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

Percentage Accurate: 99.8% → 99.8%

Time: 4.8s

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

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

FPCore

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

C

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

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return fma(Float64(3.0 * y), x, Float64(-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 * y), $MachinePrecision] * x + (-z)), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in x around 0

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

   1. *-lft-identity N/A

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

   2. metadata-eval N/A

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

   3. fp-cancel-sign-sub-inv N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f64 99.9

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

5. Applied rewrites 99.9%

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

Alternative 2: 77.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := \left(x \cdot 3\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+41} \lor \neg \left(t\_0 \leq 10^{+113}\right):\\ \;\;\;\;\left(y \cdot 3\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

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 -1e+41) (not (<= t_0 1e+113))) (* (* y 3.0) 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 <= -1e+41) || !(t_0 <= 1e+113)) {
		tmp = (y * 3.0) * 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 <= (-1d+41)) .or. (.not. (t_0 <= 1d+113))) then
        tmp = (y * 3.0d0) * 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 <= -1e+41) || !(t_0 <= 1e+113)) {
		tmp = (y * 3.0) * 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 <= -1e+41) or not (t_0 <= 1e+113):
		tmp = (y * 3.0) * 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 <= -1e+41) || !(t_0 <= 1e+113))
		tmp = Float64(Float64(y * 3.0) * 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 <= -1e+41) || ~((t_0 <= 1e+113)))
		tmp = (y * 3.0) * 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, -1e+41], N[Not[LessEqual[t$95$0, 1e+113]], $MachinePrecision]], N[(N[(y * 3.0), $MachinePrecision] * x), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x #s(literal 3 binary64)) y) < -1.00000000000000001e41 or 1e113 < (*.f64 (*.f64 x #s(literal 3 binary64)) y)

   1. Initial program 98.8%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. metadata-eval N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 83.2%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 83.7

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

   10. Applied rewrites 83.7%

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

   12. Applied rewrites 83.8%

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

   if -1.00000000000000001e41 < (*.f64 (*.f64 x #s(literal 3 binary64)) y) < 1e113

   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 79.4

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

   5. Applied rewrites 79.4%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 3\right) \cdot y \leq -1 \cdot 10^{+41} \lor \neg \left(\left(x \cdot 3\right) \cdot y \leq 10^{+113}\right):\\ \;\;\;\;\left(y \cdot 3\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := \left(x \cdot 3\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+41} \lor \neg \left(t\_0 \leq 2 \cdot 10^{+99}\right):\\ \;\;\;\;\left(3 \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

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 -1e+41) (not (<= t_0 2e+99))) (* (* 3.0 x) y) (- 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 <= -1e+41) || !(t_0 <= 2e+99)) {
		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) :: t_0
    real(8) :: tmp
    t_0 = (x * 3.0d0) * y
    if ((t_0 <= (-1d+41)) .or. (.not. (t_0 <= 2d+99))) 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 t_0 = (x * 3.0) * y;
	double tmp;
	if ((t_0 <= -1e+41) || !(t_0 <= 2e+99)) {
		tmp = (3.0 * x) * y;
	} 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 <= -1e+41) or not (t_0 <= 2e+99):
		tmp = (3.0 * x) * y
	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 <= -1e+41) || !(t_0 <= 2e+99))
		tmp = Float64(Float64(3.0 * 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)
	t_0 = (x * 3.0) * y;
	tmp = 0.0;
	if ((t_0 <= -1e+41) || ~((t_0 <= 2e+99)))
		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_] := Block[{t$95$0 = N[(N[(x * 3.0), $MachinePrecision] * y), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e+41], N[Not[LessEqual[t$95$0, 2e+99]], $MachinePrecision]], N[(N[(3.0 * x), $MachinePrecision] * y), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x #s(literal 3 binary64)) y) < -1.00000000000000001e41 or 1.9999999999999999e99 < (*.f64 (*.f64 x #s(literal 3 binary64)) y)

   1. Initial program 98.8%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. metadata-eval N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 82.5%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 83.1

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

   10. Applied rewrites 83.1%

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

   12. Applied rewrites 83.2%

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

   if -1.00000000000000001e41 < (*.f64 (*.f64 x #s(literal 3 binary64)) y) < 1.9999999999999999e99

   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 79.7

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

   5. Applied rewrites 79.7%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 3\right) \cdot y \leq -1 \cdot 10^{+41} \lor \neg \left(\left(x \cdot 3\right) \cdot y \leq 2 \cdot 10^{+99}\right):\\ \;\;\;\;\left(3 \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 50.5% 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 55.3

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

5. Applied rewrites 55.3%

   \[\leadsto \color{blue}{-z} \]
6. 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 2024339 
(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))