Numeric.Interval.Internal:bisect from intervals-0.7.1, A

Percentage Accurate: 100.0% → 100.0%

Time: 2.9s

Alternatives: 3

Speedup: 1.4×

Specification

Math

\[\begin{array}{l} \\ x + \frac{y - x}{2} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x (/ (- y x) 2.0)))

C

double code(double x, double y) {
	return x + ((y - x) / 2.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + ((y - x) / 2.0d0)
end function

Java

public static double code(double x, double y) {
	return x + ((y - x) / 2.0);
}

Python

def code(x, y):
	return x + ((y - x) / 2.0)

Julia

function code(x, y)
	return Float64(x + Float64(Float64(y - x) / 2.0))
end

MATLAB

function tmp = code(x, y)
	tmp = x + ((y - x) / 2.0);
end

Wolfram

code[x_, y_] := N[(x + N[(N[(y - x), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{y - x}{2} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x (/ (- y x) 2.0)))

C

double code(double x, double y) {
	return x + ((y - x) / 2.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + ((y - x) / 2.0d0)
end function

Java

public static double code(double x, double y) {
	return x + ((y - x) / 2.0);
}

Python

def code(x, y):
	return x + ((y - x) / 2.0)

Julia

function code(x, y)
	return Float64(x + Float64(Float64(y - x) / 2.0))
end

MATLAB

function tmp = code(x, y)
	tmp = x + ((y - x) / 2.0);
end

Wolfram

code[x_, y_] := N[(x + N[(N[(y - x), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(x + y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* 0.5 (+ x y)))

C

double code(double x, double y) {
	return 0.5 * (x + y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.5d0 * (x + y)
end function

Java

public static double code(double x, double y) {
	return 0.5 * (x + y);
}

Python

def code(x, y):
	return 0.5 * (x + y)

Julia

function code(x, y)
	return Float64(0.5 * Float64(x + y))
end

MATLAB

function tmp = code(x, y)
	tmp = 0.5 * (x + y);
end

Wolfram

code[x_, y_] := N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x + \frac{y - x}{2} \]
2. Step-by-step derivation

   1. div-sub N/A

      \[\leadsto x + \left(\frac{y}{2} - \color{blue}{\frac{x}{2}}\right) \]

   2. sub-neg N/A

      \[\leadsto x + \left(\frac{y}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{2}\right)\right)}\right) \]

   3. +-commutative N/A

      \[\leadsto x + \left(\left(\mathsf{neg}\left(\frac{x}{2}\right)\right) + \color{blue}{\frac{y}{2}}\right) \]

   4. associate-+r+ N/A

      \[\leadsto \left(x + \left(\mathsf{neg}\left(\frac{x}{2}\right)\right)\right) + \color{blue}{\frac{y}{2}} \]

   5. *-rgt-identity N/A

      \[\leadsto \left(x \cdot 1 + \left(\mathsf{neg}\left(\frac{x}{2}\right)\right)\right) + \frac{y}{2} \]

   6. metadata-eval N/A

      \[\leadsto \left(x \cdot \left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(\frac{x}{2}\right)\right)\right) + \frac{y}{2} \]

   7. distribute-neg-frac N/A

      \[\leadsto \left(x \cdot \left(\mathsf{neg}\left(-1\right)\right) + \frac{\mathsf{neg}\left(x\right)}{2}\right) + \frac{y}{2} \]

   8. neg-mul-1 N/A

      \[\leadsto \left(x \cdot \left(\mathsf{neg}\left(-1\right)\right) + \frac{-1 \cdot x}{2}\right) + \frac{y}{2} \]

   9. *-commutative N/A

      \[\leadsto \left(x \cdot \left(\mathsf{neg}\left(-1\right)\right) + \frac{x \cdot -1}{2}\right) + \frac{y}{2} \]

   10. associate-/l* N/A

       \[\leadsto \left(x \cdot \left(\mathsf{neg}\left(-1\right)\right) + x \cdot \frac{-1}{2}\right) + \frac{y}{2} \]

   11. distribute-lft-out N/A

       \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + \frac{\color{blue}{y}}{2} \]

   12. *-rgt-identity N/A

       \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + \frac{y}{2} \cdot \color{blue}{1} \]

   13. metadata-eval N/A

       \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + \frac{y}{2} \cdot \left(\mathsf{neg}\left(-1\right)\right) \]

