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 99.8%

   \[\left(x \cdot 3\right) \cdot y - z \]
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. Final simplification 99.8%

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

Alternative 2: 69.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+101} \lor \neg \left(z \leq -4 \cdot 10^{-63}\right) \land \left(z \leq -3.7 \cdot 10^{-101} \lor \neg \left(z \leq 7.6 \cdot 10^{-40}\right)\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.6e+101)
         (and (not (<= z -4e-63)) (or (<= z -3.7e-101) (not (<= z 7.6e-40)))))
   (- z)
   (* 3.0 (* x y))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.6e+101) || (!(z <= -4e-63) && ((z <= -3.7e-101) || !(z <= 7.6e-40)))) {
		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 ((z <= (-1.6d+101)) .or. (.not. (z <= (-4d-63))) .and. (z <= (-3.7d-101)) .or. (.not. (z <= 7.6d-40))) 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 ((z <= -1.6e+101) || (!(z <= -4e-63) && ((z <= -3.7e-101) || !(z <= 7.6e-40)))) {
		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 (z <= -1.6e+101) or (not (z <= -4e-63) and ((z <= -3.7e-101) or not (z <= 7.6e-40))):
		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 ((z <= -1.6e+101) || (!(z <= -4e-63) && ((z <= -3.7e-101) || !(z <= 7.6e-40))))
		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 ((z <= -1.6e+101) || (~((z <= -4e-63)) && ((z <= -3.7e-101) || ~((z <= 7.6e-40)))))
		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[Or[LessEqual[z, -1.6e+101], And[N[Not[LessEqual[z, -4e-63]], $MachinePrecision], Or[LessEqual[z, -3.7e-101], N[Not[LessEqual[z, 7.6e-40]], $MachinePrecision]]]], (-z), N[(3.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.60000000000000003e101 or -4.00000000000000027e-63 < z < -3.70000000000000005e-101 or 7.5999999999999998e-40 < z

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 79.6%

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

      1. neg-mul-1 79.6%

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

   7. Simplified 79.6%

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

   if -1.60000000000000003e101 < z < -4.00000000000000027e-63 or -3.70000000000000005e-101 < z < 7.5999999999999998e-40

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around inf 84.5%

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

   6. Taylor expanded in z around 0 79.2%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+101} \lor \neg \left(z \leq -4 \cdot 10^{-63}\right) \land \left(z \leq -3.7 \cdot 10^{-101} \lor \neg \left(z \leq 7.6 \cdot 10^{-40}\right)\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.18 \cdot 10^{+101}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{-63}:\\ \;\;\;\;x \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{-101} \lor \neg \left(z \leq 5.6 \cdot 10^{-41}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z -1.18e+101)
   (- z)
   (if (<= z -3.7e-63)
     (* x (* 3.0 y))
     (if (or (<= z -4.8e-101) (not (<= z 5.6e-41))) (- z) (* 3.0 (* x y))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.18e+101) {
		tmp = -z;
	} else if (z <= -3.7e-63) {
		tmp = x * (3.0 * y);
	} else if ((z <= -4.8e-101) || !(z <= 5.6e-41)) {
		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 (z <= (-1.18d+101)) then
        tmp = -z
    else if (z <= (-3.7d-63)) then
        tmp = x * (3.0d0 * y)
    else if ((z <= (-4.8d-101)) .or. (.not. (z <= 5.6d-41))) 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 (z <= -1.18e+101) {
		tmp = -z;
	} else if (z <= -3.7e-63) {
		tmp = x * (3.0 * y);
	} else if ((z <= -4.8e-101) || !(z <= 5.6e-41)) {
		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 z <= -1.18e+101:
		tmp = -z
	elif z <= -3.7e-63:
		tmp = x * (3.0 * y)
	elif (z <= -4.8e-101) or not (z <= 5.6e-41):
		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 (z <= -1.18e+101)
		tmp = Float64(-z);
	elseif (z <= -3.7e-63)
		tmp = Float64(x * Float64(3.0 * y));
	elseif ((z <= -4.8e-101) || !(z <= 5.6e-41))
		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 (z <= -1.18e+101)
		tmp = -z;
	elseif (z <= -3.7e-63)
		tmp = x * (3.0 * y);
	elseif ((z <= -4.8e-101) || ~((z <= 5.6e-41)))
		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[z, -1.18e+101], (-z), If[LessEqual[z, -3.7e-63], N[(x * N[(3.0 * y), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, -4.8e-101], N[Not[LessEqual[z, 5.6e-41]], $MachinePrecision]], (-z), N[(3.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.18000000000000005e101 or -3.70000000000000012e-63 < z < -4.8e-101 or 5.6000000000000003e-41 < z

