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

Percentage Accurate: 100.0% → 100.0%

Time: 3.3s

Alternatives: 3

Speedup: 2.1×

Specification

Math

\[\begin{array}{l} \\ x - \frac{y}{4} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- x (/ y 4.0)))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x - (y / 4.0d0)
end function

Java

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

Python

def code(x, y):
	return x - (y / 4.0)

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - \frac{y}{4} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- x (/ y 4.0)))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x - (y / 4.0d0)
end function

Java

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

Python

def code(x, y):
	return x - (y / 4.0)

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, -0.25, x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma y -0.25 x))

C

double code(double x, double y) {
	return fma(y, -0.25, x);
}

Julia

function code(x, y)
	return fma(y, -0.25, x)
end

Wolfram

code[x_, y_] := N[(y * -0.25 + x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x - \frac{y}{4} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(\frac{y}{4}\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y}{4}\right)\right) + x} \]

   3. distribute-neg-frac2 N/A

      \[\leadsto \color{blue}{\frac{y}{\mathsf{neg}\left(4\right)}} + x \]

   4. div-inv N/A

      \[\leadsto \color{blue}{y \cdot \frac{1}{\mathsf{neg}\left(4\right)}} + x \]

   5. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{1}{\mathsf{neg}\left(4\right)}, x\right)} \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(y, \frac{1}{\color{blue}{-4}}, x\right) \]

   7. metadata-eval 100.0

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{-0.25}, x\right) \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, -0.25, x\right)} \]
5. Add Preprocessing

Alternative 2: 75.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y}{4} \leq -0.5:\\ \;\;\;\;y \cdot -0.25\\ \mathbf{elif}\;\frac{y}{4} \leq 4 \cdot 10^{-29}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ y 4.0) -0.5) (* y -0.25) (if (<= (/ y 4.0) 4e-29) x (* y -0.25))))

C

double code(double x, double y) {
	double tmp;
	if ((y / 4.0) <= -0.5) {
		tmp = y * -0.25;
	} else if ((y / 4.0) <= 4e-29) {
		tmp = x;
	} else {
		tmp = y * -0.25;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y / 4.0d0) <= (-0.5d0)) then
        tmp = y * (-0.25d0)
    else if ((y / 4.0d0) <= 4d-29) then
        tmp = x
    else
        tmp = y * (-0.25d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y / 4.0) <= -0.5) {
		tmp = y * -0.25;
	} else if ((y / 4.0) <= 4e-29) {
		tmp = x;
	} else {
		tmp = y * -0.25;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y / 4.0) <= -0.5:
		tmp = y * -0.25
	elif (y / 4.0) <= 4e-29:
		tmp = x
	else:
		tmp = y * -0.25
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y / 4.0) <= -0.5)
		tmp = Float64(y * -0.25);
	elseif (Float64(y / 4.0) <= 4e-29)
		tmp = x;
	else
		tmp = Float64(y * -0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y / 4.0) <= -0.5)
		tmp = y * -0.25;
	elseif ((y / 4.0) <= 4e-29)
		tmp = x;
	else
		tmp = y * -0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y / 4.0), $MachinePrecision], -0.5], N[(y * -0.25), $MachinePrecision], If[LessEqual[N[(y / 4.0), $MachinePrecision], 4e-29], x, N[(y * -0.25), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 y #s(literal 4 binary64)) < -0.5 or 3.99999999999999977e-29 < (/.f64 y #s(literal 4 binary64))

   1. Initial program 100.0%

      \[x - \frac{y}{4} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \frac{-1}{4}} \]

      2. *-lowering-*.f64 74.7

         \[\leadsto \color{blue}{y \cdot -0.25} \]

   5. Simplified 74.7%

      \[\leadsto \color{blue}{y \cdot -0.25} \]

   if -0.5 < (/.f64 y #s(literal 4 binary64)) < 3.99999999999999977e-29

   1. Initial program 100.0%

      \[x - \frac{y}{4} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 84.0%

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

Alternative 3: 51.0% accurate, 15.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x
end function

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 100.0%

   \[x - \frac{y}{4} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Simplified 54.5%

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

Reproduce

herbie shell --seed 2024198 
(FPCore (x y)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, E"
  :precision binary64
  (- x (/ y 4.0)))