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

Percentage Accurate: 99.8% → 99.8%

Time: 5.1s

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

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

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(x * Float64(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[(x * N[(3.0 * y), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. associate-*l* 99.8%

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

3. Simplified 99.8%

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

Alternative 2: 71.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+57} \lor \neg \left(x \leq 6 \cdot 10^{-33}\right):\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3e+57) (not (<= x 6e-33))) (* 3.0 (* x y)) (- z)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3e+57) || !(x <= 6e-33)) {
		tmp = 3.0 * (x * y);
	} 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) :: tmp
    if ((x <= (-3d+57)) .or. (.not. (x <= 6d-33))) then
        tmp = 3.0d0 * (x * y)
    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 tmp;
	if ((x <= -3e+57) || !(x <= 6e-33)) {
		tmp = 3.0 * (x * y);
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if (x <= -3e+57) or not (x <= 6e-33):
		tmp = 3.0 * (x * y)
	else:
		tmp = -z
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if ((x <= -3e+57) || !(x <= 6e-33))
		tmp = Float64(3.0 * Float64(x * y));
	else
		tmp = Float64(-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 ((x <= -3e+57) || ~((x <= 6e-33)))
		tmp = 3.0 * (x * y);
	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_] := If[Or[LessEqual[x, -3e+57], N[Not[LessEqual[x, 6e-33]], $MachinePrecision]], N[(3.0 * N[(x * y), $MachinePrecision]), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if x < -3e57 or 6.0000000000000003e-33 < x

   1. Initial program 99.8%

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

      1. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around inf 90.3%

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

      1. *-commutative 90.3%

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

      2. fma-neg 90.3%

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

      3. associate-/l* 90.3%

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

      4. metadata-eval 90.3%

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

   7. Simplified 90.3%

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

   8. Taylor expanded in x around inf 69.8%

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

   if -3e57 < x < 6.0000000000000003e-33

   1. Initial program 99.9%

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

      1. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 65.8%

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

      1. mul-1-neg 65.8%

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

   7. Simplified 65.8%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+57} \lor \neg \left(x \leq 6 \cdot 10^{-33}\right):\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+56}:\\ \;\;\;\;x \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-31}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -1.8e+56)
   (* x (* 3.0 y))
   (if (<= x 1.35e-31) (- z) (* y (* x 3.0)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.8e+56) {
		tmp = x * (3.0 * y);
	} else if (x <= 1.35e-31) {
		tmp = -z;
	} else {
		tmp = y * (x * 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) :: tmp
    if (x <= (-1.8d+56)) then
        tmp = x * (3.0d0 * y)
    else if (x <= 1.35d-31) then
        tmp = -z
    else
        tmp = y * (x * 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 tmp;
	if (x <= -1.8e+56) {
		tmp = x * (3.0 * y);
	} else if (x <= 1.35e-31) {
		tmp = -z;
	} else {
		tmp = y * (x * 3.0);
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if x <= -1.8e+56:
		tmp = x * (3.0 * y)
	elif x <= 1.35e-31:
		tmp = -z
	else:
		tmp = y * (x * 3.0)
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (x <= -1.8e+56)
		tmp = Float64(x * Float64(3.0 * y));
	elseif (x <= 1.35e-31)
		tmp = Float64(-z);
	else
		tmp = Float64(y * Float64(x * 3.0));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.8e+56)
		tmp = x * (3.0 * y);
	elseif (x <= 1.35e-31)
		tmp = -z;
	else
		tmp = y * (x * 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_] := If[LessEqual[x, -1.8e+56], N[(x * N[(3.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.35e-31], (-z), N[(y * N[(x * 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.79999999999999999e56

