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

Percentage Accurate: 99.8% → 99.8%

Time: 7.2s

Alternatives: 7

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

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

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 2: 78.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := \left(x \cdot 3\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-78}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(3 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* x 3.0) y)))
   (if (<= t_0 -5e+21) t_0 (if (<= t_0 2e-78) (- 0.0 z) (* x (* 3.0 y))))))

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 <= -5e+21) {
		tmp = t_0;
	} else if (t_0 <= 2e-78) {
		tmp = 0.0 - z;
	} else {
		tmp = x * (3.0 * 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) :: t_0
    real(8) :: tmp
    t_0 = (x * 3.0d0) * y
    if (t_0 <= (-5d+21)) then
        tmp = t_0
    else if (t_0 <= 2d-78) then
        tmp = 0.0d0 - z
    else
        tmp = x * (3.0d0 * y)
    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 <= -5e+21) {
		tmp = t_0;
	} else if (t_0 <= 2e-78) {
		tmp = 0.0 - z;
	} else {
		tmp = x * (3.0 * y);
	}
	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 <= -5e+21:
		tmp = t_0
	elif t_0 <= 2e-78:
		tmp = 0.0 - z
	else:
		tmp = x * (3.0 * y)
	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 <= -5e+21)
		tmp = t_0;
	elseif (t_0 <= 2e-78)
		tmp = Float64(0.0 - z);
	else
		tmp = Float64(x * Float64(3.0 * y));
	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 <= -5e+21)
		tmp = t_0;
	elseif (t_0 <= 2e-78)
		tmp = 0.0 - z;
	else
		tmp = x * (3.0 * 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_] := Block[{t$95$0 = N[(N[(x * 3.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+21], t$95$0, If[LessEqual[t$95$0, 2e-78], N[(0.0 - z), $MachinePrecision], N[(x * N[(3.0 * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 3 binary64)) y) < -5e21

   1. Initial program 99.7%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(x \cdot 3\right) \cdot y\right), \color{blue}{z}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \left(3 \cdot y\right)\right), z\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \left(3 \cdot y\right)\right), z\right) \]

      4. *-lowering-*.f64 99.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(3, y\right)\right), z\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(3, \color{blue}{\left(x \cdot y\right)}\right) \]

      2. *-lowering-*.f64 80.6%

         \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right) \]

   7. Simplified 80.6%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot 3\right), \color{blue}{y}\right) \]

      4. *-lowering-*.f64 80.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, 3\right), y\right) \]

   9. Applied egg-rr 80.6%

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

   if -5e21 < (*.f64 (*.f64 x #s(literal 3 binary64)) y) < 2e-78

   1. Initial program 99.9%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(x \cdot 3\right) \cdot y\right), \color{blue}{z}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \left(3 \cdot y\right)\right), z\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \left(3 \cdot y\right)\right), z\right) \]

      4. *-lowering-*.f64 99.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(3, y\right)\right), z\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{z} \]

      3. --lowering--.f64 81.8%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{z}\right) \]

   7. Simplified 81.8%

      \[\leadsto \color{blue}{0 - z} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 81.8%

         \[\leadsto \mathsf{neg.f64}\left(z\right) \]

   9. Applied egg-rr 81.8%

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

   if 2e-78 < (*.f64 (*.f64 x #s(literal 3 binary64)) y)

   1. Initial program 99.8%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(x \cdot 3\right) \cdot y\right), \color{blue}{z}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \left(3 \cdot y\right)\right), z\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \left(3 \cdot y\right)\right), z\right) \]

      4. *-lowering-*.f64 99.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(3, y\right)\right), z\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(3, \color{blue}{\left(x \cdot y\right)}\right) \]

      2. *-lowering-*.f64 77.2%

         \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right) \]

   7. Simplified 77.2%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot 3\right), \color{blue}{x}\right) \]

      5. *-lowering-*.f64 77.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 3\right), x\right) \]

