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

Percentage Accurate: 96.1% → 99.7%

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: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := \left(\left(-y\right) \cdot x\right) \cdot z\\ \mathbf{if}\;z \cdot y \leq -2 \cdot 10^{+260}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \cdot y \leq 10^{+202}:\\ \;\;\;\;\left(1 - z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 (* (* (- y) x) z)))
   (if (<= (* z y) -2e+260)
     t_0
     (if (<= (* z y) 1e+202) (* (- 1.0 (* z y)) x) t_0))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = (-y * x) * z;
	double tmp;
	if ((z * y) <= -2e+260) {
		tmp = t_0;
	} else if ((z * y) <= 1e+202) {
		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 = (-y * x) * z
    if ((z * y) <= (-2d+260)) then
        tmp = t_0
    else if ((z * y) <= 1d+202) 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 = (-y * x) * z;
	double tmp;
	if ((z * y) <= -2e+260) {
		tmp = t_0;
	} else if ((z * y) <= 1e+202) {
		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 = (-y * x) * z
	tmp = 0
	if (z * y) <= -2e+260:
		tmp = t_0
	elif (z * y) <= 1e+202:
		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(-y) * x) * z)
	tmp = 0.0
	if (Float64(z * y) <= -2e+260)
		tmp = t_0;
	elseif (Float64(z * y) <= 1e+202)
		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 = (-y * x) * z;
	tmp = 0.0;
	if ((z * y) <= -2e+260)
		tmp = t_0;
	elseif ((z * y) <= 1e+202)
		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[((-y) * x), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[N[(z * y), $MachinePrecision], -2e+260], t$95$0, If[LessEqual[N[(z * y), $MachinePrecision], 1e+202], 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) < -2.00000000000000013e260 or 9.999999999999999e201 < (*.f64 y z)

   1. Initial program 80.4%

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

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

      5. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(x \cdot 1\right)}^{3} + {\left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)}^{3}}{\left(x \cdot 1\right) \cdot \left(x \cdot 1\right) + \left(\left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) - \left(x \cdot 1\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)\right)}} \]

      6. div-inv N/A

         \[\leadsto \color{blue}{\left({\left(x \cdot 1\right)}^{3} + {\left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)}^{3}\right) \cdot \frac{1}{\left(x \cdot 1\right) \cdot \left(x \cdot 1\right) + \left(\left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) - \left(x \cdot 1\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(x \cdot 1\right)}^{3} + {\left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)}^{3}\right) \cdot \frac{1}{\left(x \cdot 1\right) \cdot \left(x \cdot 1\right) + \left(\left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) - \left(x \cdot 1\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)\right)}} \]

   4. Applied rewrites 9.6%

      \[\leadsto \color{blue}{\left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(x, x, {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{2} - x \cdot \left(\left(x \cdot \left(-z\right)\right) \cdot y\right)\right)}} \]

   5. Taylor expanded in z around inf

      \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\color{blue}{{x}^{2} \cdot \left({y}^{2} \cdot {z}^{2}\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\color{blue}{\left({y}^{2} \cdot {z}^{2}\right) \cdot {x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left({y}^{2} \cdot {z}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\color{blue}{\left(\left({y}^{2} \cdot {z}^{2}\right) \cdot x\right) \cdot x}} \]

      4. lower-*.f64 N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\color{blue}{\left(\left({y}^{2} \cdot {z}^{2}\right) \cdot x\right) \cdot x}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\color{blue}{\left(\left({y}^{2} \cdot {z}^{2}\right) \cdot x\right)} \cdot x} \]

      6. unpow2 N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left(\left({y}^{2} \cdot \color{blue}{\left(z \cdot z\right)}\right) \cdot x\right) \cdot x} \]

      7. associate-*r* N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left(\color{blue}{\left(\left({y}^{2} \cdot z\right) \cdot z\right)} \cdot x\right) \cdot x} \]

      8. lower-*.f64 N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left(\color{blue}{\left(\left({y}^{2} \cdot z\right) \cdot z\right)} \cdot x\right) \cdot x} \]

      9. lower-*.f64 N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left(\left(\color{blue}{\left({y}^{2} \cdot z\right)} \cdot z\right) \cdot x\right) \cdot x} \]

