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

Percentage Accurate: 100.0% → 100.0%

Time: 4.5s

Alternatives: 3

Speedup: 2.0×

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, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x + \frac{y - x}{2} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x + \left(\frac{-1}{2} \cdot x + \frac{1}{2} \cdot y\right)} \]
4. Step-by-step derivation

   1. associate-+r+ N/A

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

   2. distribute-rgt1-in N/A

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

   3. metadata-eval N/A

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot x + \frac{1}{2} \cdot y \]

   4. distribute-lft-out N/A

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

   5. +-commutative N/A

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

   6. *-rgt-identity N/A

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

   7. *-inverses N/A

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

   8. associate-/l* N/A

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

   9. associate-*l/ N/A

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

   10. distribute-lft-out N/A

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

   11. associate-*l* N/A

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

   12. distribute-rgt-in N/A

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

   13. *-commutative N/A

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

   14. distribute-lft1-in N/A

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

   15. associate-*r* N/A

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

   16. lower-*.f64 N/A

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

   17. distribute-rgt-in N/A

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

   18. associate-*l/ N/A

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

   19. associate-/l* N/A

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

   20. *-inverses N/A

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

   21. *-rgt-identity N/A

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

   22. *-lft-identity N/A

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

   23. lower-+.f64 100.0

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

5. Applied rewrites 100.0%

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

6. Final simplification 100.0%

   \[\leadsto 0.5 \cdot \left(y + x\right) \]
7. Add Preprocessing

Alternative 2: 49.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y - x}{2} + x \leq -5 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (/ (- y x) 2.0) x) -5e-276) (* 0.5 x) (* 0.5 y)))

C

double code(double x, double y) {
	double tmp;
	if ((((y - x) / 2.0) + x) <= -5e-276) {
		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 - x) / 2.0d0) + x) <= (-5d-276)) 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 - x) / 2.0) + x) <= -5e-276) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (((y - x) / 2.0) + x) <= -5e-276:
		tmp = 0.5 * x
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(Float64(y - x) / 2.0) + x) <= -5e-276)
		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 - x) / 2.0) + x) <= -5e-276)
		tmp = 0.5 * x;
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[(y - x), $MachinePrecision] / 2.0), $MachinePrecision] + x), $MachinePrecision], -5e-276], N[(0.5 * x), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (-.f64 y x) #s(literal 2 binary64))) < -4.99999999999999967e-276

   1. Initial program 100.0%

      \[x + \frac{y - x}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{1}{2}} \cdot x \]

      3. lower-*.f64 48.8

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

   5. Applied rewrites 48.8%

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

   if -4.99999999999999967e-276 < (+.f64 x (/.f64 (-.f64 y x) #s(literal 2 binary64)))

   1. Initial program 100.0%

      \[x + \frac{y - x}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \frac{1}{2}} \]

      2. lower-*.f64 54.4

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

   5. Applied rewrites 54.4%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{2} + x \leq -5 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.6% accurate, 3.0× 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. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. distribute-rgt1-in N/A

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

   2. metadata-eval N/A

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot x \]

   3. lower-*.f64 47.9

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

5. Applied rewrites 47.9%

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

Developer Target 1: 100.0% accurate, 2.0× 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 2024250 
(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)))