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

Percentage Accurate: 100.0% → 100.0%

Time: 3.1s

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

\[\begin{array}{l} \\ \mathsf{fma}\left(y, \frac{x}{2}, \frac{z}{-8}\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-/l* 100.0%

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

   3. fma-neg 100.0%

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

   4. distribute-neg-frac2 100.0%

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

   5. metadata-eval 100.0%

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

3. Simplified 100.0%

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

Alternative 2: 68.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-43} \lor \neg \left(x \leq 4.3 \cdot 10^{-114}\right):\\ \;\;\;\;x \cdot \left(y \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot -0.125\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.8e-43) (not (<= x 4.3e-114))) (* x (* y 0.5)) (* z -0.125)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.8e-43) || !(x <= 4.3e-114)) {
		tmp = x * (y * 0.5);
	} else {
		tmp = z * -0.125;
	}
	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 ((x <= (-1.8d-43)) .or. (.not. (x <= 4.3d-114))) then
        tmp = x * (y * 0.5d0)
    else
        tmp = z * (-0.125d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.8e-43) || !(x <= 4.3e-114)) {
		tmp = x * (y * 0.5);
	} else {
		tmp = z * -0.125;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.8e-43) or not (x <= 4.3e-114):
		tmp = x * (y * 0.5)
	else:
		tmp = z * -0.125
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.8e-43) || !(x <= 4.3e-114))
		tmp = Float64(x * Float64(y * 0.5));
	else
		tmp = Float64(z * -0.125);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.8e-43) || ~((x <= 4.3e-114)))
		tmp = x * (y * 0.5);
	else
		tmp = z * -0.125;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.8e-43], N[Not[LessEqual[x, 4.3e-114]], $MachinePrecision]], N[(x * N[(y * 0.5), $MachinePrecision]), $MachinePrecision], N[(z * -0.125), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.7999999999999999e-43 or 4.3e-114 < x

   1. Initial program 99.5%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 75.6%

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

      1. *-commutative 75.6%

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

      2. associate-*r* 76.1%

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

      3. *-commutative 76.1%

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

   7. Simplified 76.1%

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

   if -1.7999999999999999e-43 < x < 4.3e-114

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 69.3%

      \[\leadsto \color{blue}{-0.125 \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-43} \lor \neg \left(x \leq 4.3 \cdot 10^{-114}\right):\\ \;\;\;\;x \cdot \left(y \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot -0.125\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

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(x * Float64(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[(x * N[(y / 2.0), $MachinePrecision]), $MachinePrecision] - N[(z / 8.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. associate-/l* 100.0%

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

3. Simplified 100.0%

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

Alternative 4: 51.0% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

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 * (-0.125d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate-/l* 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 39.9%

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

6. Final simplification 39.9%

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

Reproduce

herbie shell --seed 2024100 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, D"
  :precision binary64
  (- (/ (* x y) 2.0) (/ z 8.0)))