      10. unpow2 N/A

          \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left(\left(\left(\color{blue}{\left(y \cdot y\right)} \cdot z\right) \cdot z\right) \cdot x\right) \cdot x} \]

      11. lower-*.f64 0.3

          \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left(\left(\left(\color{blue}{\left(y \cdot y\right)} \cdot z\right) \cdot z\right) \cdot x\right) \cdot x} \]

   7. Applied rewrites 0.3%

      \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\color{blue}{\left(\left(\left(\left(y \cdot y\right) \cdot z\right) \cdot z\right) \cdot x\right) \cdot x}} \]

   8. Taylor expanded in z around inf

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

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

      10. lower-*.f64 99.8

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

   10. Applied rewrites 99.8%

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

   if -2.00000000000000013e260 < (*.f64 y z) < 9.999999999999999e201

   1. Initial program 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot y \leq -2 \cdot 10^{+260}:\\ \;\;\;\;\left(\left(-y\right) \cdot x\right) \cdot z\\ \mathbf{elif}\;z \cdot y \leq 10^{+202}:\\ \;\;\;\;\left(1 - z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-y\right) \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot y \leq -2000:\\ \;\;\;\;\left(\left(-x\right) \cdot z\right) \cdot y\\ \mathbf{elif}\;z \cdot y \leq 1:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;z \cdot y \leq 10^{+202}:\\ \;\;\;\;\left(\left(-z\right) \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-y\right) \cdot x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z y) -2000.0)
   (* (* (- x) z) y)
   (if (<= (* z y) 1.0)
     (* 1.0 x)
     (if (<= (* z y) 1e+202) (* (* (- z) y) x) (* (* (- y) x) z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((z * y) <= -2000.0) {
		tmp = (-x * z) * y;
	} else if ((z * y) <= 1.0) {
		tmp = 1.0 * x;
	} else if ((z * y) <= 1e+202) {
		tmp = (-z * y) * x;
	} else {
		tmp = (-y * x) * z;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * y) <= (-2000.0d0)) then
        tmp = (-x * z) * y
    else if ((z * y) <= 1.0d0) then
        tmp = 1.0d0 * x
    else if ((z * y) <= 1d+202) then
        tmp = (-z * y) * x
    else
        tmp = (-y * x) * z
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * y) <= -2000.0) {
		tmp = (-x * z) * y;
	} else if ((z * y) <= 1.0) {
		tmp = 1.0 * x;
	} else if ((z * y) <= 1e+202) {
		tmp = (-z * y) * x;
	} else {
		tmp = (-y * x) * z;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if (z * y) <= -2000.0:
		tmp = (-x * z) * y
	elif (z * y) <= 1.0:
		tmp = 1.0 * x
	elif (z * y) <= 1e+202:
		tmp = (-z * y) * x
	else:
		tmp = (-y * x) * z
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * y) <= -2000.0)
		tmp = Float64(Float64(Float64(-x) * z) * y);
	elseif (Float64(z * y) <= 1.0)
		tmp = Float64(1.0 * x);
	elseif (Float64(z * y) <= 1e+202)
		tmp = Float64(Float64(Float64(-z) * y) * x);
	else
		tmp = Float64(Float64(Float64(-y) * x) * z);
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * y) <= -2000.0)
		tmp = (-x * z) * y;
	elseif ((z * y) <= 1.0)
		tmp = 1.0 * x;
	elseif ((z * y) <= 1e+202)
		tmp = (-z * y) * x;
	else
		tmp = (-y * x) * z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * y), $MachinePrecision], -2000.0], N[(N[((-x) * z), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[N[(z * y), $MachinePrecision], 1.0], N[(1.0 * x), $MachinePrecision], If[LessEqual[N[(z * y), $MachinePrecision], 1e+202], N[(N[((-z) * y), $MachinePrecision] * x), $MachinePrecision], N[(N[((-y) * x), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 y z) < -2e3

   1. Initial program 86.5%

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

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

      5. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(x \cdot 1\right)}^{3} + {\left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)}^{3}}{\left(x \cdot 1\right) \cdot \left(x \cdot 1\right) + \left(\left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) - \left(x \cdot 1\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)\right)}} \]

