Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, B

Percentage Accurate: 99.8% → 99.8%

Time: 6.2s

Alternatives: 6

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot 3\right) \cdot y - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (* (* x 3.0) y) z))

C

double code(double x, double y, double z) {
	return ((x * 3.0) * 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 * 3.0d0) * y) - z
end function

Java

public static double code(double x, double y, double z) {
	return ((x * 3.0) * y) - z;
}

Python

def code(x, y, z):
	return ((x * 3.0) * y) - z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x * 3.0) * y) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x * 3.0) * y) - z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x * 3.0), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot 3\right) \cdot y - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (* (* x 3.0) y) z))

C

double code(double x, double y, double z) {
	return ((x * 3.0) * 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 * 3.0d0) * y) - z
end function

Java

public static double code(double x, double y, double z) {
	return ((x * 3.0) * y) - z;
}

Python

def code(x, y, z):
	return ((x * 3.0) * y) - z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x * 3.0) * y) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x * 3.0) * y) - z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x * 3.0), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \left(y \cdot 3\right) \cdot x - z \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (- (* (* y 3.0) x) z))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return ((y * 3.0) * x) - z;
}

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 = ((y * 3.0d0) * x) - z
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	return ((y * 3.0) * x) - z;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return ((y * 3.0) * x) - z

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(Float64(Float64(y * 3.0) * x) - z)
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = ((y * 3.0) * x) - z;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(N[(N[(y * 3.0), $MachinePrecision] * x), $MachinePrecision] - z), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 99.9

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

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(y \cdot 3\right) \cdot x} - z \]
5. Add Preprocessing

Alternative 2: 79.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := \left(x \cdot 3\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -780000000000 \lor \neg \left(t\_0 \leq 7.6 \cdot 10^{-57}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* x 3.0) y)))
   (if (or (<= t_0 -780000000000.0) (not (<= t_0 7.6e-57))) t_0 (- z))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = (x * 3.0) * y;
	double tmp;
	if ((t_0 <= -780000000000.0) || !(t_0 <= 7.6e-57)) {
		tmp = t_0;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * 3.0d0) * y
    if ((t_0 <= (-780000000000.0d0)) .or. (.not. (t_0 <= 7.6d-57))) then
        tmp = t_0
    else
        tmp = -z
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double t_0 = (x * 3.0) * y;
	double tmp;
	if ((t_0 <= -780000000000.0) || !(t_0 <= 7.6e-57)) {
		tmp = t_0;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	t_0 = (x * 3.0) * y
	tmp = 0
	if (t_0 <= -780000000000.0) or not (t_0 <= 7.6e-57):
		tmp = t_0
	else:
		tmp = -z
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(Float64(x * 3.0) * y)
	tmp = 0.0
	if ((t_0 <= -780000000000.0) || !(t_0 <= 7.6e-57))
		tmp = t_0;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	t_0 = (x * 3.0) * y;
	tmp = 0.0;
	if ((t_0 <= -780000000000.0) || ~((t_0 <= 7.6e-57)))
		tmp = t_0;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * 3.0), $MachinePrecision] * y), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -780000000000.0], N[Not[LessEqual[t$95$0, 7.6e-57]], $MachinePrecision]], t$95$0, (-z)]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x #s(literal 3 binary64)) y) < -7.8e11 or 7.5999999999999995e-57 < (*.f64 (*.f64 x #s(literal 3 binary64)) y)

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 97.8

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

   7. Applied rewrites 97.8%

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

   8. Taylor expanded in x around inf

      \[\leadsto 3 \cdot \color{blue}{\left(x \cdot y\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 79.1%

       \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{3} \]
   11. Step-by-step derivation

   12. Applied rewrites 79.2%

       \[\leadsto \left(x \cdot 3\right) \cdot y \]

   if -7.8e11 < (*.f64 (*.f64 x #s(literal 3 binary64)) y) < 7.5999999999999995e-57

   1. Initial program 100.0%

      \[\left(x \cdot 3\right) \cdot y - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 87.7

         \[\leadsto \color{blue}{-z} \]

