Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, A

Percentage Accurate: 99.9% → 100.0%

Time: 4.2s

Alternatives: 4

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ x - \left(y \cdot 4\right) \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- x (* (* y 4.0) z)))

C

double code(double x, double y, double z) {
	return x - ((y * 4.0) * 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 - ((y * 4.0d0) * z)
end function

Java

public static double code(double x, double y, double z) {
	return x - ((y * 4.0) * z);
}

Python

def code(x, y, z):
	return x - ((y * 4.0) * z)

Julia

function code(x, y, z)
	return Float64(x - Float64(Float64(y * 4.0) * z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x - ((y * 4.0) * z);
end

Wolfram

code[x_, y_, z_] := N[(x - N[(N[(y * 4.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - \left(y \cdot 4\right) \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- x (* (* y 4.0) z)))

C

double code(double x, double y, double z) {
	return x - ((y * 4.0) * 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 - ((y * 4.0d0) * z)
end function

Java

public static double code(double x, double y, double z) {
	return x - ((y * 4.0) * z);
}

Python

def code(x, y, z):
	return x - ((y * 4.0) * z)

Julia

function code(x, y, z)
	return Float64(x - Float64(Float64(y * 4.0) * z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x - ((y * 4.0) * z);
end

Wolfram

code[x_, y_, z_] := N[(x - N[(N[(y * 4.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, z \cdot -4, x\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma y (* z -4.0) x))

C

double code(double x, double y, double z) {
	return fma(y, (z * -4.0), x);
}

Julia

function code(x, y, z)
	return fma(y, Float64(z * -4.0), x)
end

Wolfram

code[x_, y_, z_] := N[(y * N[(z * -4.0), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x - \left(y \cdot 4\right) \cdot z \]
2. Step-by-step derivation

   1. sub-neg 100.0%

      \[\leadsto \color{blue}{x + \left(-\left(y \cdot 4\right) \cdot z\right)} \]

   2. distribute-rgt-neg-out 100.0%

      \[\leadsto x + \color{blue}{\left(y \cdot 4\right) \cdot \left(-z\right)} \]

   3. +-commutative 100.0%

      \[\leadsto \color{blue}{\left(y \cdot 4\right) \cdot \left(-z\right) + x} \]

   4. associate-*l* 100.0%

      \[\leadsto \color{blue}{y \cdot \left(4 \cdot \left(-z\right)\right)} + x \]

   5. distribute-rgt-neg-in 100.0%

      \[\leadsto y \cdot \color{blue}{\left(-4 \cdot z\right)} + x \]

   6. *-commutative 100.0%

      \[\leadsto y \cdot \left(-\color{blue}{z \cdot 4}\right) + x \]

   7. fma-define 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, -z \cdot 4, x\right)} \]

   8. distribute-rgt-neg-in 100.0%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{z \cdot \left(-4\right)}, x\right) \]

   9. metadata-eval 100.0%

      \[\leadsto \mathsf{fma}\left(y, z \cdot \color{blue}{-4}, x\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, z \cdot -4, x\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 68.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-115} \lor \neg \left(z \leq 8.5 \cdot 10^{+60}\right):\\ \;\;\;\;y \cdot \left(z \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.4e-115) (not (<= z 8.5e+60))) (* y (* z -4.0)) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.4e-115) || !(z <= 8.5e+60)) {
		tmp = y * (z * -4.0);
	} else {
		tmp = 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) :: tmp
    if ((z <= (-1.4d-115)) .or. (.not. (z <= 8.5d+60))) then
        tmp = y * (z * (-4.0d0))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.4e-115) || !(z <= 8.5e+60)) {
		tmp = y * (z * -4.0);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.4e-115) or not (z <= 8.5e+60):
		tmp = y * (z * -4.0)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.4e-115) || !(z <= 8.5e+60))
		tmp = Float64(y * Float64(z * -4.0));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.4e-115) || ~((z <= 8.5e+60)))
		tmp = y * (z * -4.0);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.4e-115], N[Not[LessEqual[z, 8.5e+60]], $MachinePrecision]], N[(y * N[(z * -4.0), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -1.39999999999999994e-115 or 8.50000000000000064e60 < z

   1. Initial program 100.0%

      \[x - \left(y \cdot 4\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 69.0%

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

      1. associate-*r* 69.0%

         \[\leadsto \color{blue}{\left(-4 \cdot y\right) \cdot z} \]

      2. *-commutative 69.0%

         \[\leadsto \color{blue}{\left(y \cdot -4\right)} \cdot z \]

      3. associate-*r* 69.0%

         \[\leadsto \color{blue}{y \cdot \left(-4 \cdot z\right)} \]

   5. Simplified 69.0%

      \[\leadsto \color{blue}{y \cdot \left(-4 \cdot z\right)} \]

   if -1.39999999999999994e-115 < z < 8.50000000000000064e60

   1. Initial program 100.0%

      \[x - \left(y \cdot 4\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 69.6%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-115} \lor \neg \left(z \leq 8.5 \cdot 10^{+60}\right):\\ \;\;\;\;y \cdot \left(z \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - z \cdot \left(y \cdot 4\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- x (* z (* y 4.0))))

C

double code(double x, double y, double z) {
	return x - (z * (y * 4.0));
}

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 - (z * (y * 4.0d0))
end function

Java

public static double code(double x, double y, double z) {
	return x - (z * (y * 4.0));
}

Python

def code(x, y, z):
	return x - (z * (y * 4.0))

Julia

function code(x, y, z)
	return Float64(x - Float64(z * Float64(y * 4.0)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x - (z * (y * 4.0));
end

Wolfram

code[x_, y_, z_] := N[(x - N[(z * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x - \left(y \cdot 4\right) \cdot z \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto x - z \cdot \left(y \cdot 4\right) \]
4. Add Preprocessing

Alternative 4: 51.2% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 100.0%

   \[x - \left(y \cdot 4\right) \cdot z \]
2. Add Preprocessing

3. Taylor expanded in x around inf 47.7%

   \[\leadsto \color{blue}{x} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024116 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, A"
  :precision binary64
  (- x (* (* y 4.0) z)))