      6. div-inv N/A

         \[\leadsto \color{blue}{\left({\left(x \cdot 1\right)}^{3} + {\left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)}^{3}\right) \cdot \frac{1}{\left(x \cdot 1\right) \cdot \left(x \cdot 1\right) + \left(\left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) - \left(x \cdot 1\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(x \cdot 1\right)}^{3} + {\left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)}^{3}\right) \cdot \frac{1}{\left(x \cdot 1\right) \cdot \left(x \cdot 1\right) + \left(\left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) - \left(x \cdot 1\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)\right)}} \]

   4. Applied rewrites 19.9%

      \[\leadsto \color{blue}{\left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(x, x, {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{2} - x \cdot \left(\left(x \cdot \left(-z\right)\right) \cdot y\right)\right)}} \]

   5. Taylor expanded in z around inf

      \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\color{blue}{{x}^{2} \cdot \left({y}^{2} \cdot {z}^{2}\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\color{blue}{\left({y}^{2} \cdot {z}^{2}\right) \cdot {x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left({y}^{2} \cdot {z}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\color{blue}{\left(\left({y}^{2} \cdot {z}^{2}\right) \cdot x\right) \cdot x}} \]

      4. lower-*.f64 N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\color{blue}{\left(\left({y}^{2} \cdot {z}^{2}\right) \cdot x\right) \cdot x}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\color{blue}{\left(\left({y}^{2} \cdot {z}^{2}\right) \cdot x\right)} \cdot x} \]

      6. unpow2 N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left(\left({y}^{2} \cdot \color{blue}{\left(z \cdot z\right)}\right) \cdot x\right) \cdot x} \]

      7. associate-*r* N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left(\color{blue}{\left(\left({y}^{2} \cdot z\right) \cdot z\right)} \cdot x\right) \cdot x} \]

      8. lower-*.f64 N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left(\color{blue}{\left(\left({y}^{2} \cdot z\right) \cdot z\right)} \cdot x\right) \cdot x} \]

      9. lower-*.f64 N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left(\left(\color{blue}{\left({y}^{2} \cdot z\right)} \cdot z\right) \cdot x\right) \cdot x} \]

      10. unpow2 N/A

          \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left(\left(\left(\color{blue}{\left(y \cdot y\right)} \cdot z\right) \cdot z\right) \cdot x\right) \cdot x} \]

      11. lower-*.f64 2.6

          \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left(\left(\left(\color{blue}{\left(y \cdot y\right)} \cdot z\right) \cdot z\right) \cdot x\right) \cdot x} \]

   7. Applied rewrites 2.6%

      \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\color{blue}{\left(\left(\left(\left(y \cdot y\right) \cdot z\right) \cdot z\right) \cdot x\right) \cdot x}} \]

   8. Taylor expanded in z around inf

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

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

      10. lower-*.f64 91.9

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

   10. Applied rewrites 91.9%

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

   12. Applied rewrites 97.0%

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

   if -2e3 < (*.f64 y z) < 1

   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 96.8%

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

   if 1 < (*.f64 y z) < 9.999999999999999e201

   1. Initial program 99.8%

      \[x \cdot \left(1 - y \cdot z\right) \]
   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 92.3

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

   5. Applied rewrites 92.3%

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

   if 9.999999999999999e201 < (*.f64 y z)

   1. Initial program 88.1%

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

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

      5. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(x \cdot 1\right)}^{3} + {\left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)}^{3}}{\left(x \cdot 1\right) \cdot \left(x \cdot 1\right) + \left(\left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) - \left(x \cdot 1\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)\right)}} \]

      6. div-inv N/A

         \[\leadsto \color{blue}{\left({\left(x \cdot 1\right)}^{3} + {\left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)}^{3}\right) \cdot \frac{1}{\left(x \cdot 1\right) \cdot \left(x \cdot 1\right) + \left(\left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) - \left(x \cdot 1\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(x \cdot 1\right)}^{3} + {\left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)}^{3}\right) \cdot \frac{1}{\left(x \cdot 1\right) \cdot \left(x \cdot 1\right) + \left(\left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) - \left(x \cdot 1\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)\right)}} \]

