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

Percentage Accurate: 100.0% → 100.0%

Time: 3.9s

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. fp-cancel-sign-sub-inv N/A

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

   3. distribute-lft-neg-out N/A

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

   4. *-commutative N/A

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

   5. distribute-lft-neg-in N/A

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

   6. *-rgt-identity N/A

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

   7. *-inverses N/A

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

   8. associate-/l* N/A

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

   9. associate-*l/ N/A

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

   10. metadata-eval N/A

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

   11. associate-*r/ N/A

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

   12. *-commutative N/A

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

   13. *-commutative N/A

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

   14. associate-*l* N/A

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

   15. metadata-eval N/A

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

   16. fp-cancel-sign-sub-inv N/A

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

   17. distribute-lft-out N/A

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

   18. remove-double-neg N/A

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

   19. distribute-lft-neg-out N/A

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

   20. mul-1-neg N/A

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

   21. *-commutative N/A

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

   22. distribute-lft-neg-in N/A

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

5. Applied rewrites 100.0%

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

Alternative 2: 74.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+39} \lor \neg \left(y \leq 10^{-13}\right):\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.7e+39) (not (<= y 1e-13))) (* 0.5 y) (* 0.5 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.7e+39) || !(y <= 1e-13)) {
		tmp = 0.5 * y;
	} else {
		tmp = 0.5 * 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 <= (-1.7d+39)) .or. (.not. (y <= 1d-13))) then
        tmp = 0.5d0 * y
    else
        tmp = 0.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.7e+39) || !(y <= 1e-13)) {
		tmp = 0.5 * y;
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.7e+39) or not (y <= 1e-13):
		tmp = 0.5 * y
	else:
		tmp = 0.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.7e+39) || !(y <= 1e-13))
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(0.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.7e+39) || ~((y <= 1e-13)))
		tmp = 0.5 * y;
	else
		tmp = 0.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.7e+39], N[Not[LessEqual[y, 1e-13]], $MachinePrecision]], N[(0.5 * y), $MachinePrecision], N[(0.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6999999999999999e39 or 1e-13 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 78.4

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

   5. Applied rewrites 78.4%

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

   if -1.6999999999999999e39 < y < 1e-13

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 74.8

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

   5. Applied rewrites 74.8%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+39} \lor \neg \left(y \leq 10^{-13}\right):\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.7% 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 x around inf

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

   1. lower-*.f64 50.2

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

5. Applied rewrites 50.2%

   \[\leadsto \color{blue}{0.5 \cdot x} \]
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 2024329 
(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)))