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

Percentage Accurate: 96.0% → 99.8%

Time: 7.4s

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: 96.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := \left(\left(-x\right) \cdot y\right) \cdot z\\ \mathbf{if}\;z \cdot y \leq -1 \cdot 10^{+245}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \cdot y \leq 10^{+236}:\\ \;\;\;\;x - x \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 (<= (* z y) -1e+245)
     t_0
     (if (<= (* z y) 1e+236) (- x (* x (* z y))) 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 ((z * y) <= -1e+245) {
		tmp = t_0;
	} else if ((z * y) <= 1e+236) {
		tmp = x - (x * (z * y));
	} 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 ((z * y) <= (-1d+245)) then
        tmp = t_0
    else if ((z * y) <= 1d+236) then
        tmp = x - (x * (z * y))
    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 ((z * y) <= -1e+245) {
		tmp = t_0;
	} else if ((z * y) <= 1e+236) {
		tmp = x - (x * (z * y));
	} 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 (z * y) <= -1e+245:
		tmp = t_0
	elif (z * y) <= 1e+236:
		tmp = x - (x * (z * y))
	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(z * y) <= -1e+245)
		tmp = t_0;
	elseif (Float64(z * y) <= 1e+236)
		tmp = Float64(x - Float64(x * Float64(z * y)));
	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 ((z * y) <= -1e+245)
		tmp = t_0;
	elseif ((z * y) <= 1e+236)
		tmp = x - (x * (z * y));
	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[(z * y), $MachinePrecision], -1e+245], t$95$0, If[LessEqual[N[(z * y), $MachinePrecision], 1e+236], N[(x - N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -1.00000000000000004e245 or 1.00000000000000005e236 < (*.f64 y z)

   1. Initial program 78.5%

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

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

      8. associate-*r* N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(0 - y\right)} \cdot x, z, x\right) \]

      5. flip3-- N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{0}^{3} - {y}^{3}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}} \cdot x, z, x\right) \]

      6. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}}, z, x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{\color{blue}{0} + \left(y \cdot y + 0 \cdot y\right)}, z, x\right) \]

      8. +-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{\color{blue}{y \cdot y + 0 \cdot y}}, z, x\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{\color{blue}{y \cdot \left(y + 0\right)}}, z, x\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{y \cdot \color{blue}{\left(0 + y\right)}}, z, x\right) \]

      11. +-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{y \cdot \color{blue}{y}}, z, x\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{y \cdot y}}, z, x\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left({0}^{3} - {y}^{3}\right) \cdot x}}{y \cdot y}, z, x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(\color{blue}{0} - {y}^{3}\right) \cdot x}{y \cdot y}, z, x\right) \]

      15. sub0-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(\mathsf{neg}\left({y}^{3}\right)\right)} \cdot x}{y \cdot y}, z, x\right) \]

      16. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(-{y}^{3}\right)} \cdot x}{y \cdot y}, z, x\right) \]

      17. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(-\color{blue}{{y}^{3}}\right) \cdot x}{y \cdot y}, z, x\right) \]

      18. lower-*.f64 34.0

          \[\leadsto \mathsf{fma}\left(\frac{\left(-{y}^{3}\right) \cdot x}{\color{blue}{y \cdot y}}, z, x\right) \]

   6. Applied rewrites 34.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\left(-{y}^{3}\right) \cdot x}{y \cdot y}}, z, x\right) \]

   7. Taylor expanded in z around inf

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

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

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

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

      5. mul-1-neg 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. *-commutative N/A

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

      10. lower-*.f64 99.9

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

   9. Applied rewrites 99.9%

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

   if -1.00000000000000004e245 < (*.f64 y z) < 1.00000000000000005e236

   1. Initial program 99.8%

      \[x \cdot \left(1 - y \cdot z\right) \]
   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. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