   9. Applied egg-rr 77.3%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 3\right) \cdot y \leq -5 \cdot 10^{+21}:\\ \;\;\;\;\left(x \cdot 3\right) \cdot y\\ \mathbf{elif}\;\left(x \cdot 3\right) \cdot y \leq 2 \cdot 10^{-78}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(3 \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := \left(x \cdot 3\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-78}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* x 3.0) y)))
   (if (<= t_0 -5e+21) t_0 (if (<= t_0 2e-78) (- 0.0 z) (* 3.0 (* x y))))))

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 <= -5e+21) {
		tmp = t_0;
	} else if (t_0 <= 2e-78) {
		tmp = 0.0 - 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) :: t_0
    real(8) :: tmp
    t_0 = (x * 3.0d0) * y
    if (t_0 <= (-5d+21)) then
        tmp = t_0
    else if (t_0 <= 2d-78) then
        tmp = 0.0d0 - 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 t_0 = (x * 3.0) * y;
	double tmp;
	if (t_0 <= -5e+21) {
		tmp = t_0;
	} else if (t_0 <= 2e-78) {
		tmp = 0.0 - z;
	} else {
		tmp = 3.0 * (x * y);
	}
	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 <= -5e+21:
		tmp = t_0
	elif t_0 <= 2e-78:
		tmp = 0.0 - z
	else:
		tmp = 3.0 * (x * y)
	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 <= -5e+21)
		tmp = t_0;
	elseif (t_0 <= 2e-78)
		tmp = Float64(0.0 - 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)
	t_0 = (x * 3.0) * y;
	tmp = 0.0;
	if (t_0 <= -5e+21)
		tmp = t_0;
	elseif (t_0 <= 2e-78)
		tmp = 0.0 - 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_] := Block[{t$95$0 = N[(N[(x * 3.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+21], t$95$0, If[LessEqual[t$95$0, 2e-78], N[(0.0 - z), $MachinePrecision], N[(3.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 3 binary64)) y) < -5e21

   1. Initial program 99.7%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(x \cdot 3\right) \cdot y\right), \color{blue}{z}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \left(3 \cdot y\right)\right), z\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \left(3 \cdot y\right)\right), z\right) \]

      4. *-lowering-*.f64 99.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(3, y\right)\right), z\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(3, \color{blue}{\left(x \cdot y\right)}\right) \]

      2. *-lowering-*.f64 80.6%

         \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right) \]

   7. Simplified 80.6%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot 3\right), \color{blue}{y}\right) \]

      4. *-lowering-*.f64 80.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, 3\right), y\right) \]

   9. Applied egg-rr 80.6%

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

   if -5e21 < (*.f64 (*.f64 x #s(literal 3 binary64)) y) < 2e-78

   1. Initial program 99.9%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(x \cdot 3\right) \cdot y\right), \color{blue}{z}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \left(3 \cdot y\right)\right), z\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \left(3 \cdot y\right)\right), z\right) \]

      4. *-lowering-*.f64 99.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(3, y\right)\right), z\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{z} \]

      3. --lowering--.f64 81.8%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{z}\right) \]

   7. Simplified 81.8%

      \[\leadsto \color{blue}{0 - z} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 81.8%

         \[\leadsto \mathsf{neg.f64}\left(z\right) \]

   9. Applied egg-rr 81.8%

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

   if 2e-78 < (*.f64 (*.f64 x #s(literal 3 binary64)) y)

   1. Initial program 99.8%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(x \cdot 3\right) \cdot y\right), \color{blue}{z}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \left(3 \cdot y\right)\right), z\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \left(3 \cdot y\right)\right), z\right) \]

      4. *-lowering-*.f64 99.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(3, y\right)\right), z\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(3, \color{blue}{\left(x \cdot y\right)}\right) \]

      2. *-lowering-*.f64 77.2%

         \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right) \]

