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

Percentage Accurate: 99.8% → 99.8%

Time: 6.6s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.9%

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

Alternative 2: 79.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot 3\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-20} \lor \neg \left(t\_0 \leq 2 \cdot 10^{+18}\right):\\ \;\;\;\;\left(3 \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* x 3.0) y)))
   (if (or (<= t_0 -5e-20) (not (<= t_0 2e+18))) (* (* 3.0 x) y) (- z))))

C

double code(double x, double y, double z) {
	double t_0 = (x * 3.0) * y;
	double tmp;
	if ((t_0 <= -5e-20) || !(t_0 <= 2e+18)) {
		tmp = (3.0 * x) * y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

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-20)) .or. (.not. (t_0 <= 2d+18))) then
        tmp = (3.0d0 * x) * y
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x * 3.0) * y;
	double tmp;
	if ((t_0 <= -5e-20) || !(t_0 <= 2e+18)) {
		tmp = (3.0 * x) * y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x * 3.0) * y
	tmp = 0
	if (t_0 <= -5e-20) or not (t_0 <= 2e+18):
		tmp = (3.0 * x) * y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * 3.0) * y)
	tmp = 0.0
	if ((t_0 <= -5e-20) || !(t_0 <= 2e+18))
		tmp = Float64(Float64(3.0 * x) * y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x * 3.0) * y;
	tmp = 0.0;
	if ((t_0 <= -5e-20) || ~((t_0 <= 2e+18)))
		tmp = (3.0 * x) * y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * 3.0), $MachinePrecision] * y), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e-20], N[Not[LessEqual[t$95$0, 2e+18]], $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) < -4.9999999999999999e-20 or 2e18 < (*.f64 (*.f64 x #s(literal 3 binary64)) y)

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-neg-in N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 84.4

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

   9. Applied rewrites 84.4%

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

   11. Applied rewrites 84.5%

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

   if -4.9999999999999999e-20 < (*.f64 (*.f64 x #s(literal 3 binary64)) y) < 2e18

   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 83.1

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

   5. Applied rewrites 83.1%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 3\right) \cdot y \leq -5 \cdot 10^{-20} \lor \neg \left(\left(x \cdot 3\right) \cdot y \leq 2 \cdot 10^{+18}\right):\\ \;\;\;\;\left(3 \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot 3\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-20}:\\ \;\;\;\;\left(3 \cdot x\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+18}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* x 3.0) y)))
   (if (<= t_0 -5e-20)
     (* (* 3.0 x) y)
     (if (<= t_0 2e+18) (- z) (* (* 3.0 y) x)))))

C

double code(double x, double y, double z) {
	double t_0 = (x * 3.0) * y;
	double tmp;
	if (t_0 <= -5e-20) {
		tmp = (3.0 * x) * y;
	} else if (t_0 <= 2e+18) {
		tmp = -z;
	} else {
		tmp = (3.0 * y) * x;
	}
	return tmp;
}

Fortran

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-20)) then
        tmp = (3.0d0 * x) * y
    else if (t_0 <= 2d+18) then
        tmp = -z
    else
        tmp = (3.0d0 * y) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x * 3.0) * y;
	double tmp;
	if (t_0 <= -5e-20) {
		tmp = (3.0 * x) * y;
	} else if (t_0 <= 2e+18) {
		tmp = -z;
	} else {
		tmp = (3.0 * y) * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x * 3.0) * y
	tmp = 0
	if t_0 <= -5e-20:
		tmp = (3.0 * x) * y
	elif t_0 <= 2e+18:
		tmp = -z
	else:
		tmp = (3.0 * y) * x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * 3.0) * y)
	tmp = 0.0
	if (t_0 <= -5e-20)
		tmp = Float64(Float64(3.0 * x) * y);
	elseif (t_0 <= 2e+18)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(3.0 * y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x * 3.0) * y;
	tmp = 0.0;
	if (t_0 <= -5e-20)
		tmp = (3.0 * x) * y;
	elseif (t_0 <= 2e+18)
		tmp = -z;
	else
		tmp = (3.0 * y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * 3.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-20], N[(N[(3.0 * x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 2e+18], (-z), N[(N[(3.0 * y), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 3 binary64)) y) < -4.9999999999999999e-20

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-neg-in N/A

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

   6. Applied rewrites 99.2%

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

   7. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 81.3

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

   9. Applied rewrites 81.3%

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

   11. Applied rewrites 81.4%

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

   if -4.9999999999999999e-20 < (*.f64 (*.f64 x #s(literal 3 binary64)) y) < 2e18

   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 83.1

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

   5. Applied rewrites 83.1%

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

   if 2e18 < (*.f64 (*.f64 x #s(literal 3 binary64)) y)

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-neg-in N/A

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 87.8

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

   9. Applied rewrites 87.8%

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

   11. Applied rewrites 87.7%

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

Alternative 4: 50.3% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ -z \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return -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 = -z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := (-z)

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

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

   2. lower-neg.f64 49.4

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

5. Applied rewrites 49.4%

   \[\leadsto \color{blue}{-z} \]
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 2024337 
(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))