   4. Applied rewrites 15.0%

      \[\leadsto \color{blue}{\left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(x, x, {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{2} - x \cdot \left(\left(x \cdot \left(-z\right)\right) \cdot y\right)\right)}} \]

   5. Taylor expanded in z around inf

      \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\color{blue}{{x}^{2} \cdot \left({y}^{2} \cdot {z}^{2}\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\color{blue}{\left({y}^{2} \cdot {z}^{2}\right) \cdot {x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left({y}^{2} \cdot {z}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\color{blue}{\left(\left({y}^{2} \cdot {z}^{2}\right) \cdot x\right) \cdot x}} \]

      4. lower-*.f64 N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\color{blue}{\left(\left({y}^{2} \cdot {z}^{2}\right) \cdot x\right) \cdot x}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\color{blue}{\left(\left({y}^{2} \cdot {z}^{2}\right) \cdot x\right)} \cdot x} \]

      6. unpow2 N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left(\left({y}^{2} \cdot \color{blue}{\left(z \cdot z\right)}\right) \cdot x\right) \cdot x} \]

      7. associate-*r* N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left(\color{blue}{\left(\left({y}^{2} \cdot z\right) \cdot z\right)} \cdot x\right) \cdot x} \]

      8. lower-*.f64 N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left(\color{blue}{\left(\left({y}^{2} \cdot z\right) \cdot z\right)} \cdot x\right) \cdot x} \]

      9. lower-*.f64 N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left(\left(\color{blue}{\left({y}^{2} \cdot z\right)} \cdot z\right) \cdot x\right) \cdot x} \]

      10. unpow2 N/A

          \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left(\left(\left(\color{blue}{\left(y \cdot y\right)} \cdot z\right) \cdot z\right) \cdot x\right) \cdot x} \]

      11. lower-*.f64 0.4

          \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left(\left(\left(\color{blue}{\left(y \cdot y\right)} \cdot z\right) \cdot z\right) \cdot x\right) \cdot x} \]

   7. Applied rewrites 0.4%

      \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\color{blue}{\left(\left(\left(\left(y \cdot y\right) \cdot z\right) \cdot z\right) \cdot x\right) \cdot x}} \]

   8. Taylor expanded in z around inf

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

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

      10. lower-*.f64 99.8

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

   10. Applied rewrites 99.8%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot y \leq -2000:\\ \;\;\;\;\left(\left(-x\right) \cdot z\right) \cdot y\\ \mathbf{elif}\;z \cdot y \leq 1:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;z \cdot y \leq 10^{+202}:\\ \;\;\;\;\left(\left(-z\right) \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-y\right) \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot y \leq -2000:\\ \;\;\;\;\left(\left(-x\right) \cdot z\right) \cdot y\\ \mathbf{elif}\;z \cdot y \leq 1:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-y\right) \cdot x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z y) -2000.0)
   (* (* (- x) z) y)
   (if (<= (* z y) 1.0) (* 1.0 x) (* (* (- y) x) z))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((z * y) <= -2000.0) {
		tmp = (-x * z) * y;
	} else if ((z * y) <= 1.0) {
		tmp = 1.0 * x;
	} else {
		tmp = (-y * x) * z;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * y) <= (-2000.0d0)) then
        tmp = (-x * z) * y
    else if ((z * y) <= 1.0d0) then
        tmp = 1.0d0 * x
    else
        tmp = (-y * x) * z
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * y) <= -2000.0) {
		tmp = (-x * z) * y;
	} else if ((z * y) <= 1.0) {
		tmp = 1.0 * x;
	} else {
		tmp = (-y * x) * z;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if (z * y) <= -2000.0:
		tmp = (-x * z) * y
	elif (z * y) <= 1.0:
		tmp = 1.0 * x
	else:
		tmp = (-y * x) * z
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * y) <= -2000.0)
		tmp = Float64(Float64(Float64(-x) * z) * y);
	elseif (Float64(z * y) <= 1.0)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(Float64(Float64(-y) * x) * z);
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * y) <= -2000.0)
		tmp = (-x * z) * y;
	elseif ((z * y) <= 1.0)
		tmp = 1.0 * x;
	else
		tmp = (-y * x) * z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * y), $MachinePrecision], -2000.0], N[(N[((-x) * z), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[N[(z * y), $MachinePrecision], 1.0], N[(1.0 * x), $MachinePrecision], N[(N[((-y) * x), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y z) < -2e3

