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

Percentage Accurate: 99.8% → 99.8%

Time: 6.4s

Alternatives: 4

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 99.5%

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

Alternative 2: 78.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := \left(x \cdot 3\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-71} \lor \neg \left(t\_0 \leq 5 \cdot 10^{+80}\right):\\ \;\;\;\;\left(3 \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

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 -5e-71) (not (<= t_0 5e+80))) (* (* 3.0 y) x) (- 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 <= -5e-71) || !(t_0 <= 5e+80)) {
		tmp = (3.0 * y) * x;
	} 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 <= (-5d-71)) .or. (.not. (t_0 <= 5d+80))) then
        tmp = (3.0d0 * y) * x
    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 <= -5e-71) || !(t_0 <= 5e+80)) {
		tmp = (3.0 * y) * x;
	} 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 <= -5e-71) or not (t_0 <= 5e+80):
		tmp = (3.0 * y) * x
	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 <= -5e-71) || !(t_0 <= 5e+80))
		tmp = Float64(Float64(3.0 * y) * x);
	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 <= -5e-71) || ~((t_0 <= 5e+80)))
		tmp = (3.0 * y) * x;
	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, -5e-71], N[Not[LessEqual[t$95$0, 5e+80]], $MachinePrecision]], N[(N[(3.0 * y), $MachinePrecision] * x), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x #s(literal 3 binary64)) y) < -4.99999999999999998e-71 or 4.99999999999999961e80 < (*.f64 (*.f64 x #s(literal 3 binary64)) y)

   1. Initial program 99.0%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. sqrt-prod N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 19.4

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

   4. Applied rewrites 19.4%

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

   5. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(y \cdot {\left(\sqrt{-3}\right)}^{2}\right)\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(y \cdot {\left(\sqrt{-3}\right)}^{2}\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-3}\right)}^{2} \cdot y}\right)\right) \]

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. distribute-lft-neg-in N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 81.7

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

   7. Applied rewrites 81.7%

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

   9. Applied rewrites 81.8%

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

   11. Applied rewrites 81.8%

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

   if -4.99999999999999998e-71 < (*.f64 (*.f64 x #s(literal 3 binary64)) y) < 4.99999999999999961e80

   1. Initial program 99.9%

      \[\left(x \cdot 3\right) \cdot y - z \]
   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 83.4

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

   5. Applied rewrites 83.4%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 3\right) \cdot y \leq -5 \cdot 10^{-71} \lor \neg \left(\left(x \cdot 3\right) \cdot y \leq 5 \cdot 10^{+80}\right):\\ \;\;\;\;\left(3 \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := \left(x \cdot 3\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-71}:\\ \;\;\;\;\left(3 \cdot y\right) \cdot x\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+80}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* x 3.0) y)))
   (if (<= t_0 -5e-71) (* (* 3.0 y) x) (if (<= t_0 5e+80) (- z) t_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 <= -5e-71) {
		tmp = (3.0 * y) * x;
	} else if (t_0 <= 5e+80) {
		tmp = -z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * 3.0d0) * y
    if (t_0 <= (-5d-71)) then
        tmp = (3.0d0 * y) * x
    else if (t_0 <= 5d+80) then
        tmp = -z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double t_0 = (x * 3.0) * y;
	double tmp;
	if (t_0 <= -5e-71) {
		tmp = (3.0 * y) * x;
	} else if (t_0 <= 5e+80) {
		tmp = -z;
	} else {
		tmp = t_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 <= -5e-71:
		tmp = (3.0 * y) * x
	elif t_0 <= 5e+80:
		tmp = -z
	else:
		tmp = t_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 <= -5e-71)
		tmp = Float64(Float64(3.0 * y) * x);
	elseif (t_0 <= 5e+80)
		tmp = Float64(-z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	t_0 = (x * 3.0) * y;
	tmp = 0.0;
	if (t_0 <= -5e-71)
		tmp = (3.0 * y) * x;
	elseif (t_0 <= 5e+80)
		tmp = -z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * 3.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-71], N[(N[(3.0 * y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$0, 5e+80], (-z), t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 3 binary64)) y) < -4.99999999999999998e-71

   1. Initial program 99.7%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. sqrt-prod N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 8.1

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

   4. Applied rewrites 8.1%

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

   5. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(y \cdot {\left(\sqrt{-3}\right)}^{2}\right)\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(y \cdot {\left(\sqrt{-3}\right)}^{2}\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-3}\right)}^{2} \cdot y}\right)\right) \]

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. distribute-lft-neg-in N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 80.4

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

   7. Applied rewrites 80.4%

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

   9. Applied rewrites 80.4%

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

   11. Applied rewrites 80.4%

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

   if -4.99999999999999998e-71 < (*.f64 (*.f64 x #s(literal 3 binary64)) y) < 4.99999999999999961e80

   1. Initial program 99.9%

      \[\left(x \cdot 3\right) \cdot y - z \]
   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 83.4

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

   5. Applied rewrites 83.4%

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

   if 4.99999999999999961e80 < (*.f64 (*.f64 x #s(literal 3 binary64)) y)

   1. Initial program 97.8%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. sqrt-prod N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 36.2

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

   4. Applied rewrites 36.2%

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

   5. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(y \cdot {\left(\sqrt{-3}\right)}^{2}\right)\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(y \cdot {\left(\sqrt{-3}\right)}^{2}\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-3}\right)}^{2} \cdot y}\right)\right) \]

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. distribute-lft-neg-in N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 83.7

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

   7. Applied rewrites 83.7%

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

   9. Applied rewrites 81.8%

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

Alternative 4: 51.1% 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 99.5%

   \[\left(x \cdot 3\right) \cdot y - z \]
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 55.5

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

5. Applied rewrites 55.5%

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

Developer Target 1: 99.7% 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 2024338 
(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))