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

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 99.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot y \leq -1 \cdot 10^{+245}:\\ \;\;\;\;\left(\left(-x\right) \cdot y\right) \cdot z\\ \mathbf{elif}\;z \cdot y \leq 10^{+236}:\\ \;\;\;\;x - x \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x\right) \cdot y\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := \left(\left(-x\right) \cdot y\right) \cdot z\\ \mathbf{if}\;z \cdot y \leq -1 \cdot 10^{+245}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \cdot y \leq 10^{+236}:\\ \;\;\;\;\left(1 - z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 (<= (* z y) -1e+245)
     t_0
     (if (<= (* z y) 1e+236) (* (- 1.0 (* z y)) 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 ((z * y) <= -1e+245) {
		tmp = t_0;
	} else if ((z * y) <= 1e+236) {
		tmp = (1.0 - (z * y)) * 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 ((z * y) <= (-1d+245)) then
        tmp = t_0
    else if ((z * y) <= 1d+236) then
        tmp = (1.0d0 - (z * y)) * 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 ((z * y) <= -1e+245) {
		tmp = t_0;
	} else if ((z * y) <= 1e+236) {
		tmp = (1.0 - (z * y)) * 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 (z * y) <= -1e+245:
		tmp = t_0
	elif (z * y) <= 1e+236:
		tmp = (1.0 - (z * y)) * 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(z * y) <= -1e+245)
		tmp = t_0;
	elseif (Float64(z * y) <= 1e+236)
		tmp = Float64(Float64(1.0 - Float64(z * y)) * 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 ((z * y) <= -1e+245)
		tmp = t_0;
	elseif ((z * y) <= 1e+236)
		tmp = (1.0 - (z * y)) * 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[(z * y), $MachinePrecision], -1e+245], t$95$0, If[LessEqual[N[(z * y), $MachinePrecision], 1e+236], N[(N[(1.0 - N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -1.00000000000000004e245 or 1.00000000000000005e236 < (*.f64 y z)

   1. Initial program 78.5%

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

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

      8. associate-*r* N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(0 - y\right)} \cdot x, z, x\right) \]

      5. flip3-- N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{0}^{3} - {y}^{3}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}} \cdot x, z, x\right) \]

      6. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}}, z, x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{\color{blue}{0} + \left(y \cdot y + 0 \cdot y\right)}, z, x\right) \]

      8. +-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{\color{blue}{y \cdot y + 0 \cdot y}}, z, x\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{\color{blue}{y \cdot \left(y + 0\right)}}, z, x\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{y \cdot \color{blue}{\left(0 + y\right)}}, z, x\right) \]

      11. +-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{y \cdot \color{blue}{y}}, z, x\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{y \cdot y}}, z, x\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left({0}^{3} - {y}^{3}\right) \cdot x}}{y \cdot y}, z, x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(\color{blue}{0} - {y}^{3}\right) \cdot x}{y \cdot y}, z, x\right) \]

      15. sub0-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(\mathsf{neg}\left({y}^{3}\right)\right)} \cdot x}{y \cdot y}, z, x\right) \]

      16. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(-{y}^{3}\right)} \cdot x}{y \cdot y}, z, x\right) \]

      17. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(-\color{blue}{{y}^{3}}\right) \cdot x}{y \cdot y}, z, x\right) \]

      18. lower-*.f64 34.0

          \[\leadsto \mathsf{fma}\left(\frac{\left(-{y}^{3}\right) \cdot x}{\color{blue}{y \cdot y}}, z, x\right) \]

   6. Applied rewrites 34.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\left(-{y}^{3}\right) \cdot x}{y \cdot y}}, z, x\right) \]

   7. Taylor expanded in z around inf

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

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

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

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

      5. mul-1-neg 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. *-commutative N/A

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

      10. lower-*.f64 99.9

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

   9. Applied rewrites 99.9%

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

   if -1.00000000000000004e245 < (*.f64 y z) < 1.00000000000000005e236

   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 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot y \leq -1 \cdot 10^{+245}:\\ \;\;\;\;\left(\left(-x\right) \cdot y\right) \cdot z\\ \mathbf{elif}\;z \cdot y \leq 10^{+236}:\\ \;\;\;\;\left(1 - z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x\right) \cdot y\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot y \leq -10000:\\ \;\;\;\;\left(\left(-z\right) \cdot x\right) \cdot y\\ \mathbf{elif}\;z \cdot y \leq 0.005:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x\right) \cdot y\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z y) -10000.0)
   (* (* (- z) x) y)
   (if (<= (* z y) 0.005) (* 1.0 x) (* (* (- x) y) z))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((z * y) <= -10000.0) {
		tmp = (-z * x) * y;
	} else if ((z * y) <= 0.005) {
		tmp = 1.0 * x;
	} else {
		tmp = (-x * y) * 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) :: tmp
    if ((z * y) <= (-10000.0d0)) then
        tmp = (-z * x) * y
    else if ((z * y) <= 0.005d0) then
        tmp = 1.0d0 * x
    else
        tmp = (-x * y) * z
    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 ((z * y) <= -10000.0) {
		tmp = (-z * x) * y;
	} else if ((z * y) <= 0.005) {
		tmp = 1.0 * x;
	} else {
		tmp = (-x * y) * z;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if (z * y) <= -10000.0:
		tmp = (-z * x) * y
	elif (z * y) <= 0.005:
		tmp = 1.0 * x
	else:
		tmp = (-x * y) * z
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * y) <= -10000.0)
		tmp = Float64(Float64(Float64(-z) * x) * y);
	elseif (Float64(z * y) <= 0.005)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(Float64(Float64(-x) * 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 ((z * y) <= -10000.0)
		tmp = (-z * x) * y;
	elseif ((z * y) <= 0.005)
		tmp = 1.0 * x;
	else
		tmp = (-x * 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[LessEqual[N[(z * y), $MachinePrecision], -10000.0], N[(N[((-z) * x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[N[(z * y), $MachinePrecision], 0.005], N[(1.0 * x), $MachinePrecision], N[(N[((-x) * y), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y z) < -1e4

