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

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

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. 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

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

   2. metadata-eval N/A

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

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

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

   4. metadata-eval N/A

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

   5. associate-*r* N/A

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

   6. distribute-rgt-neg-out N/A

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

   7. mul-1-neg N/A

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

   8. distribute-lft-out N/A

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

   9. *-lft-identity N/A

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

   10. *-inverses N/A

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

   11. associate-*l/ N/A

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

   12. associate-*r/ N/A

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

   13. associate-*r/ N/A

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

   14. associate-*r* N/A

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

   15. *-commutative N/A

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

   16. *-rgt-identity N/A

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

   17. distribute-lft-in N/A

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

   18. mul-1-neg N/A

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

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

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

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

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

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

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

   22. metadata-eval N/A

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

   23. lower-*.f64 N/A

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

   24. +-commutative N/A

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

5. Simplified 100.0%

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

Alternative 2: 51.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + \frac{y - x}{2} \leq -1 \cdot 10^{-300}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ x (/ (- y x) 2.0)) -1e-300) (* 0.5 x) (* 0.5 y)))

C

double code(double x, double y) {
	double tmp;
	if ((x + ((y - x) / 2.0)) <= -1e-300) {
		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 ((x + ((y - x) / 2.0d0)) <= (-1d-300)) 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 ((x + ((y - x) / 2.0)) <= -1e-300) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x + ((y - x) / 2.0)) <= -1e-300:
		tmp = 0.5 * x
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x + Float64(Float64(y - x) / 2.0)) <= -1e-300)
		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 ((x + ((y - x) / 2.0)) <= -1e-300)
		tmp = 0.5 * x;
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x + N[(N[(y - x), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], -1e-300], 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))) < -1.00000000000000003e-300

   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. *-commutative N/A

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

      2. lower-*.f64 51.6

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

   5. Simplified 51.6%

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

   if -1.00000000000000003e-300 < (+.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 x around 0

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

      1. lower-*.f64 52.5

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

   5. Simplified 52.5%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y - x}{2} \leq -1 \cdot 10^{-300}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.1% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 51.4

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

5. Simplified 51.4%

   \[\leadsto \color{blue}{0.5 \cdot y} \]
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 2024207 
(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)))