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

Percentage Accurate: 99.8% → 99.8%

Time: 7.7s

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 99.8%

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

Alternative 2: 79.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := \left(x \cdot 3\right) \cdot y\\ t_1 := x \cdot \left(3 \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-39}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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)) (t_1 (* x (* 3.0 y))))
   (if (<= t_0 -0.1) t_1 (if (<= t_0 1e-39) (- 0.0 z) t_1))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = (x * 3.0) * y;
	double t_1 = x * (3.0 * y);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = t_1;
	} else if (t_0 <= 1e-39) {
		tmp = 0.0 - z;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (x * 3.0d0) * y
    t_1 = x * (3.0d0 * y)
    if (t_0 <= (-0.1d0)) then
        tmp = t_1
    else if (t_0 <= 1d-39) then
        tmp = 0.0d0 - z
    else
        tmp = t_1
    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 t_1 = x * (3.0 * y);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = t_1;
	} else if (t_0 <= 1e-39) {
		tmp = 0.0 - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	t_0 = (x * 3.0) * y
	t_1 = x * (3.0 * y)
	tmp = 0
	if t_0 <= -0.1:
		tmp = t_1
	elif t_0 <= 1e-39:
		tmp = 0.0 - z
	else:
		tmp = t_1
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(Float64(x * 3.0) * y)
	t_1 = Float64(x * Float64(3.0 * y))
	tmp = 0.0
	if (t_0 <= -0.1)
		tmp = t_1;
	elseif (t_0 <= 1e-39)
		tmp = Float64(0.0 - z);
	else
		tmp = t_1;
	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;
	t_1 = x * (3.0 * y);
	tmp = 0.0;
	if (t_0 <= -0.1)
		tmp = t_1;
	elseif (t_0 <= 1e-39)
		tmp = 0.0 - z;
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(x * N[(3.0 * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.1], t$95$1, If[LessEqual[t$95$0, 1e-39], N[(0.0 - z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x #s(literal 3 binary64)) y) < -0.10000000000000001 or 9.99999999999999929e-40 < (*.f64 (*.f64 x #s(literal 3 binary64)) y)

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{3 \cdot \left(x \cdot y\right)} \]
   4. 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. metadata-eval N/A

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

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

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

      5. *-commutative N/A

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

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

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

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

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 83.8%

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

   5. Simplified 83.8%

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

   if -0.10000000000000001 < (*.f64 (*.f64 x #s(literal 3 binary64)) y) < 9.99999999999999929e-40

   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

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 82.7%

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

   5. Simplified 82.7%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 82.7%

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

   7. Applied egg-rr 82.7%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 3\right) \cdot y \leq -0.1:\\ \;\;\;\;x \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;\left(x \cdot 3\right) \cdot y \leq 10^{-39}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(3 \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.7% 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. 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

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

   2. neg-sub0 N/A

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

   3. --lowering--.f64 51.7%

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

5. Simplified 51.7%

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

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 51.7%

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

7. Applied egg-rr 51.7%

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

8. Final simplification 51.7%

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

Alternative 4: 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. Add Preprocessing

4. Applied egg-rr 5.2%

   \[\leadsto \color{blue}{\frac{\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 27\right) \cdot \left(y \cdot \left(y \cdot y\right)\right) + z \cdot \left(z \cdot z\right)}{y \cdot \left(y \cdot \left(\left(x \cdot x\right) \cdot 9\right)\right) + z \cdot \left(\left(x \cdot 3\right) \cdot y + z\right)}} \]

5. Taylor expanded in x around 0

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

7. Simplified 2.5%

   \[\leadsto \color{blue}{z} \]
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 2024192 
(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))