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

Percentage Accurate: 99.9% → 100.0%

Time: 3.4s

Alternatives: 4

Speedup: 1.4×

Specification

Math

\[\begin{array}{l} \\ x - \frac{3}{8} \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- x (* (/ 3.0 8.0) y)))

C

double code(double x, double y) {
	return x - ((3.0 / 8.0) * y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x - ((3.0d0 / 8.0d0) * y)
end function

Java

public static double code(double x, double y) {
	return x - ((3.0 / 8.0) * y);
}

Python

def code(x, y):
	return x - ((3.0 / 8.0) * y)

Julia

function code(x, y)
	return Float64(x - Float64(Float64(3.0 / 8.0) * y))
end

MATLAB

function tmp = code(x, y)
	tmp = x - ((3.0 / 8.0) * y);
end

Wolfram

code[x_, y_] := N[(x - N[(N[(3.0 / 8.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - \frac{3}{8} \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- x (* (/ 3.0 8.0) y)))

C

double code(double x, double y) {
	return x - ((3.0 / 8.0) * y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x - ((3.0d0 / 8.0d0) * y)
end function

Java

public static double code(double x, double y) {
	return x - ((3.0 / 8.0) * y);
}

Python

def code(x, y):
	return x - ((3.0 / 8.0) * y)

Julia

function code(x, y)
	return Float64(x - Float64(Float64(3.0 / 8.0) * y))
end

MATLAB

function tmp = code(x, y)
	tmp = x - ((3.0 / 8.0) * y);
end

Wolfram

code[x_, y_] := N[(x - N[(N[(3.0 / 8.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, -0.375, x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma y -0.375 x))

C

double code(double x, double y) {
	return fma(y, -0.375, x);
}

Julia

function code(x, y)
	return fma(y, -0.375, x)
end

Wolfram

code[x_, y_] := N[(y * -0.375 + x), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x - \frac{3}{8} \cdot y \]
2. Step-by-step derivation

   1. sub-neg 99.9%

      \[\leadsto \color{blue}{x + \left(-\frac{3}{8} \cdot y\right)} \]

   2. distribute-rgt-neg-out 99.9%

      \[\leadsto x + \color{blue}{\frac{3}{8} \cdot \left(-y\right)} \]

   3. +-commutative 99.9%

      \[\leadsto \color{blue}{\frac{3}{8} \cdot \left(-y\right) + x} \]

   4. distribute-rgt-neg-out 99.9%

      \[\leadsto \color{blue}{\left(-\frac{3}{8} \cdot y\right)} + x \]

   5. *-commutative 99.9%

      \[\leadsto \left(-\color{blue}{y \cdot \frac{3}{8}}\right) + x \]

   6. distribute-rgt-neg-in 99.9%

      \[\leadsto \color{blue}{y \cdot \left(-\frac{3}{8}\right)} + x \]

   7. fma-define 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, -\frac{3}{8}, x\right)} \]

   8. metadata-eval 100.0%

      \[\leadsto \mathsf{fma}\left(y, -\color{blue}{0.375}, x\right) \]

   9. metadata-eval 100.0%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{-0.375}, x\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, -0.375, x\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 74.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+110} \lor \neg \left(y \leq -3.3 \cdot 10^{+90}\right) \land \left(y \leq -3 \cdot 10^{-50} \lor \neg \left(y \leq 1.2 \cdot 10^{-25}\right)\right):\\ \;\;\;\;y \cdot -0.375\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -6.5e+110)
         (and (not (<= y -3.3e+90)) (or (<= y -3e-50) (not (<= y 1.2e-25)))))
   (* y -0.375)
   x))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -6.5e+110) || (!(y <= -3.3e+90) && ((y <= -3e-50) || !(y <= 1.2e-25)))) {
		tmp = y * -0.375;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-6.5d+110)) .or. (.not. (y <= (-3.3d+90))) .and. (y <= (-3d-50)) .or. (.not. (y <= 1.2d-25))) then
        tmp = y * (-0.375d0)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -6.5e+110) || (!(y <= -3.3e+90) && ((y <= -3e-50) || !(y <= 1.2e-25)))) {
		tmp = y * -0.375;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -6.5e+110) or (not (y <= -3.3e+90) and ((y <= -3e-50) or not (y <= 1.2e-25))):
		tmp = y * -0.375
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -6.5e+110) || (!(y <= -3.3e+90) && ((y <= -3e-50) || !(y <= 1.2e-25))))
		tmp = Float64(y * -0.375);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -6.5e+110) || (~((y <= -3.3e+90)) && ((y <= -3e-50) || ~((y <= 1.2e-25)))))
		tmp = y * -0.375;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -6.5e+110], And[N[Not[LessEqual[y, -3.3e+90]], $MachinePrecision], Or[LessEqual[y, -3e-50], N[Not[LessEqual[y, 1.2e-25]], $MachinePrecision]]]], N[(y * -0.375), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -6.4999999999999997e110 or -3.30000000000000008e90 < y < -2.9999999999999999e-50 or 1.20000000000000005e-25 < y