   1. Initial program 86.5%

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

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

      5. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(x \cdot 1\right)}^{3} + {\left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)}^{3}}{\left(x \cdot 1\right) \cdot \left(x \cdot 1\right) + \left(\left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) - \left(x \cdot 1\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)\right)}} \]

      6. div-inv N/A

         \[\leadsto \color{blue}{\left({\left(x \cdot 1\right)}^{3} + {\left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)}^{3}\right) \cdot \frac{1}{\left(x \cdot 1\right) \cdot \left(x \cdot 1\right) + \left(\left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) - \left(x \cdot 1\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(x \cdot 1\right)}^{3} + {\left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)}^{3}\right) \cdot \frac{1}{\left(x \cdot 1\right) \cdot \left(x \cdot 1\right) + \left(\left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) - \left(x \cdot 1\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)\right)}} \]

   4. Applied rewrites 19.9%

      \[\leadsto \color{blue}{\left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(x, x, {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{2} - x \cdot \left(\left(x \cdot \left(-z\right)\right) \cdot y\right)\right)}} \]

   5. Taylor expanded in z around inf

      \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\color{blue}{{x}^{2} \cdot \left({y}^{2} \cdot {z}^{2}\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\color{blue}{\left({y}^{2} \cdot {z}^{2}\right) \cdot {x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left({y}^{2} \cdot {z}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\color{blue}{\left(\left({y}^{2} \cdot {z}^{2}\right) \cdot x\right) \cdot x}} \]

      4. lower-*.f64 N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\color{blue}{\left(\left({y}^{2} \cdot {z}^{2}\right) \cdot x\right) \cdot x}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\color{blue}{\left(\left({y}^{2} \cdot {z}^{2}\right) \cdot x\right)} \cdot x} \]

      6. unpow2 N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left(\left({y}^{2} \cdot \color{blue}{\left(z \cdot z\right)}\right) \cdot x\right) \cdot x} \]

      7. associate-*r* N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left(\color{blue}{\left(\left({y}^{2} \cdot z\right) \cdot z\right)} \cdot x\right) \cdot x} \]

      8. lower-*.f64 N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left(\color{blue}{\left(\left({y}^{2} \cdot z\right) \cdot z\right)} \cdot x\right) \cdot x} \]

      9. lower-*.f64 N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left(\left(\color{blue}{\left({y}^{2} \cdot z\right)} \cdot z\right) \cdot x\right) \cdot x} \]

      10. unpow2 N/A

          \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left(\left(\left(\color{blue}{\left(y \cdot y\right)} \cdot z\right) \cdot z\right) \cdot x\right) \cdot x} \]

      11. lower-*.f64 2.6

          \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left(\left(\left(\color{blue}{\left(y \cdot y\right)} \cdot z\right) \cdot z\right) \cdot x\right) \cdot x} \]

   7. Applied rewrites 2.6%

      \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\color{blue}{\left(\left(\left(\left(y \cdot y\right) \cdot z\right) \cdot z\right) \cdot x\right) \cdot x}} \]

   8. Taylor expanded in z around inf

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

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

      10. lower-*.f64 91.9

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

   10. Applied rewrites 91.9%

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

   12. Applied rewrites 97.0%

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

   if -2e3 < (*.f64 y z) < 1

   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 96.8%

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

   if 1 < (*.f64 y z)

   1. Initial program 94.1%

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

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

      5. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(x \cdot 1\right)}^{3} + {\left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)}^{3}}{\left(x \cdot 1\right) \cdot \left(x \cdot 1\right) + \left(\left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) - \left(x \cdot 1\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)\right)}} \]

