Diagrams.Solve.Polynomial:quartForm from diagrams-solve-0.1, D

Percentage Accurate: 100.0% → 100.0%

Time: 4.0s

Alternatives: 4

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot y}{2} - \frac{z}{8} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (/ (* x y) 2.0) (/ z 8.0)))

C

double code(double x, double y, double z) {
	return ((x * y) / 2.0) - (z / 8.0);
}

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 * y) / 2.0d0) - (z / 8.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return ((x * y) / 2.0) - (z / 8.0);
}

Python

def code(x, y, z):
	return ((x * y) / 2.0) - (z / 8.0)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x * y) / 2.0) - Float64(z / 8.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x * y) / 2.0) - (z / 8.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x * y), $MachinePrecision] / 2.0), $MachinePrecision] - N[(z / 8.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot y}{2} - \frac{z}{8} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (/ (* x y) 2.0) (/ z 8.0)))

C

double code(double x, double y, double z) {
	return ((x * y) / 2.0) - (z / 8.0);
}

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 * y) / 2.0d0) - (z / 8.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return ((x * y) / 2.0) - (z / 8.0);
}

Python

def code(x, y, z):
	return ((x * y) / 2.0) - (z / 8.0)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x * y) / 2.0) - Float64(z / 8.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x * y) / 2.0) - (z / 8.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x * y), $MachinePrecision] / 2.0), $MachinePrecision] - N[(z / 8.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{x}{2}, y, -0.125 \cdot z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (/ x 2.0) y (* -0.125 z)))

C

double code(double x, double y, double z) {
	return fma((x / 2.0), y, (-0.125 * z));
}

Julia

function code(x, y, z)
	return fma(Float64(x / 2.0), y, Float64(-0.125 * z))
end

Wolfram

code[x_, y_, z_] := N[(N[(x / 2.0), $MachinePrecision] * y + N[(-0.125 * z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{x \cdot y}{2} - \frac{z}{8} \]
2. Step-by-step derivation

   1. associate-*l/ 100.0%

      \[\leadsto \color{blue}{\frac{x}{2} \cdot y} - \frac{z}{8} \]

   2. fma-neg 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{2}, y, -\frac{z}{8}\right)} \]

   3. distribute-frac-neg 100.0%

      \[\leadsto \mathsf{fma}\left(\frac{x}{2}, y, \color{blue}{\frac{-z}{8}}\right) \]

   4. neg-mul-1 100.0%

      \[\leadsto \mathsf{fma}\left(\frac{x}{2}, y, \frac{\color{blue}{-1 \cdot z}}{8}\right) \]

   5. associate-/l* 99.9%

      \[\leadsto \mathsf{fma}\left(\frac{x}{2}, y, \color{blue}{\frac{-1}{\frac{8}{z}}}\right) \]

   6. associate-/r/ 100.0%

      \[\leadsto \mathsf{fma}\left(\frac{x}{2}, y, \color{blue}{\frac{-1}{8} \cdot z}\right) \]

   7. metadata-eval 100.0%

      \[\leadsto \mathsf{fma}\left(\frac{x}{2}, y, \color{blue}{-0.125} \cdot z\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{2}, y, -0.125 \cdot z\right)} \]
4. Add Preprocessing

5. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(\frac{x}{2}, y, -0.125 \cdot z\right) \]
6. Add Preprocessing

Alternative 2: 71.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-5} \lor \neg \left(z \leq 180000000000\right):\\ \;\;\;\;-0.125 \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2e-5) (not (<= z 180000000000.0)))
   (* -0.125 z)
   (* x (* y 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2e-5) || !(z <= 180000000000.0)) {
		tmp = -0.125 * z;
	} else {
		tmp = x * (y * 0.5);
	}
	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 <= (-2d-5)) .or. (.not. (z <= 180000000000.0d0))) then
        tmp = (-0.125d0) * z
    else
        tmp = x * (y * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2e-5) || !(z <= 180000000000.0)) {
		tmp = -0.125 * z;
	} else {
		tmp = x * (y * 0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2e-5) or not (z <= 180000000000.0):
		tmp = -0.125 * z
	else:
		tmp = x * (y * 0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2e-5) || !(z <= 180000000000.0))
		tmp = Float64(-0.125 * z);
	else
		tmp = Float64(x * Float64(y * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2e-5) || ~((z <= 180000000000.0)))
		tmp = -0.125 * z;
	else
		tmp = x * (y * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2e-5], N[Not[LessEqual[z, 180000000000.0]], $MachinePrecision]], N[(-0.125 * z), $MachinePrecision], N[(x * N[(y * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.00000000000000016e-5 or 1.8e11 < z

   1. Initial program 100.0%

      \[\frac{x \cdot y}{2} - \frac{z}{8} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 75.2%

      \[\leadsto \color{blue}{-0.125 \cdot z} \]

   if -2.00000000000000016e-5 < z < 1.8e11

   1. Initial program 100.0%

      \[\frac{x \cdot y}{2} - \frac{z}{8} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 74.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(x \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-commutative 74.4%

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot 0.5} \]

      2. associate-*r* 74.4%

         \[\leadsto \color{blue}{x \cdot \left(y \cdot 0.5\right)} \]

      3. *-commutative 74.4%

         \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot y\right)} \]

   5. Simplified 74.4%

      \[\leadsto \color{blue}{x \cdot \left(0.5 \cdot y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-5} \lor \neg \left(z \leq 180000000000\right):\\ \;\;\;\;-0.125 \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot y}{2} - \frac{z}{8} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (/ (* x y) 2.0) (/ z 8.0)))

C

double code(double x, double y, double z) {
	return ((x * y) / 2.0) - (z / 8.0);
}

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 * y) / 2.0d0) - (z / 8.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return ((x * y) / 2.0) - (z / 8.0);
}

Python

def code(x, y, z):
	return ((x * y) / 2.0) - (z / 8.0)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x * y) / 2.0) - Float64(z / 8.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x * y) / 2.0) - (z / 8.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x * y), $MachinePrecision] / 2.0), $MachinePrecision] - N[(z / 8.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{x \cdot y}{2} - \frac{z}{8} \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto \frac{x \cdot y}{2} - \frac{z}{8} \]
4. Add Preprocessing

Alternative 4: 50.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ -0.125 \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* -0.125 z))

C

double code(double x, double y, double z) {
	return -0.125 * 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 = (-0.125d0) * z
end function

Java

public static double code(double x, double y, double z) {
	return -0.125 * z;
}

Python

def code(x, y, z):
	return -0.125 * z

Julia

function code(x, y, z)
	return Float64(-0.125 * z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = -0.125 * z;
end

Wolfram

code[x_, y_, z_] := N[(-0.125 * z), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{x \cdot y}{2} - \frac{z}{8} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 51.4%

   \[\leadsto \color{blue}{-0.125 \cdot z} \]

4. Final simplification 51.4%

   \[\leadsto -0.125 \cdot z \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024013 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, D"
  :precision binary64
  (- (/ (* x y) 2.0) (/ z 8.0)))