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

Percentage Accurate: 99.8% → 99.8%

Time: 3.2s

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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(x * N[(3.0 * y), $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%

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

   2. fma-neg 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 68.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{-9}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -5.4 \cdot 10^{-44} \lor \neg \left(z \leq -9.4 \cdot 10^{-94}\right) \land z \leq 1.18 \cdot 10^{+87}:\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z -6e-9)
   (- z)
   (if (or (<= z -5.4e-44) (and (not (<= z -9.4e-94)) (<= z 1.18e+87)))
     (* 3.0 (* x y))
     (- z))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -6e-9) {
		tmp = -z;
	} else if ((z <= -5.4e-44) || (!(z <= -9.4e-94) && (z <= 1.18e+87))) {
		tmp = 3.0 * (x * y);
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

NOTE: x and y 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 <= (-6d-9)) then
        tmp = -z
    else if ((z <= (-5.4d-44)) .or. (.not. (z <= (-9.4d-94))) .and. (z <= 1.18d+87)) then
        tmp = 3.0d0 * (x * y)
    else
        tmp = -z
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -6e-9) {
		tmp = -z;
	} else if ((z <= -5.4e-44) || (!(z <= -9.4e-94) && (z <= 1.18e+87))) {
		tmp = 3.0 * (x * y);
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if z <= -6e-9:
		tmp = -z
	elif (z <= -5.4e-44) or (not (z <= -9.4e-94) and (z <= 1.18e+87)):
		tmp = 3.0 * (x * y)
	else:
		tmp = -z
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (z <= -6e-9)
		tmp = Float64(-z);
	elseif ((z <= -5.4e-44) || (!(z <= -9.4e-94) && (z <= 1.18e+87)))
		tmp = Float64(3.0 * Float64(x * y));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -6e-9)
		tmp = -z;
	elseif ((z <= -5.4e-44) || (~((z <= -9.4e-94)) && (z <= 1.18e+87)))
		tmp = 3.0 * (x * y);
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[z, -6e-9], (-z), If[Or[LessEqual[z, -5.4e-44], And[N[Not[LessEqual[z, -9.4e-94]], $MachinePrecision], LessEqual[z, 1.18e+87]]], N[(3.0 * N[(x * y), $MachinePrecision]), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -5.99999999999999996e-9 or -5.3999999999999998e-44 < z < -9.40000000000000007e-94 or 1.1799999999999999e87 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 83.7%

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

      1. mul-1-neg 83.7%

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

   4. Simplified 83.7%

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

   if -5.99999999999999996e-9 < z < -5.3999999999999998e-44 or -9.40000000000000007e-94 < z < 1.1799999999999999e87

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 99.7%

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

   3. Taylor expanded in y around inf 81.0%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{-9}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -5.4 \cdot 10^{-44} \lor \neg \left(z \leq -9.4 \cdot 10^{-94}\right) \land z \leq 1.18 \cdot 10^{+87}:\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x and y 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);
double code(double x, double y, double z) {
	return (3.0 * (x * y)) - z;
}

Fortran

NOTE: x and y 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;
public static double code(double x, double y, double z) {
	return (3.0 * (x * y)) - z;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	return (3.0 * (x * y)) - z

Julia

x, y = sort([x, y])
function code(x, y, z)
	return Float64(Float64(3.0 * Float64(x * y)) - z)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y, z)
	tmp = (3.0 * (x * y)) - z;
end

Wolfram

NOTE: x and y 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. Taylor expanded in x around 0 99.8%

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

3. Final simplification 99.8%

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

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x and y 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);
double code(double x, double y, double z) {
	return (x * (3.0 * y)) - z;
}

Fortran

NOTE: x and y 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;
public static double code(double x, double y, double z) {
	return (x * (3.0 * y)) - z;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	return (x * (3.0 * y)) - z

Julia

x, y = sort([x, y])
function code(x, y, z)
	return Float64(Float64(x * Float64(3.0 * y)) - z)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y, z)
	tmp = (x * (3.0 * y)) - z;
end

Wolfram

NOTE: x and y 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%

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

   2. fma-neg 99.8%

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

3. Simplified 99.8%

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

   1. fma-neg 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 5: 51.3% accurate, 3.5× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y, double z) {
	return -z;
}

Fortran

NOTE: x and y 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;
public static double code(double x, double y, double z) {
	return -z;
}

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x and y 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. Taylor expanded in x around 0 53.8%

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

   1. mul-1-neg 53.8%

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

4. Simplified 53.8%

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

5. Final simplification 53.8%

   \[\leadsto -z \]

Alternative 6: 2.3% accurate, 7.0× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y, double z) {
	return z;
}

Fortran

NOTE: x and y 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;
public static double code(double x, double y, double z) {
	return z;
}

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x and y 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. *-commutative 99.8%

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

   2. associate-*r* 99.8%

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

   3. fma-neg 99.8%

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

   4. add-sqr-sqrt 53.2%

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

   5. sqrt-unprod 61.6%

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

   6. sqr-neg 61.6%

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

   7. sqrt-unprod 22.8%

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

   8. add-sqr-sqrt 47.6%

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

3. Applied egg-rr 47.6%

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

4. Taylor expanded in y around 0 2.0%

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

5. Final simplification 2.0%

   \[\leadsto z \]

Developer target: 99.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (- (* x (* 3.0 y)) z))

C

double code(double x, double y, double z) {
	return (x * (3.0 * y)) - z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * (3.0d0 * y)) - z
end function

Java

public static double code(double x, double y, double z) {
	return (x * (3.0 * y)) - z;
}

Python

def code(x, y, z):
	return (x * (3.0 * y)) - z

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(3.0 * y)) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * (3.0 * y)) - z;
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(3.0 * y), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]

Reproduce

herbie shell --seed 2023252 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (- (* x (* 3.0 y)) z)

  (- (* (* x 3.0) y) z))