   14. associate-*l/ N/A

       \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + \frac{y \cdot \left(\mathsf{neg}\left(-1\right)\right)}{\color{blue}{2}} \]

   15. associate-/l* N/A

       \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + y \cdot \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{2}} \]

   16. metadata-eval N/A

       \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + y \cdot \frac{1}{2} \]

   17. metadata-eval N/A

       \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + y \cdot \frac{1}{2} \]

   18. metadata-eval N/A

       \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + y \cdot \left(1 + \color{blue}{\frac{-1}{2}}\right) \]

   19. metadata-eval N/A

       \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + y \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) \]

   20. metadata-eval N/A

       \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + y \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{\color{blue}{2}}\right) \]

   21. distribute-rgt-out N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) \cdot \color{blue}{\left(x + y\right)} \]

   22. +-commutative N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) \cdot \left(y + \color{blue}{x}\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right), \color{blue}{\left(y + x\right)}\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{0.5 \cdot \left(x + y\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 61.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{-94}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 3.7e-94) (* 0.5 x) (* 0.5 y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.7e-94) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.7d-94) then
        tmp = 0.5d0 * x
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.7e-94) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.7e-94:
		tmp = 0.5 * x
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.7e-94)
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.7e-94)
		tmp = 0.5 * x;
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.7e-94], N[(0.5 * x), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.6999999999999998e-94

   1. Initial program 100.0%

      \[x + \frac{y - x}{2} \]
   2. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto x + \left(\frac{y}{2} - \color{blue}{\frac{x}{2}}\right) \]

      2. sub-neg N/A

         \[\leadsto x + \left(\frac{y}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{2}\right)\right)}\right) \]

      3. +-commutative N/A

         \[\leadsto x + \left(\left(\mathsf{neg}\left(\frac{x}{2}\right)\right) + \color{blue}{\frac{y}{2}}\right) \]

      4. associate-+r+ N/A

         \[\leadsto \left(x + \left(\mathsf{neg}\left(\frac{x}{2}\right)\right)\right) + \color{blue}{\frac{y}{2}} \]

      5. *-rgt-identity N/A

         \[\leadsto \left(x \cdot 1 + \left(\mathsf{neg}\left(\frac{x}{2}\right)\right)\right) + \frac{y}{2} \]

      6. metadata-eval N/A

         \[\leadsto \left(x \cdot \left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(\frac{x}{2}\right)\right)\right) + \frac{y}{2} \]

      7. distribute-neg-frac N/A

         \[\leadsto \left(x \cdot \left(\mathsf{neg}\left(-1\right)\right) + \frac{\mathsf{neg}\left(x\right)}{2}\right) + \frac{y}{2} \]

      8. neg-mul-1 N/A

         \[\leadsto \left(x \cdot \left(\mathsf{neg}\left(-1\right)\right) + \frac{-1 \cdot x}{2}\right) + \frac{y}{2} \]

      9. *-commutative N/A

         \[\leadsto \left(x \cdot \left(\mathsf{neg}\left(-1\right)\right) + \frac{x \cdot -1}{2}\right) + \frac{y}{2} \]

      10. associate-/l* N/A

          \[\leadsto \left(x \cdot \left(\mathsf{neg}\left(-1\right)\right) + x \cdot \frac{-1}{2}\right) + \frac{y}{2} \]

      11. distribute-lft-out N/A

          \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + \frac{\color{blue}{y}}{2} \]

      12. *-rgt-identity N/A

          \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + \frac{y}{2} \cdot \color{blue}{1} \]

      13. metadata-eval N/A

          \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + \frac{y}{2} \cdot \left(\mathsf{neg}\left(-1\right)\right) \]

      14. associate-*l/ N/A

          \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + \frac{y \cdot \left(\mathsf{neg}\left(-1\right)\right)}{\color{blue}{2}} \]

      15. associate-/l* N/A

          \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + y \cdot \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{2}} \]

      16. metadata-eval N/A

          \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + y \cdot \frac{1}{2} \]

      17. metadata-eval N/A

          \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + y \cdot \frac{1}{2} \]

      18. metadata-eval N/A

          \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + y \cdot \left(1 + \color{blue}{\frac{-1}{2}}\right) \]

      19. metadata-eval N/A

          \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + y \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) \]