   1. Initial program 99.8%

      \[x - \frac{3}{8} \cdot y \]
   2. Step-by-step derivation

      1. sub-neg 99.8%

         \[\leadsto \color{blue}{x + \left(-\frac{3}{8} \cdot y\right)} \]

      2. *-commutative 99.8%

         \[\leadsto x + \left(-\color{blue}{y \cdot \frac{3}{8}}\right) \]

      3. distribute-rgt-neg-in 99.8%

         \[\leadsto x + \color{blue}{y \cdot \left(-\frac{3}{8}\right)} \]

      4. metadata-eval 99.8%

         \[\leadsto x + y \cdot \left(-\color{blue}{0.375}\right) \]

      5. metadata-eval 99.8%

         \[\leadsto x + y \cdot \color{blue}{-0.375} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x + y \cdot -0.375} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 76.5%

      \[\leadsto \color{blue}{-0.375 \cdot y} \]

   if -6.4999999999999997e110 < y < -3.30000000000000008e90 or -2.9999999999999999e-50 < y < 1.20000000000000005e-25

   1. Initial program 100.0%

      \[x - \frac{3}{8} \cdot y \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \color{blue}{x + \left(-\frac{3}{8} \cdot y\right)} \]

      2. *-commutative 100.0%

         \[\leadsto x + \left(-\color{blue}{y \cdot \frac{3}{8}}\right) \]

      3. distribute-rgt-neg-in 100.0%

         \[\leadsto x + \color{blue}{y \cdot \left(-\frac{3}{8}\right)} \]

      4. metadata-eval 100.0%

         \[\leadsto x + y \cdot \left(-\color{blue}{0.375}\right) \]

      5. metadata-eval 100.0%

         \[\leadsto x + y \cdot \color{blue}{-0.375} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x + y \cdot -0.375} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 77.8%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+110} \lor \neg \left(y \leq -3.3 \cdot 10^{+90}\right) \land \left(y \leq -3 \cdot 10^{-50} \lor \neg \left(y \leq 1.2 \cdot 10^{-25}\right)\right):\\ \;\;\;\;y \cdot -0.375\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ x + y \cdot -0.375 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x (* y -0.375)))

C

double code(double x, double y) {
	return x + (y * -0.375);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + (y * (-0.375d0))
end function

Java

public static double code(double x, double y) {
	return x + (y * -0.375);
}

Python

def code(x, y):
	return x + (y * -0.375)

Julia

function code(x, y)
	return Float64(x + Float64(y * -0.375))
end

MATLAB

function tmp = code(x, y)
	tmp = x + (y * -0.375);
end

Wolfram

code[x_, y_] := N[(x + N[(y * -0.375), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x - \frac{3}{8} \cdot y \]
2. Step-by-step derivation

   1. sub-neg 99.9%

      \[\leadsto \color{blue}{x + \left(-\frac{3}{8} \cdot y\right)} \]

   2. *-commutative 99.9%

      \[\leadsto x + \left(-\color{blue}{y \cdot \frac{3}{8}}\right) \]

   3. distribute-rgt-neg-in 99.9%

      \[\leadsto x + \color{blue}{y \cdot \left(-\frac{3}{8}\right)} \]

   4. metadata-eval 99.9%

      \[\leadsto x + y \cdot \left(-\color{blue}{0.375}\right) \]

   5. metadata-eval 99.9%

      \[\leadsto x + y \cdot \color{blue}{-0.375} \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{x + y \cdot -0.375} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 4: 50.6% accurate, 7.0× 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 99.9%

   \[x - \frac{3}{8} \cdot y \]
2. Step-by-step derivation

   1. sub-neg 99.9%

      \[\leadsto \color{blue}{x + \left(-\frac{3}{8} \cdot y\right)} \]

   2. *-commutative 99.9%

      \[\leadsto x + \left(-\color{blue}{y \cdot \frac{3}{8}}\right) \]

   3. distribute-rgt-neg-in 99.9%

      \[\leadsto x + \color{blue}{y \cdot \left(-\frac{3}{8}\right)} \]

   4. metadata-eval 99.9%

      \[\leadsto x + y \cdot \left(-\color{blue}{0.375}\right) \]

   5. metadata-eval 99.9%

      \[\leadsto x + y \cdot \color{blue}{-0.375} \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{x + y \cdot -0.375} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 46.1%

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

Reproduce

herbie shell --seed 2024107 
(FPCore (x y)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, A"
  :precision binary64
  (- x (* (/ 3.0 8.0) y)))