   1. Initial program 92.3%

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

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

      8. associate-*r* N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 95.3

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

   4. Applied rewrites 95.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(0 - y\right)} \cdot x, z, x\right) \]

      5. flip3-- N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{0}^{3} - {y}^{3}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}} \cdot x, z, x\right) \]

      6. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}}, z, x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{\color{blue}{0} + \left(y \cdot y + 0 \cdot y\right)}, z, x\right) \]

      8. +-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{\color{blue}{y \cdot y + 0 \cdot y}}, z, x\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{\color{blue}{y \cdot \left(y + 0\right)}}, z, x\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{y \cdot \color{blue}{\left(0 + y\right)}}, z, x\right) \]

      11. +-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{y \cdot \color{blue}{y}}, z, x\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{y \cdot y}}, z, x\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left({0}^{3} - {y}^{3}\right) \cdot x}}{y \cdot y}, z, x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(\color{blue}{0} - {y}^{3}\right) \cdot x}{y \cdot y}, z, x\right) \]

      15. sub0-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(\mathsf{neg}\left({y}^{3}\right)\right)} \cdot x}{y \cdot y}, z, x\right) \]

      16. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(-{y}^{3}\right)} \cdot x}{y \cdot y}, z, x\right) \]

      17. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(-\color{blue}{{y}^{3}}\right) \cdot x}{y \cdot y}, z, x\right) \]

      18. lower-*.f64 34.4

          \[\leadsto \mathsf{fma}\left(\frac{\left(-{y}^{3}\right) \cdot x}{\color{blue}{y \cdot y}}, z, x\right) \]

   6. Applied rewrites 34.4%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\left(-{y}^{3}\right) \cdot x}{y \cdot y}}, z, x\right) \]

   7. Taylor expanded in z around inf

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

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

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

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

      5. mul-1-neg 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. *-commutative N/A

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

      10. lower-*.f64 93.8

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

   9. Applied rewrites 93.8%

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

   11. Applied rewrites 86.6%

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

   if -1e4 < (*.f64 y z) < 0.0050000000000000001

   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 97.9%

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

   if 0.0050000000000000001 < (*.f64 y z)

   1. Initial program 92.1%

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

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

      8. associate-*r* N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 91.6

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

   4. Applied rewrites 91.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(0 - y\right)} \cdot x, z, x\right) \]

      5. flip3-- N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{0}^{3} - {y}^{3}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}} \cdot x, z, x\right) \]

      6. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}}, z, x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{\color{blue}{0} + \left(y \cdot y + 0 \cdot y\right)}, z, x\right) \]

      8. +-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{\color{blue}{y \cdot y + 0 \cdot y}}, z, x\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{\color{blue}{y \cdot \left(y + 0\right)}}, z, x\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{y \cdot \color{blue}{\left(0 + y\right)}}, z, x\right) \]

      11. +-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{y \cdot \color{blue}{y}}, z, x\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{y \cdot y}}, z, x\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left({0}^{3} - {y}^{3}\right) \cdot x}}{y \cdot y}, z, x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(\color{blue}{0} - {y}^{3}\right) \cdot x}{y \cdot y}, z, x\right) \]

      15. sub0-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(\mathsf{neg}\left({y}^{3}\right)\right)} \cdot x}{y \cdot y}, z, x\right) \]

      16. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(-{y}^{3}\right)} \cdot x}{y \cdot y}, z, x\right) \]

      17. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(-\color{blue}{{y}^{3}}\right) \cdot x}{y \cdot y}, z, x\right) \]