      20. metadata-eval N/A

          \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + y \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{\color{blue}{2}}\right) \]

      21. distribute-rgt-out N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) \cdot \color{blue}{\left(x + y\right)} \]

      22. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) \cdot \left(y + \color{blue}{x}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right), \color{blue}{\left(y + x\right)}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(x + y\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 60.6%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{x}\right) \]

   7. Simplified 60.6%

      \[\leadsto \color{blue}{0.5 \cdot x} \]

   if 3.6999999999999998e-94 < y

   1. Initial program 100.0%

      \[x + \frac{y - x}{2} \]
   2. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto x + \left(\frac{y}{2} - \color{blue}{\frac{x}{2}}\right) \]

      2. sub-neg N/A

         \[\leadsto x + \left(\frac{y}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{2}\right)\right)}\right) \]

      3. +-commutative N/A

         \[\leadsto x + \left(\left(\mathsf{neg}\left(\frac{x}{2}\right)\right) + \color{blue}{\frac{y}{2}}\right) \]

      4. associate-+r+ N/A

         \[\leadsto \left(x + \left(\mathsf{neg}\left(\frac{x}{2}\right)\right)\right) + \color{blue}{\frac{y}{2}} \]

      5. *-rgt-identity N/A

         \[\leadsto \left(x \cdot 1 + \left(\mathsf{neg}\left(\frac{x}{2}\right)\right)\right) + \frac{y}{2} \]

      6. metadata-eval N/A

         \[\leadsto \left(x \cdot \left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(\frac{x}{2}\right)\right)\right) + \frac{y}{2} \]

      7. distribute-neg-frac N/A

         \[\leadsto \left(x \cdot \left(\mathsf{neg}\left(-1\right)\right) + \frac{\mathsf{neg}\left(x\right)}{2}\right) + \frac{y}{2} \]

      8. neg-mul-1 N/A

         \[\leadsto \left(x \cdot \left(\mathsf{neg}\left(-1\right)\right) + \frac{-1 \cdot x}{2}\right) + \frac{y}{2} \]

      9. *-commutative N/A

         \[\leadsto \left(x \cdot \left(\mathsf{neg}\left(-1\right)\right) + \frac{x \cdot -1}{2}\right) + \frac{y}{2} \]

      10. associate-/l* N/A

          \[\leadsto \left(x \cdot \left(\mathsf{neg}\left(-1\right)\right) + x \cdot \frac{-1}{2}\right) + \frac{y}{2} \]

      11. distribute-lft-out N/A

          \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + \frac{\color{blue}{y}}{2} \]

      12. *-rgt-identity N/A

          \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + \frac{y}{2} \cdot \color{blue}{1} \]

      13. metadata-eval N/A

          \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + \frac{y}{2} \cdot \left(\mathsf{neg}\left(-1\right)\right) \]

      14. associate-*l/ N/A

          \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + \frac{y \cdot \left(\mathsf{neg}\left(-1\right)\right)}{\color{blue}{2}} \]

      15. associate-/l* N/A

          \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + y \cdot \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{2}} \]

      16. metadata-eval N/A

          \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + y \cdot \frac{1}{2} \]

      17. metadata-eval N/A

          \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + y \cdot \frac{1}{2} \]

      18. metadata-eval N/A

          \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + y \cdot \left(1 + \color{blue}{\frac{-1}{2}}\right) \]

      19. metadata-eval N/A

          \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + y \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) \]

      20. metadata-eval N/A

          \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + y \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{\color{blue}{2}}\right) \]

      21. distribute-rgt-out N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) \cdot \color{blue}{\left(x + y\right)} \]

      22. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) \cdot \left(y + \color{blue}{x}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right), \color{blue}{\left(y + x\right)}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(x + y\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot y} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 64.3%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{y}\right) \]

   7. Simplified 64.3%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 50.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* 0.5 x))

C

double code(double x, double y) {
	return 0.5 * x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.5d0 * x
end function

Java

public static double code(double x, double y) {
	return 0.5 * x;
}

Python

def code(x, y):
	return 0.5 * x

Julia

function code(x, y)
	return Float64(0.5 * x)
end

MATLAB

function tmp = code(x, y)
	tmp = 0.5 * x;
end

Wolfram

code[x_, y_] := N[(0.5 * x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x + \frac{y - x}{2} \]
2. Step-by-step derivation