      6. div-inv N/A

         \[\leadsto \color{blue}{\left({\left(x \cdot 1\right)}^{3} + {\left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)}^{3}\right) \cdot \frac{1}{\left(x \cdot 1\right) \cdot \left(x \cdot 1\right) + \left(\left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) - \left(x \cdot 1\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(x \cdot 1\right)}^{3} + {\left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)}^{3}\right) \cdot \frac{1}{\left(x \cdot 1\right) \cdot \left(x \cdot 1\right) + \left(\left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) - \left(x \cdot 1\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)\right)}} \]

   4. Applied rewrites 31.6%

      \[\leadsto \color{blue}{\left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(x, x, {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{2} - x \cdot \left(\left(x \cdot \left(-z\right)\right) \cdot y\right)\right)}} \]

   5. Taylor expanded in z around inf

      \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\color{blue}{{x}^{2} \cdot \left({y}^{2} \cdot {z}^{2}\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\color{blue}{\left({y}^{2} \cdot {z}^{2}\right) \cdot {x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left({y}^{2} \cdot {z}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\color{blue}{\left(\left({y}^{2} \cdot {z}^{2}\right) \cdot x\right) \cdot x}} \]

      4. lower-*.f64 N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\color{blue}{\left(\left({y}^{2} \cdot {z}^{2}\right) \cdot x\right) \cdot x}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\color{blue}{\left(\left({y}^{2} \cdot {z}^{2}\right) \cdot x\right)} \cdot x} \]

      6. unpow2 N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left(\left({y}^{2} \cdot \color{blue}{\left(z \cdot z\right)}\right) \cdot x\right) \cdot x} \]

      7. associate-*r* N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left(\color{blue}{\left(\left({y}^{2} \cdot z\right) \cdot z\right)} \cdot x\right) \cdot x} \]

      8. lower-*.f64 N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left(\color{blue}{\left(\left({y}^{2} \cdot z\right) \cdot z\right)} \cdot x\right) \cdot x} \]

      9. lower-*.f64 N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left(\left(\color{blue}{\left({y}^{2} \cdot z\right)} \cdot z\right) \cdot x\right) \cdot x} \]

      10. unpow2 N/A

          \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left(\left(\left(\color{blue}{\left(y \cdot y\right)} \cdot z\right) \cdot z\right) \cdot x\right) \cdot x} \]

      11. lower-*.f64 15.5

          \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left(\left(\left(\color{blue}{\left(y \cdot y\right)} \cdot z\right) \cdot z\right) \cdot x\right) \cdot x} \]

   7. Applied rewrites 15.5%

      \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\color{blue}{\left(\left(\left(\left(y \cdot y\right) \cdot z\right) \cdot z\right) \cdot x\right) \cdot x}} \]

   8. Taylor expanded in z around inf

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

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

      10. lower-*.f64 87.4

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

   10. Applied rewrites 87.4%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot y \leq -2000:\\ \;\;\;\;\left(\left(-x\right) \cdot z\right) \cdot y\\ \mathbf{elif}\;z \cdot y \leq 1:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-y\right) \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := \left(\left(-y\right) \cdot x\right) \cdot z\\ \mathbf{if}\;z \cdot y \leq -4:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \cdot y \leq 1:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 (* (* (- y) x) z)))
   (if (<= (* z y) -4.0) t_0 (if (<= (* z y) 1.0) (* 1.0 x) t_0))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = (-y * x) * z;
	double tmp;
	if ((z * y) <= -4.0) {
		tmp = t_0;
	} else if ((z * y) <= 1.0) {
		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 = (-y * x) * z
    if ((z * y) <= (-4.0d0)) then
        tmp = t_0
    else if ((z * y) <= 1.0d0) 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 = (-y * x) * z;
	double tmp;
	if ((z * y) <= -4.0) {
		tmp = t_0;
	} else if ((z * y) <= 1.0) {
		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 = (-y * x) * z
	tmp = 0
	if (z * y) <= -4.0:
		tmp = t_0
	elif (z * y) <= 1.0:
		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(-y) * x) * z)
	tmp = 0.0
	if (Float64(z * y) <= -4.0)
		tmp = t_0;
	elseif (Float64(z * y) <= 1.0)
		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 = (-y * x) * z;
	tmp = 0.0;
	if ((z * y) <= -4.0)
		tmp = t_0;
	elseif ((z * y) <= 1.0)
		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[((-y) * x), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[N[(z * y), $MachinePrecision], -4.0], t$95$0, If[LessEqual[N[(z * y), $MachinePrecision], 1.0], N[(1.0 * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -4 or 1 < (*.f64 y z)