   7. Simplified 77.2%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 3\right) \cdot y \leq -5 \cdot 10^{+21}:\\ \;\;\;\;\left(x \cdot 3\right) \cdot y\\ \mathbf{elif}\;\left(x \cdot 3\right) \cdot y \leq 2 \cdot 10^{-78}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -5.1 \cdot 10^{-18}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+50}:\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z -5.1e-18) (- 0.0 z) (if (<= z 1.9e+50) (* 3.0 (* x y)) (- 0.0 z))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.1e-18) {
		tmp = 0.0 - z;
	} else if (z <= 1.9e+50) {
		tmp = 3.0 * (x * y);
	} else {
		tmp = 0.0 - 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 <= (-5.1d-18)) then
        tmp = 0.0d0 - z
    else if (z <= 1.9d+50) then
        tmp = 3.0d0 * (x * y)
    else
        tmp = 0.0d0 - 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 <= -5.1e-18) {
		tmp = 0.0 - z;
	} else if (z <= 1.9e+50) {
		tmp = 3.0 * (x * y);
	} else {
		tmp = 0.0 - z;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if z <= -5.1e-18:
		tmp = 0.0 - z
	elif z <= 1.9e+50:
		tmp = 3.0 * (x * y)
	else:
		tmp = 0.0 - z
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (z <= -5.1e-18)
		tmp = Float64(0.0 - z);
	elseif (z <= 1.9e+50)
		tmp = Float64(3.0 * Float64(x * y));
	else
		tmp = Float64(0.0 - 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 <= -5.1e-18)
		tmp = 0.0 - z;
	elseif (z <= 1.9e+50)
		tmp = 3.0 * (x * y);
	else
		tmp = 0.0 - 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[z, -5.1e-18], N[(0.0 - z), $MachinePrecision], If[LessEqual[z, 1.9e+50], N[(3.0 * N[(x * y), $MachinePrecision]), $MachinePrecision], N[(0.0 - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.09999999999999983e-18 or 1.89999999999999994e50 < z

   1. Initial program 100.0%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(x \cdot 3\right) \cdot y\right), \color{blue}{z}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \left(3 \cdot y\right)\right), z\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \left(3 \cdot y\right)\right), z\right) \]

      4. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(3, y\right)\right), z\right) \]

   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

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{z} \]

      3. --lowering--.f64 77.0%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{z}\right) \]

   7. Simplified 77.0%

      \[\leadsto \color{blue}{0 - z} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 77.0%

         \[\leadsto \mathsf{neg.f64}\left(z\right) \]

   9. Applied egg-rr 77.0%

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

   if -5.09999999999999983e-18 < z < 1.89999999999999994e50

   1. Initial program 99.7%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(x \cdot 3\right) \cdot y\right), \color{blue}{z}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \left(3 \cdot y\right)\right), z\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \left(3 \cdot y\right)\right), z\right) \]

      4. *-lowering-*.f64 99.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(3, y\right)\right), z\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(3, \color{blue}{\left(x \cdot y\right)}\right) \]

      2. *-lowering-*.f64 73.2%

         \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right) \]

   7. Simplified 73.2%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.1 \cdot 10^{-18}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+50}:\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 1.0× speedup

Math

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

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. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\left(x \cdot 3\right) \cdot y\right), \color{blue}{z}\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \left(3 \cdot y\right)\right), z\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \left(3 \cdot y\right)\right), z\right) \]

   4. *-lowering-*.f64 99.8%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(3, y\right)\right), z\right) \]

3. Simplified 99.8%

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

Alternative 6: 50.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ 0 - z \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (- 0.0 z))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return 0.0 - 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 = 0.0d0 - z
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	return 0.0 - z;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return 0.0 - z

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(0.0 - z)
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = 0.0 - z;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(0.0 - z), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\left(x \cdot 3\right) \cdot y\right), \color{blue}{z}\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \left(3 \cdot y\right)\right), z\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \left(3 \cdot y\right)\right), z\right) \]