   1. div-sub N/A

      \[\leadsto x + \left(\frac{y}{2} - \color{blue}{\frac{x}{2}}\right) \]

   2. sub-neg N/A

      \[\leadsto x + \left(\frac{y}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{2}\right)\right)}\right) \]

   3. +-commutative N/A

      \[\leadsto x + \left(\left(\mathsf{neg}\left(\frac{x}{2}\right)\right) + \color{blue}{\frac{y}{2}}\right) \]

   4. associate-+r+ N/A

      \[\leadsto \left(x + \left(\mathsf{neg}\left(\frac{x}{2}\right)\right)\right) + \color{blue}{\frac{y}{2}} \]

   5. *-rgt-identity N/A

      \[\leadsto \left(x \cdot 1 + \left(\mathsf{neg}\left(\frac{x}{2}\right)\right)\right) + \frac{y}{2} \]

   6. metadata-eval N/A

      \[\leadsto \left(x \cdot \left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(\frac{x}{2}\right)\right)\right) + \frac{y}{2} \]

   7. distribute-neg-frac N/A

      \[\leadsto \left(x \cdot \left(\mathsf{neg}\left(-1\right)\right) + \frac{\mathsf{neg}\left(x\right)}{2}\right) + \frac{y}{2} \]

   8. neg-mul-1 N/A

      \[\leadsto \left(x \cdot \left(\mathsf{neg}\left(-1\right)\right) + \frac{-1 \cdot x}{2}\right) + \frac{y}{2} \]

   9. *-commutative N/A

      \[\leadsto \left(x \cdot \left(\mathsf{neg}\left(-1\right)\right) + \frac{x \cdot -1}{2}\right) + \frac{y}{2} \]

   10. associate-/l* N/A

       \[\leadsto \left(x \cdot \left(\mathsf{neg}\left(-1\right)\right) + x \cdot \frac{-1}{2}\right) + \frac{y}{2} \]

   11. distribute-lft-out N/A

       \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + \frac{\color{blue}{y}}{2} \]

   12. *-rgt-identity N/A

       \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + \frac{y}{2} \cdot \color{blue}{1} \]

   13. metadata-eval N/A

       \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + \frac{y}{2} \cdot \left(\mathsf{neg}\left(-1\right)\right) \]

   14. associate-*l/ N/A

       \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + \frac{y \cdot \left(\mathsf{neg}\left(-1\right)\right)}{\color{blue}{2}} \]

   15. associate-/l* N/A

       \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + y \cdot \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{2}} \]

   16. metadata-eval N/A

       \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + y \cdot \frac{1}{2} \]

   17. metadata-eval N/A

       \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + y \cdot \frac{1}{2} \]

   18. metadata-eval N/A

       \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + y \cdot \left(1 + \color{blue}{\frac{-1}{2}}\right) \]

   19. metadata-eval N/A

       \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + y \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) \]

   20. metadata-eval N/A

       \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) + y \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{\color{blue}{2}}\right) \]

   21. distribute-rgt-out N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) \cdot \color{blue}{\left(x + y\right)} \]

   22. +-commutative N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right) \cdot \left(y + \color{blue}{x}\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(-1\right)\right) + \frac{-1}{2}\right), \color{blue}{\left(y + x\right)}\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{0.5 \cdot \left(x + y\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot x} \]
6. Step-by-step derivation

   1. *-lowering-*.f64 54.3%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{x}\right) \]

7. Simplified 54.3%

   \[\leadsto \color{blue}{0.5 \cdot x} \]
8. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(x + y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* 0.5 (+ x y)))

C

double code(double x, double y) {
	return 0.5 * (x + y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.5d0 * (x + y)
end function

Java

public static double code(double x, double y) {
	return 0.5 * (x + y);
}

Python

def code(x, y):
	return 0.5 * (x + y)

Julia

function code(x, y)
	return Float64(0.5 * Float64(x + y))
end

MATLAB

function tmp = code(x, y)
	tmp = 0.5 * (x + y);
end

Wolfram

code[x_, y_] := N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024161 
(FPCore (x y)
  :name "Numeric.Interval.Internal:bisect from intervals-0.7.1, A"
  :precision binary64

  :alt
  (! :herbie-platform default (* 1/2 (+ x y)))

  (+ x (/ (- y x) 2.0)))