      18. lower-*.f64 30.6

          \[\leadsto \mathsf{fma}\left(\frac{\left(-{y}^{3}\right) \cdot x}{\color{blue}{y \cdot y}}, z, x\right) \]

   6. Applied rewrites 30.6%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\left(-{y}^{3}\right) \cdot x}{y \cdot y}}, z, x\right) \]

   7. Taylor expanded in z around inf

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

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

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

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

      5. mul-1-neg 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. *-commutative N/A

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

      10. lower-*.f64 88.8

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

   9. Applied rewrites 88.8%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot y \leq -10000:\\ \;\;\;\;\left(\left(-z\right) \cdot x\right) \cdot y\\ \mathbf{elif}\;z \cdot y \leq 0.005:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x\right) \cdot y\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := \left(\left(-x\right) \cdot y\right) \cdot z\\ \mathbf{if}\;z \cdot y \leq -100000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \cdot y \leq 0.005:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 (<= (* z y) -100000.0) t_0 (if (<= (* z y) 0.005) (* 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 ((z * y) <= -100000.0) {
		tmp = t_0;
	} else if ((z * y) <= 0.005) {
		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 ((z * y) <= (-100000.0d0)) then
        tmp = t_0
    else if ((z * y) <= 0.005d0) 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 ((z * y) <= -100000.0) {
		tmp = t_0;
	} else if ((z * y) <= 0.005) {
		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 (z * y) <= -100000.0:
		tmp = t_0
	elif (z * y) <= 0.005:
		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(z * y) <= -100000.0)
		tmp = t_0;
	elseif (Float64(z * y) <= 0.005)
		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 ((z * y) <= -100000.0)
		tmp = t_0;
	elseif ((z * y) <= 0.005)
		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[(z * y), $MachinePrecision], -100000.0], t$95$0, If[LessEqual[N[(z * y), $MachinePrecision], 0.005], N[(1.0 * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -1e5 or 0.0050000000000000001 < (*.f64 y z)

   1. Initial program 92.1%

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

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

      8. associate-*r* N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 93.1

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

   4. Applied rewrites 93.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(0 - y\right)} \cdot x, z, x\right) \]

      5. flip3-- N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{0}^{3} - {y}^{3}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}} \cdot x, z, x\right) \]

      6. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}}, z, x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{\color{blue}{0} + \left(y \cdot y + 0 \cdot y\right)}, z, x\right) \]

      8. +-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{\color{blue}{y \cdot y + 0 \cdot y}}, z, x\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{\color{blue}{y \cdot \left(y + 0\right)}}, z, x\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{y \cdot \color{blue}{\left(0 + y\right)}}, z, x\right) \]

      11. +-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{y \cdot \color{blue}{y}}, z, x\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{y \cdot y}}, z, x\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left({0}^{3} - {y}^{3}\right) \cdot x}}{y \cdot y}, z, x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(\color{blue}{0} - {y}^{3}\right) \cdot x}{y \cdot y}, z, x\right) \]

      15. sub0-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(\mathsf{neg}\left({y}^{3}\right)\right)} \cdot x}{y \cdot y}, z, x\right) \]

      16. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(-{y}^{3}\right)} \cdot x}{y \cdot y}, z, x\right) \]

      17. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(-\color{blue}{{y}^{3}}\right) \cdot x}{y \cdot y}, z, x\right) \]

      18. lower-*.f64 32.4

          \[\leadsto \mathsf{fma}\left(\frac{\left(-{y}^{3}\right) \cdot x}{\color{blue}{y \cdot y}}, z, x\right) \]

   6. Applied rewrites 32.4%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\left(-{y}^{3}\right) \cdot x}{y \cdot y}}, z, x\right) \]

   7. Taylor expanded in z around inf

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

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

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

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

      5. mul-1-neg 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. *-commutative N/A

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

      10. lower-*.f64 91.2

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

   9. Applied rewrites 91.2%

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

   if -1e5 < (*.f64 y z) < 0.0050000000000000001

   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 97.1%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot y \leq -100000:\\ \;\;\;\;\left(\left(-x\right) \cdot y\right) \cdot z\\ \mathbf{elif}\;z \cdot y \leq 0.005:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x\right) \cdot y\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.6% 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 95.4%

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

3. Taylor expanded in z around 0

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

5. Applied rewrites 42.8%

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

6. Final simplification 42.8%

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

Reproduce

herbie shell --seed 2024276 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))