Data.Histogram.Bin.LogBinD:$cbinSizeN from histogram-fill-0.8.4.1

Percentage Accurate: 100.0% → 100.0%

Time: 3.8s

Alternatives: 5

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y)
	return fma(y, x, Float64(-x))
end

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x \cdot y - x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. neg-lowering-neg.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) (* y x) (if (<= y 1.0) (- x) (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = y * x;
	} else if (y <= 1.0) {
		tmp = -x;
	} else {
		tmp = y * 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 <= (-1.0d0)) then
        tmp = y * x
    else if (y <= 1.0d0) then
        tmp = -x
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = y * x;
	} else if (y <= 1.0) {
		tmp = -x;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = y * x
	elif y <= 1.0:
		tmp = -x
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(y * x);
	elseif (y <= 1.0)
		tmp = Float64(-x);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = y * x;
	elseif (y <= 1.0)
		tmp = -x;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], N[(y * x), $MachinePrecision], If[LessEqual[y, 1.0], (-x), N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 100.0%

      \[x \cdot y - x \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 98.6

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

   5. Simplified 98.6%

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

   if -1 < y < 1

   1. Initial program 100.0%

      \[x \cdot y - x \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

      2. neg-lowering-neg.f64 98.3

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

   5. Simplified 98.3%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x \cdot y - x \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto y \cdot x - x \]
4. Add Preprocessing

Alternative 4: 50.2% accurate, 3.0× 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 Float64(-x)
end

MATLAB

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

Wolfram

code[x_, y_] := (-x)

Derivation

1. Initial program 100.0%

   \[x \cdot y - x \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

   2. neg-lowering-neg.f64 46.7

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

5. Simplified 46.7%

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

Alternative 5: 3.0% accurate, 9.0× 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%

   \[x \cdot y - x \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

   2. neg-lowering-neg.f64 46.7

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

5. Simplified 46.7%

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

   1. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - x} \]

   2. flip3-- N/A

      \[\leadsto \color{blue}{\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   4. neg-sub0 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left({x}^{3}\right)}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   5. cube-neg N/A

      \[\leadsto \frac{\color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{3}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   6. sqr-pow N/A

      \[\leadsto \frac{\color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(x\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   7. pow-prod-down N/A

      \[\leadsto \frac{\color{blue}{{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   8. sqr-neg N/A

      \[\leadsto \frac{{\color{blue}{\left(x \cdot x\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   9. unpow-prod-down N/A

      \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{3}{2}\right)} \cdot {x}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   10. sqr-pow N/A

       \[\leadsto \frac{\color{blue}{{x}^{3}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   11. remove-double-neg N/A

       \[\leadsto \frac{{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)}}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   12. neg-mul-1 N/A

       \[\leadsto \frac{{\color{blue}{\left(-1 \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   13. *-commutative N/A

       \[\leadsto \frac{{\color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot -1\right)}}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   14. unpow-prod-down N/A

       \[\leadsto \frac{\color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{3} \cdot {-1}^{3}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   15. metadata-eval N/A

       \[\leadsto \frac{{\left(\mathsf{neg}\left(x\right)\right)}^{3} \cdot \color{blue}{-1}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   16. associate-*l/ N/A

       \[\leadsto \color{blue}{\frac{{\left(\mathsf{neg}\left(x\right)\right)}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \cdot -1} \]

   17. cube-neg N/A

       \[\leadsto \frac{\color{blue}{\mathsf{neg}\left({x}^{3}\right)}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \cdot -1 \]

   18. neg-sub0 N/A

       \[\leadsto \frac{\color{blue}{0 - {x}^{3}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \cdot -1 \]

   19. metadata-eval N/A

       \[\leadsto \frac{\color{blue}{{0}^{3}} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \cdot -1 \]

   20. flip3-- N/A

       \[\leadsto \color{blue}{\left(0 - x\right)} \cdot -1 \]

   21. neg-sub0 N/A

       \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot -1 \]

   22. *-commutative N/A

       \[\leadsto \color{blue}{-1 \cdot \left(\mathsf{neg}\left(x\right)\right)} \]

   23. neg-mul-1 N/A

       \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)} \]

7. Applied egg-rr 2.8%

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

Reproduce

herbie shell --seed 2024204 
(FPCore (x y)
  :name "Data.Histogram.Bin.LogBinD:$cbinSizeN from histogram-fill-0.8.4.1"
  :precision binary64
  (- (* x y) x))