   1. Initial program 99.8%

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

      1. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around inf 88.2%

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

      1. *-commutative 88.2%

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

      2. fma-neg 88.2%

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

      3. associate-/l* 88.1%

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

      4. metadata-eval 88.1%

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

   7. Simplified 88.1%

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

   8. Taylor expanded in x around inf 75.8%

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

      1. *-commutative 75.8%

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

      2. associate-*r* 75.8%

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

   10. Simplified 75.8%

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

   if -1.79999999999999999e56 < x < 1.35000000000000007e-31

   1. Initial program 99.9%

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

      1. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 66.0%

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

      1. mul-1-neg 66.0%

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

   7. Simplified 66.0%

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

   if 1.35000000000000007e-31 < x

   1. Initial program 99.9%

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

      1. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around inf 91.9%

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

      1. *-commutative 91.9%

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

      2. fma-neg 91.9%

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

      3. associate-/l* 91.9%

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

      4. metadata-eval 91.9%

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

   7. Simplified 91.9%

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

   8. Taylor expanded in x around inf 65.9%

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

      1. *-commutative 65.9%

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

      2. associate-*r* 65.8%

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

   10. Simplified 65.8%

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

   11. Taylor expanded in x around 0 65.9%

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

       1. *-commutative 65.9%

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

       2. associate-*r* 65.8%

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

       3. *-commutative 65.8%

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

       4. associate-*r* 65.9%

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

   13. Simplified 65.9%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+56}:\\ \;\;\;\;x \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-31}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+56}:\\ \;\;\;\;x \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-36}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -1.05e+56)
   (* x (* 3.0 y))
   (if (<= x 2.7e-36) (- z) (* 3.0 (* x y)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.05e+56) {
		tmp = x * (3.0 * y);
	} else if (x <= 2.7e-36) {
		tmp = -z;
	} else {
		tmp = 3.0 * (x * y);
	}
	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 (x <= (-1.05d+56)) then
        tmp = x * (3.0d0 * y)
    else if (x <= 2.7d-36) then
        tmp = -z
    else
        tmp = 3.0d0 * (x * y)
    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 (x <= -1.05e+56) {
		tmp = x * (3.0 * y);
	} else if (x <= 2.7e-36) {
		tmp = -z;
	} else {
		tmp = 3.0 * (x * y);
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if x <= -1.05e+56:
		tmp = x * (3.0 * y)
	elif x <= 2.7e-36:
		tmp = -z
	else:
		tmp = 3.0 * (x * y)
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (x <= -1.05e+56)
		tmp = Float64(x * Float64(3.0 * y));
	elseif (x <= 2.7e-36)
		tmp = Float64(-z);
	else
		tmp = Float64(3.0 * Float64(x * y));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.05e+56)
		tmp = x * (3.0 * y);
	elseif (x <= 2.7e-36)
		tmp = -z;
	else
		tmp = 3.0 * (x * y);
	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[x, -1.05e+56], N[(x * N[(3.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.7e-36], (-z), N[(3.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000009e56

   1. Initial program 99.8%

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

      1. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around inf 88.2%

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

      1. *-commutative 88.2%

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

      2. fma-neg 88.2%

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

      3. associate-/l* 88.1%

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

      4. metadata-eval 88.1%

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

   7. Simplified 88.1%

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

   8. Taylor expanded in x around inf 75.8%

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

      1. *-commutative 75.8%

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

      2. associate-*r* 75.8%

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

   10. Simplified 75.8%

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

   if -1.05000000000000009e56 < x < 2.70000000000000007e-36

   1. Initial program 99.9%

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

      1. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 65.8%

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

      1. mul-1-neg 65.8%

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

   7. Simplified 65.8%

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

   if 2.70000000000000007e-36 < x

   1. Initial program 99.9%

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

      1. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around inf 92.0%

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

      1. *-commutative 92.0%

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

      2. fma-neg 92.0%

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

      3. associate-/l* 92.0%

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

      4. metadata-eval 92.0%

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

   7. Simplified 92.0%

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

   8. Taylor expanded in x around inf 64.8%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+56}:\\ \;\;\;\;x \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-36}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return (3.0 * (x * 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 = (3.0d0 * (x * y)) - z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

   1. associate-*l* 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 99.8%

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

Alternative 6: 51.1% accurate, 3.5× 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 99.9%

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

   1. associate-*l* 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 51.0%

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

   1. mul-1-neg 51.0%

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

7. Simplified 51.0%

   \[\leadsto \color{blue}{-z} \]
8. Add Preprocessing

Developer target: 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 2024108 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, B"
  :precision binary64

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

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