tmp = t_0;
	} else if ((y * z) <= 0.02) {
		tmp = 1.0 * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	t_0 = (-x * y) * z
	tmp = 0
	if (y * z) <= -5000000000.0:
		tmp = t_0
	elif (y * z) <= 0.02:
		tmp = 1.0 * x
	else:
		tmp = t_0
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(Float64(Float64(-x) * y) * z)
	tmp = 0.0
	if (Float64(y * z) <= -5000000000.0)
		tmp = t_0;
	elseif (Float64(y * z) <= 0.02)
		tmp = Float64(1.0 * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	t_0 = (-x * y) * z;
	tmp = 0.0;
	if ((y * z) <= -5000000000.0)
		tmp = t_0;
	elseif ((y * z) <= 0.02)
		tmp = 1.0 * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(N[((-x) * y), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[N[(y * z), $MachinePrecision], -5000000000.0], t$95$0, If[LessEqual[N[(y * z), $MachinePrecision], 0.02], N[(1.0 * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -5e9 or 0.0200000000000000004 < (*.f64 y z)

   1. Initial program 90.5%

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

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-*r* N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 92.0

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

   4. Applied rewrites 92.0%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-neg.f64 N/A

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

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

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

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

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

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

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

      9. lift-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. flip-+ N/A

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

      15. clear-num N/A

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

      16. lower-/.f64 N/A

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

      17. clear-num N/A

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

   6. Applied rewrites 94.6%

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

   7. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. associate-/r* N/A

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

      3. metadata-eval N/A

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

      4. distribute-neg-frac N/A

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

      5. lower-/.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 92.0

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

   9. Applied rewrites 92.0%

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

   10. Taylor expanded in z around inf

       \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right)} \]
   11. 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. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. mul-1-neg N/A

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

       8. lower-neg.f64 N/A

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

       9. lower-*.f64 93.1

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

   12. Applied rewrites 93.1%

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

   if -5e9 < (*.f64 y z) < 0.0200000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 96.0%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -5000000000:\\ \;\;\;\;\left(\left(-x\right) \cdot y\right) \cdot z\\ \mathbf{elif}\;y \cdot z \leq 0.02:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x\right) \cdot y\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 3.9 \cdot 10^{-132}:\\ \;\;\;\;\mathsf{fma}\left(\left(-z\right) \cdot x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - y \cdot z\right) \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x 3.9e-132) (fma (* (- z) x) y x) (* (- 1.0 (* y z)) x)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (x <= 3.9e-132) {
		tmp = fma((-z * x), y, x);
	} else {
		tmp = (1.0 - (y * z)) * x;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (x <= 3.9e-132)
		tmp = fma(Float64(Float64(-z) * x), y, x);
	else
		tmp = Float64(Float64(1.0 - Float64(y * z)) * x);
	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, 3.9e-132], N[(N[((-z) * x), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.89999999999999982e-132

   1. Initial program 91.7%

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

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-*r* N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 91.5

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

   4. Applied rewrites 91.5%

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

   if 3.89999999999999982e-132 < 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. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.9 \cdot 10^{-132}:\\ \;\;\;\;\mathsf{fma}\left(\left(-z\right) \cdot x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - y \cdot z\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

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 = 1.0d0 * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.7%

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

3. Taylor expanded in z around 0

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

5. Applied rewrites 44.8%

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

6. Final simplification 44.8%

   \[\leadsto 1 \cdot x \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024268 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))