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 79.6%

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

      1. neg-mul-1 79.6%

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

   7. Simplified 79.6%

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

   if -1.18000000000000005e101 < z < -3.70000000000000012e-63

   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 99.6%

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

   6. Taylor expanded in z around 0 66.3%

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

      1. associate-*r* 66.5%

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

      2. *-commutative 66.5%

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

   8. Simplified 66.5%

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

   9. Taylor expanded in y around 0 66.3%

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

       1. associate-*r* 66.5%

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

       2. *-commutative 66.5%

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

       3. associate-*r* 66.3%

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

   11. Simplified 66.3%

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

   if -4.8e-101 < z < 5.6000000000000003e-41

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around inf 80.6%

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

   6. Taylor expanded in z around 0 82.6%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.18 \cdot 10^{+101}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{-63}:\\ \;\;\;\;x \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{-101} \lor \neg \left(z \leq 5.6 \cdot 10^{-41}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.28 \cdot 10^{+101}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -3.9 \cdot 10^{-63}:\\ \;\;\;\;y \cdot \left(x \cdot 3\right)\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{-101} \lor \neg \left(z \leq 1.2 \cdot 10^{-40}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z -1.28e+101)
   (- z)
   (if (<= z -3.9e-63)
     (* y (* x 3.0))
     (if (or (<= z -3.7e-101) (not (<= z 1.2e-40))) (- z) (* 3.0 (* x y))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.28e+101) {
		tmp = -z;
	} else if (z <= -3.9e-63) {
		tmp = y * (x * 3.0);
	} else if ((z <= -3.7e-101) || !(z <= 1.2e-40)) {
		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 (z <= (-1.28d+101)) then
        tmp = -z
    else if (z <= (-3.9d-63)) then
        tmp = y * (x * 3.0d0)
    else if ((z <= (-3.7d-101)) .or. (.not. (z <= 1.2d-40))) 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 (z <= -1.28e+101) {
		tmp = -z;
	} else if (z <= -3.9e-63) {
		tmp = y * (x * 3.0);
	} else if ((z <= -3.7e-101) || !(z <= 1.2e-40)) {
		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 z <= -1.28e+101:
		tmp = -z
	elif z <= -3.9e-63:
		tmp = y * (x * 3.0)
	elif (z <= -3.7e-101) or not (z <= 1.2e-40):
		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 (z <= -1.28e+101)
		tmp = Float64(-z);
	elseif (z <= -3.9e-63)
		tmp = Float64(y * Float64(x * 3.0));
	elseif ((z <= -3.7e-101) || !(z <= 1.2e-40))
		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 (z <= -1.28e+101)
		tmp = -z;
	elseif (z <= -3.9e-63)
		tmp = y * (x * 3.0);
	elseif ((z <= -3.7e-101) || ~((z <= 1.2e-40)))
		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[z, -1.28e+101], (-z), If[LessEqual[z, -3.9e-63], N[(y * N[(x * 3.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, -3.7e-101], N[Not[LessEqual[z, 1.2e-40]], $MachinePrecision]], (-z), N[(3.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.28e101 or -3.90000000000000022e-63 < z < -3.70000000000000005e-101 or 1.19999999999999996e-40 < z

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 79.6%

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

      1. neg-mul-1 79.6%

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

   7. Simplified 79.6%

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

   if -1.28e101 < z < -3.90000000000000022e-63

   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 99.6%

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

   6. Taylor expanded in z around 0 66.3%

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

      1. associate-*r* 66.5%

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

      2. *-commutative 66.5%

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

   8. Simplified 66.5%

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

   if -3.70000000000000005e-101 < z < 1.19999999999999996e-40

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around inf 80.6%

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

   6. Taylor expanded in z around 0 82.6%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.28 \cdot 10^{+101}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -3.9 \cdot 10^{-63}:\\ \;\;\;\;y \cdot \left(x \cdot 3\right)\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{-101} \lor \neg \left(z \leq 1.2 \cdot 10^{-40}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \end{array} \]
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 99.8%

   \[\left(x \cdot 3\right) \cdot y - z \]
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 x around 0 99.8%

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

6. Final simplification 99.8%

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

Alternative 6: 51.0% 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 99.8%

   \[\left(x \cdot 3\right) \cdot y - z \]
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 x around 0 50.6%

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

   1. neg-mul-1 50.6%

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

7. Simplified 50.6%

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

8. Final simplification 50.6%

   \[\leadsto -z \]
9. 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 2024081 
(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))