   5. Applied rewrites 87.7%

      \[\leadsto \color{blue}{-z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 3\right) \cdot y \leq -780000000000 \lor \neg \left(\left(x \cdot 3\right) \cdot y \leq 7.6 \cdot 10^{-57}\right):\\ \;\;\;\;\left(x \cdot 3\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := \left(x \cdot 3\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -2000000000000:\\ \;\;\;\;\left(3 \cdot y\right) \cdot x\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-57}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 3\\ \end{array} \end{array} \]

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 3.0) y)))
   (if (<= t_0 -2000000000000.0)
     (* (* 3.0 y) x)
     (if (<= t_0 4e-57) (- z) (* (* x y) 3.0)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = (x * 3.0) * y;
	double tmp;
	if (t_0 <= -2000000000000.0) {
		tmp = (3.0 * y) * x;
	} else if (t_0 <= 4e-57) {
		tmp = -z;
	} else {
		tmp = (x * y) * 3.0;
	}
	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 * 3.0d0) * y
    if (t_0 <= (-2000000000000.0d0)) then
        tmp = (3.0d0 * y) * x
    else if (t_0 <= 4d-57) then
        tmp = -z
    else
        tmp = (x * y) * 3.0d0
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double t_0 = (x * 3.0) * y;
	double tmp;
	if (t_0 <= -2000000000000.0) {
		tmp = (3.0 * y) * x;
	} else if (t_0 <= 4e-57) {
		tmp = -z;
	} else {
		tmp = (x * y) * 3.0;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	t_0 = (x * 3.0) * y
	tmp = 0
	if t_0 <= -2000000000000.0:
		tmp = (3.0 * y) * x
	elif t_0 <= 4e-57:
		tmp = -z
	else:
		tmp = (x * y) * 3.0
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(Float64(x * 3.0) * y)
	tmp = 0.0
	if (t_0 <= -2000000000000.0)
		tmp = Float64(Float64(3.0 * y) * x);
	elseif (t_0 <= 4e-57)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(x * y) * 3.0);
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	t_0 = (x * 3.0) * y;
	tmp = 0.0;
	if (t_0 <= -2000000000000.0)
		tmp = (3.0 * y) * x;
	elseif (t_0 <= 4e-57)
		tmp = -z;
	else
		tmp = (x * y) * 3.0;
	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 * 3.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -2000000000000.0], N[(N[(3.0 * y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$0, 4e-57], (-z), N[(N[(x * y), $MachinePrecision] * 3.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 3 binary64)) y) < -2e12

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in x around inf

      \[\leadsto \left(3 \cdot y\right) \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 82.2%

       \[\leadsto \left(3 \cdot y\right) \cdot x \]

   if -2e12 < (*.f64 (*.f64 x #s(literal 3 binary64)) y) < 3.99999999999999982e-57

   1. Initial program 100.0%

      \[\left(x \cdot 3\right) \cdot y - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 87.7

         \[\leadsto \color{blue}{-z} \]

   5. Applied rewrites 87.7%

      \[\leadsto \color{blue}{-z} \]

   if 3.99999999999999982e-57 < (*.f64 (*.f64 x #s(literal 3 binary64)) y)

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 95.8

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

   7. Applied rewrites 95.8%

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

   8. Taylor expanded in x around inf

      \[\leadsto 3 \cdot \color{blue}{\left(x \cdot y\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 76.4%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 3\right) \cdot y \leq -2000000000000:\\ \;\;\;\;\left(3 \cdot y\right) \cdot x\\ \mathbf{elif}\;\left(x \cdot 3\right) \cdot y \leq 4 \cdot 10^{-57}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 3\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := \left(x \cdot 3\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -2000000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-57}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 3\\ \end{array} \end{array} \]