   1. Initial program 91.0%

      \[x \cdot \left(1 - y \cdot z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. 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-lft-in N/A

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

      5. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(x \cdot 1\right)}^{3} + {\left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)}^{3}}{\left(x \cdot 1\right) \cdot \left(x \cdot 1\right) + \left(\left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) - \left(x \cdot 1\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)\right)}} \]

      6. div-inv N/A

         \[\leadsto \color{blue}{\left({\left(x \cdot 1\right)}^{3} + {\left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)}^{3}\right) \cdot \frac{1}{\left(x \cdot 1\right) \cdot \left(x \cdot 1\right) + \left(\left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) - \left(x \cdot 1\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(x \cdot 1\right)}^{3} + {\left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)}^{3}\right) \cdot \frac{1}{\left(x \cdot 1\right) \cdot \left(x \cdot 1\right) + \left(\left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) - \left(x \cdot 1\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)\right)}} \]

   4. Applied rewrites 26.6%

      \[\leadsto \color{blue}{\left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(x, x, {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{2} - x \cdot \left(\left(x \cdot \left(-z\right)\right) \cdot y\right)\right)}} \]

   5. Taylor expanded in z around inf

      \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\color{blue}{{x}^{2} \cdot \left({y}^{2} \cdot {z}^{2}\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\color{blue}{\left({y}^{2} \cdot {z}^{2}\right) \cdot {x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left({y}^{2} \cdot {z}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\color{blue}{\left(\left({y}^{2} \cdot {z}^{2}\right) \cdot x\right) \cdot x}} \]

      4. lower-*.f64 N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\color{blue}{\left(\left({y}^{2} \cdot {z}^{2}\right) \cdot x\right) \cdot x}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\color{blue}{\left(\left({y}^{2} \cdot {z}^{2}\right) \cdot x\right)} \cdot x} \]

      6. unpow2 N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left(\left({y}^{2} \cdot \color{blue}{\left(z \cdot z\right)}\right) \cdot x\right) \cdot x} \]

      7. associate-*r* N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left(\color{blue}{\left(\left({y}^{2} \cdot z\right) \cdot z\right)} \cdot x\right) \cdot x} \]

      8. lower-*.f64 N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left(\color{blue}{\left(\left({y}^{2} \cdot z\right) \cdot z\right)} \cdot x\right) \cdot x} \]

      9. lower-*.f64 N/A

         \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left(\left(\color{blue}{\left({y}^{2} \cdot z\right)} \cdot z\right) \cdot x\right) \cdot x} \]

      10. unpow2 N/A

          \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left(\left(\left(\color{blue}{\left(y \cdot y\right)} \cdot z\right) \cdot z\right) \cdot x\right) \cdot x} \]

      11. lower-*.f64 10.2

          \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\left(\left(\left(\color{blue}{\left(y \cdot y\right)} \cdot z\right) \cdot z\right) \cdot x\right) \cdot x} \]

   7. Applied rewrites 10.2%

      \[\leadsto \left({x}^{3} + {\left(\left(x \cdot \left(-z\right)\right) \cdot y\right)}^{3}\right) \cdot \frac{1}{\color{blue}{\left(\left(\left(\left(y \cdot y\right) \cdot z\right) \cdot z\right) \cdot x\right) \cdot x}} \]

   8. Taylor expanded in z around inf

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

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

      10. lower-*.f64 88.8

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

   10. Applied rewrites 88.8%

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

   if -4 < (*.f64 y z) < 1

   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 97.5%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot y \leq -4:\\ \;\;\;\;\left(\left(-y\right) \cdot x\right) \cdot z\\ \mathbf{elif}\;z \cdot y \leq 1:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-y\right) \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.4% 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 95.1%

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

3. Taylor expanded in z around 0

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

5. Applied rewrites 45.8%

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

6. Final simplification 45.8%

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

Reproduce

herbie shell --seed 2024248 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))