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

Percentage Accurate: 99.9% → 100.0%

Time: 5.0s

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 99.8%

   \[x - \frac{3}{8} \cdot y \]
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%

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

   3. *-commutative 99.8%

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

   4. distribute-rgt-neg-in 99.8%

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

   5. fma-def 100.0%

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

   6. metadata-eval 100.0%

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

   7. 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. Final simplification 100.0%

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

Alternative 2: 74.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-52}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-64}:\\ \;\;\;\;y \cdot -0.375\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-22}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+17}:\\ \;\;\;\;y \cdot -0.375\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.1e-52)
   x
   (if (<= x 3.5e-64)
     (* y -0.375)
     (if (<= x 5.5e-22) x (if (<= x 2.6e+17) (* y -0.375) x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.1e-52) {
		tmp = x;
	} else if (x <= 3.5e-64) {
		tmp = y * -0.375;
	} else if (x <= 5.5e-22) {
		tmp = x;
	} else if (x <= 2.6e+17) {
		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 (x <= (-3.1d-52)) then
        tmp = x
    else if (x <= 3.5d-64) then
        tmp = y * (-0.375d0)
    else if (x <= 5.5d-22) then
        tmp = x
    else if (x <= 2.6d+17) 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 (x <= -3.1e-52) {
		tmp = x;
	} else if (x <= 3.5e-64) {
		tmp = y * -0.375;
	} else if (x <= 5.5e-22) {
		tmp = x;
	} else if (x <= 2.6e+17) {
		tmp = y * -0.375;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.1e-52:
		tmp = x
	elif x <= 3.5e-64:
		tmp = y * -0.375
	elif x <= 5.5e-22:
		tmp = x
	elif x <= 2.6e+17:
		tmp = y * -0.375
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.1e-52)
		tmp = x;
	elseif (x <= 3.5e-64)
		tmp = Float64(y * -0.375);
	elseif (x <= 5.5e-22)
		tmp = x;
	elseif (x <= 2.6e+17)
		tmp = Float64(y * -0.375);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.1e-52)
		tmp = x;
	elseif (x <= 3.5e-64)
		tmp = y * -0.375;
	elseif (x <= 5.5e-22)
		tmp = x;
	elseif (x <= 2.6e+17)
		tmp = y * -0.375;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.1e-52], x, If[LessEqual[x, 3.5e-64], N[(y * -0.375), $MachinePrecision], If[LessEqual[x, 5.5e-22], x, If[LessEqual[x, 2.6e+17], N[(y * -0.375), $MachinePrecision], x]]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.0999999999999999e-52 or 3.5000000000000003e-64 < x < 5.5000000000000001e-22 or 2.6e17 < x

   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. Taylor expanded in x around inf 76.6%

      \[\leadsto \color{blue}{x} \]

   if -3.0999999999999999e-52 < x < 3.5000000000000003e-64 or 5.5000000000000001e-22 < x < 2.6e17

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. *-commutative 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. metadata-eval 99.7%

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

      5. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 83.3%

      \[\leadsto \color{blue}{-0.375 \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-52}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-64}:\\ \;\;\;\;y \cdot -0.375\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-22}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+17}:\\ \;\;\;\;y \cdot -0.375\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

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 99.8%

   \[x - \frac{3}{8} \cdot y \]
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. Final simplification 99.8%

   \[\leadsto x + y \cdot -0.375 \]

Alternative 4: 51.2% 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 99.8%

   \[x - \frac{3}{8} \cdot y \]
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. Taylor expanded in x around inf 48.8%

   \[\leadsto \color{blue}{x} \]

5. Final simplification 48.8%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023221 
(FPCore (x y)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, A"
  :precision binary64
  (- x (* (/ 3.0 8.0) y)))