   4. *-lowering-*.f64 99.8%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(3, y\right)\right), z\right) \]

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

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

      \[\leadsto 0 - \color{blue}{z} \]

   3. --lowering--.f64 51.5%

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{z}\right) \]

7. Simplified 51.5%

   \[\leadsto \color{blue}{0 - z} \]
8. Step-by-step derivation

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 51.5%

      \[\leadsto \mathsf{neg.f64}\left(z\right) \]

9. Applied egg-rr 51.5%

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

10. Final simplification 51.5%

    \[\leadsto 0 - z \]
11. Add Preprocessing

Alternative 7: 2.2% accurate, 7.0× 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 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. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\left(x \cdot 3\right) \cdot y\right), \color{blue}{z}\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \left(3 \cdot y\right)\right), z\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \left(3 \cdot y\right)\right), z\right) \]

   4. *-lowering-*.f64 99.8%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(3, y\right)\right), z\right) \]

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

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

      \[\leadsto 0 - \color{blue}{z} \]

   3. --lowering--.f64 51.5%

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{z}\right) \]

7. Simplified 51.5%

   \[\leadsto \color{blue}{0 - z} \]
8. Step-by-step derivation

   1. flip3-- N/A

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

   2. metadata-eval N/A

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

   3. +-lft-identity N/A

      \[\leadsto \frac{{0}^{3} - {z}^{3}}{z \cdot z + \color{blue}{0 \cdot z}} \]

   4. distribute-rgt-out N/A

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

   5. +-commutative N/A

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

   6. +-lft-identity N/A

      \[\leadsto \frac{{0}^{3} - {z}^{3}}{z \cdot z} \]

   7. metadata-eval N/A

      \[\leadsto \frac{0 - {z}^{3}}{z \cdot z} \]

   8. sub0-neg N/A

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

   9. cube-neg N/A

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

   10. sub0-neg N/A

       \[\leadsto \frac{{\left(0 - z\right)}^{3}}{z \cdot z} \]

   11. sqr-pow N/A

       \[\leadsto \frac{{\left(0 - z\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(0 - z\right)}^{\left(\frac{3}{2}\right)}}{\color{blue}{z} \cdot z} \]

   12. unpow-prod-down N/A

       \[\leadsto \frac{{\left(\left(0 - z\right) \cdot \left(0 - z\right)\right)}^{\left(\frac{3}{2}\right)}}{\color{blue}{z} \cdot z} \]

   13. sub0-neg N/A

       \[\leadsto \frac{{\left(\left(\mathsf{neg}\left(z\right)\right) \cdot \left(0 - z\right)\right)}^{\left(\frac{3}{2}\right)}}{z \cdot z} \]

   14. sub0-neg N/A

       \[\leadsto \frac{{\left(\left(\mathsf{neg}\left(z\right)\right) \cdot \left(\mathsf{neg}\left(z\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{z \cdot z} \]

   15. sqr-neg N/A

       \[\leadsto \frac{{\left(z \cdot z\right)}^{\left(\frac{3}{2}\right)}}{z \cdot z} \]

   16. unpow-prod-down N/A

       \[\leadsto \frac{{z}^{\left(\frac{3}{2}\right)} \cdot {z}^{\left(\frac{3}{2}\right)}}{\color{blue}{z} \cdot z} \]

   17. sqr-pow N/A

       \[\leadsto \frac{{z}^{3}}{\color{blue}{z} \cdot z} \]

   18. pow2 N/A

       \[\leadsto \frac{{z}^{3}}{{z}^{\color{blue}{2}}} \]

   19. pow-div N/A

       \[\leadsto {z}^{\color{blue}{\left(3 - 2\right)}} \]

   20. metadata-eval N/A

       \[\leadsto {z}^{1} \]

   21. unpow1 2.3%

       \[\leadsto z \]

9. Applied egg-rr 2.3%

   \[\leadsto \color{blue}{z} \]
10. 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 2024145 
(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))