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

Percentage Accurate: 99.9% → 100.0%

Time: 7.2s

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 N/A

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

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{3}{8} \cdot y\right)\right)}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(y \cdot \frac{3}{8}\right)\right)\right) \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{3}{8}\right)\right)}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(\frac{3}{8}\right)\right)}\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \left(\mathsf{neg}\left(\frac{3}{8}\right)\right)\right)\right) \]

   7. metadata-eval 99.9%

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \frac{-3}{8}\right)\right) \]

3. Simplified 99.9%

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

   1. +-commutative N/A

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

   2. fma-define N/A

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

   3. fma-lowering-fma.f64 100.0%

      \[\leadsto \mathsf{fma.f64}\left(y, \color{blue}{\frac{-3}{8}}, x\right) \]

6. Applied egg-rr 100.0%

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

Alternative 2: 74.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-63}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+19}:\\ \;\;\;\;y \cdot -0.375\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.2e-63) x (if (<= x 1.3e+19) (* y -0.375) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.2e-63) {
		tmp = x;
	} else if (x <= 1.3e+19) {
		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 <= (-4.2d-63)) then
        tmp = x
    else if (x <= 1.3d+19) 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 <= -4.2e-63) {
		tmp = x;
	} else if (x <= 1.3e+19) {
		tmp = y * -0.375;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.2e-63:
		tmp = x
	elif x <= 1.3e+19:
		tmp = y * -0.375
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.2e-63)
		tmp = x;
	elseif (x <= 1.3e+19)
		tmp = Float64(y * -0.375);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.2e-63)
		tmp = x;
	elseif (x <= 1.3e+19)
		tmp = y * -0.375;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.2e-63], x, If[LessEqual[x, 1.3e+19], N[(y * -0.375), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -4.2e-63 or 1.3e19 < x

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{3}{8} \cdot y\right)\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(y \cdot \frac{3}{8}\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{3}{8}\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(\frac{3}{8}\right)\right)}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \left(\mathsf{neg}\left(\frac{3}{8}\right)\right)\right)\right) \]

      7. metadata-eval 99.9%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \frac{-3}{8}\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 80.3%

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

   if -4.2e-63 < x < 1.3e19

   1. Initial program 99.8%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{3}{8} \cdot y\right)\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(y \cdot \frac{3}{8}\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{3}{8}\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(\frac{3}{8}\right)\right)}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \left(\mathsf{neg}\left(\frac{3}{8}\right)\right)\right)\right) \]

      7. metadata-eval 99.8%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \frac{-3}{8}\right)\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-3}{8} \cdot y} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 77.9%

         \[\leadsto \mathsf{*.f64}\left(\frac{-3}{8}, \color{blue}{y}\right) \]

   7. Simplified 77.9%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-63}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+19}:\\ \;\;\;\;y \cdot -0.375\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
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 N/A

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

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{3}{8} \cdot y\right)\right)}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(y \cdot \frac{3}{8}\right)\right)\right) \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{3}{8}\right)\right)}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(\frac{3}{8}\right)\right)}\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \left(\mathsf{neg}\left(\frac{3}{8}\right)\right)\right)\right) \]

   7. metadata-eval 99.9%

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \frac{-3}{8}\right)\right) \]

3. Simplified 99.9%

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

Alternative 4: 50.2% 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 99.9%

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

   1. sub-neg N/A

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

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{3}{8} \cdot y\right)\right)}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(y \cdot \frac{3}{8}\right)\right)\right) \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{3}{8}\right)\right)}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(\frac{3}{8}\right)\right)}\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \left(\mathsf{neg}\left(\frac{3}{8}\right)\right)\right)\right) \]

   7. metadata-eval 99.9%

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \frac{-3}{8}\right)\right) \]

3. Simplified 99.9%

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

5. Taylor expanded in x around inf

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

7. Simplified 54.3%

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

Reproduce

herbie shell --seed 2024161 
(FPCore (x y)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, A"
  :precision binary64
  (- x (* (/ 3.0 8.0) y)))