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 3.0) y)))
   (if (<= t_0 -2000000000000.0)
     t_0
     (if (<= t_0 4e-57) (- z) (* (* x y) 3.0)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = (x * 3.0) * y;
	double tmp;
	if (t_0 <= -2000000000000.0) {
		tmp = t_0;
	} else if (t_0 <= 4e-57) {
		tmp = -z;
	} else {
		tmp = (x * y) * 3.0;
	}
	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 * 3.0d0) * y
    if (t_0 <= (-2000000000000.0d0)) then
        tmp = t_0
    else if (t_0 <= 4d-57) then
        tmp = -z
    else
        tmp = (x * y) * 3.0d0
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double t_0 = (x * 3.0) * y;
	double tmp;
	if (t_0 <= -2000000000000.0) {
		tmp = t_0;
	} else if (t_0 <= 4e-57) {
		tmp = -z;
	} else {
		tmp = (x * y) * 3.0;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	t_0 = (x * 3.0) * y
	tmp = 0
	if t_0 <= -2000000000000.0:
		tmp = t_0
	elif t_0 <= 4e-57:
		tmp = -z
	else:
		tmp = (x * y) * 3.0
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(Float64(x * 3.0) * y)
	tmp = 0.0
	if (t_0 <= -2000000000000.0)
		tmp = t_0;
	elseif (t_0 <= 4e-57)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(x * y) * 3.0);
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	t_0 = (x * 3.0) * y;
	tmp = 0.0;
	if (t_0 <= -2000000000000.0)
		tmp = t_0;
	elseif (t_0 <= 4e-57)
		tmp = -z;
	else
		tmp = (x * y) * 3.0;
	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 * 3.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -2000000000000.0], t$95$0, If[LessEqual[t$95$0, 4e-57], (-z), N[(N[(x * y), $MachinePrecision] * 3.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 3 binary64)) y) < -2e12

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in x around inf

      \[\leadsto 3 \cdot \color{blue}{\left(x \cdot y\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 82.1%

       \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{3} \]
   11. Step-by-step derivation

   12. Applied rewrites 82.2%

       \[\leadsto \left(x \cdot 3\right) \cdot y \]

   if -2e12 < (*.f64 (*.f64 x #s(literal 3 binary64)) y) < 3.99999999999999982e-57

   1. Initial program 100.0%

      \[\left(x \cdot 3\right) \cdot y - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 87.7

         \[\leadsto \color{blue}{-z} \]

   5. Applied rewrites 87.7%

      \[\leadsto \color{blue}{-z} \]

   if 3.99999999999999982e-57 < (*.f64 (*.f64 x #s(literal 3 binary64)) y)

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 95.8

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

   7. Applied rewrites 95.8%

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

   8. Taylor expanded in x around inf

      \[\leadsto 3 \cdot \color{blue}{\left(x \cdot y\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 76.4%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 3\right) \cdot y \leq -2000000000000:\\ \;\;\;\;\left(x \cdot 3\right) \cdot y\\ \mathbf{elif}\;\left(x \cdot 3\right) \cdot y \leq 4 \cdot 10^{-57}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 3\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \left(x \cdot 3\right) \cdot y - z \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (- (* (* x 3.0) y) z))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return ((x * 3.0) * y) - z;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x * 3.0d0) * y) - z
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	return ((x * 3.0) * y) - z;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return ((x * 3.0) * y) - z

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(Float64(Float64(x * 3.0) * y) - z)
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = ((x * 3.0) * y) - z;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(N[(N[(x * 3.0), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(x \cdot 3\right) \cdot y - z \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 6: 51.6% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

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 = -z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left(x \cdot 3\right) \cdot y - z \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 52.6

      \[\leadsto \color{blue}{-z} \]

5. Applied rewrites 52.6%

   \[\leadsto \color{blue}{-z} \]

6. Final simplification 52.6%

   \[\leadsto -z \]
7. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(3 \cdot y\right) - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (* x (* 3.0 y)) z))

C

double code(double x, double y, double z) {
	return (x * (3.0 * 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 * (3.0d0 * y)) - z
end function

Java

public static double code(double x, double y, double z) {
	return (x * (3.0 * y)) - z;
}

Python

def code(x, y, z):
	return (x * (3.0 * y)) - z

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(3.0 * y)) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * (3.0 * y)) - z;
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(3.0 * y), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]

Reproduce

herbie shell --seed 2024324 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* x (* 3 y)) z))

  (- (* (* x 3.0) y) z))