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

Percentage Accurate: 99.8% → 99.8%

Time: 14.8s

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. Add Preprocessing

Alternative 2: 69.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+37} \lor \neg \left(z \leq 1.5 \cdot 10^{+122}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(3 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.1e+37) (not (<= z 1.5e+122))) (- z) (* x (* 3.0 y))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.1e+37) || !(z <= 1.5e+122)) {
		tmp = -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) :: tmp
    if ((z <= (-1.1d+37)) .or. (.not. (z <= 1.5d+122))) then
        tmp = -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 tmp;
	if ((z <= -1.1e+37) || !(z <= 1.5e+122)) {
		tmp = -z;
	} else {
		tmp = x * (3.0 * y);
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if (z <= -1.1e+37) or not (z <= 1.5e+122):
		tmp = -z
	else:
		tmp = x * (3.0 * y)
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.1e+37) || !(z <= 1.5e+122))
		tmp = Float64(-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)
	tmp = 0.0;
	if ((z <= -1.1e+37) || ~((z <= 1.5e+122)))
		tmp = -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_] := If[Or[LessEqual[z, -1.1e+37], N[Not[LessEqual[z, 1.5e+122]], $MachinePrecision]], (-z), N[(x * N[(3.0 * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.1e37 or 1.49999999999999993e122 < 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 81.2%

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

      1. mul-1-neg 81.2%

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

   7. Simplified 81.2%

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

   if -1.1e37 < z < 1.49999999999999993e122

   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 x around 0 99.7%

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

   6. Taylor expanded in x around inf 75.7%

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

      1. *-commutative 75.7%

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

      2. associate-*r* 75.7%

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

   8. Simplified 75.7%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+37} \lor \neg \left(z \leq 1.5 \cdot 10^{+122}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(3 \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+36} \lor \neg \left(z \leq 1.5 \cdot 10^{+122}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

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 -2.1e+36) (not (<= z 1.5e+122))) (- z) (* 3.0 (* x y))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.1e+36) || !(z <= 1.5e+122)) {
		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 <= (-2.1d+36)) .or. (.not. (z <= 1.5d+122))) 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 <= -2.1e+36) || !(z <= 1.5e+122)) {
		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 <= -2.1e+36) or not (z <= 1.5e+122):
		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 <= -2.1e+36) || !(z <= 1.5e+122))
		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 <= -2.1e+36) || ~((z <= 1.5e+122)))
		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, -2.1e+36], N[Not[LessEqual[z, 1.5e+122]], $MachinePrecision]], (-z), N[(3.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.10000000000000004e36 or 1.49999999999999993e122 < 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 81.2%

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

      1. mul-1-neg 81.2%

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

   7. Simplified 81.2%

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

   if -2.10000000000000004e36 < z < 1.49999999999999993e122

   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 x around 0 99.7%

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

   6. Taylor expanded in x around inf 75.7%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+36} \lor \neg \left(z \leq 1.5 \cdot 10^{+122}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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. Add Preprocessing

Alternative 5: 50.3% 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 45.3%

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

   1. mul-1-neg 45.3%

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

7. Simplified 45.3%

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

Alternative 6: 2.3% 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. 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. Step-by-step derivation

   1. associate-*r* 99.8%

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

   2. *-commutative 99.8%

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

   3. associate-*r* 99.8%

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

   4. fma-neg 99.8%

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

   5. add-sqr-sqrt 48.2%

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

   6. sqrt-unprod 62.7%

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

   7. sqr-neg 62.7%

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

   8. sqrt-unprod 27.1%

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

   9. add-sqr-sqrt 55.1%

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

6. Applied egg-rr 55.1%

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

7. Taylor expanded in y around 0 2.4%

   \[\leadsto \color{blue}{z} \]
8. 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 2024143 
(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))