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

Percentage Accurate: 100.0% → 100.0%

Time: 8.0s

Alternatives: 4

Speedup: 1.8×

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, 1.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \mathsf{fma}\left(y - x, 0.5, x\right) \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (fma (- y x) 0.5 x))

C

assert(x < y);
double code(double x, double y) {
	return fma((y - x), 0.5, x);
}

Julia

x, y = sort([x, y])
function code(x, y)
	return fma(Float64(y - x), 0.5, x)
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(N[(y - x), $MachinePrecision] * 0.5 + x), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. +-commutative N/A

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

   2. div-inv N/A

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

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

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

   4. --lowering--.f64 N/A

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

   5. metadata-eval 100.0

      \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{0.5}, x\right) \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 0.5, x\right)} \]
5. Add Preprocessing

Alternative 2: 98.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x + \frac{y - x}{2} \leq -5 \cdot 10^{-302}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= (+ x (/ (- y x) 2.0)) -5e-302) (* x 0.5) (* y 0.5)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if ((x + ((y - x) / 2.0)) <= -5e-302) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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)) <= (-5d-302)) then
        tmp = x * 0.5d0
    else
        tmp = y * 0.5d0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if ((x + ((y - x) / 2.0)) <= -5e-302) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if (x + ((y - x) / 2.0)) <= -5e-302:
		tmp = x * 0.5
	else:
		tmp = y * 0.5
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (Float64(x + Float64(Float64(y - x) / 2.0)) <= -5e-302)
		tmp = Float64(x * 0.5);
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x + ((y - x) / 2.0)) <= -5e-302)
		tmp = x * 0.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[N[(x + N[(N[(y - x), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], -5e-302], N[(x * 0.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (-.f64 y x) #s(literal 2 binary64))) < -5.00000000000000033e-302

   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. metadata-eval N/A

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

      2. associate-*r* N/A

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

      3. +-rgt-identity N/A

         \[\leadsto \color{blue}{-1 \cdot \left(\frac{-1}{2} \cdot x\right) + 0} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{-1}{2}\right) \cdot x} + 0 \]

      5. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{1}{2}} \cdot x + 0 \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{1}{2}} + 0 \]

      7. accelerator-lowering-fma.f64 50.2

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, 0\right)} \]

   5. Simplified 50.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 50.2

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

   7. Applied egg-rr 50.2%

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

   if -5.00000000000000033e-302 < (+.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. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot y + 0} \]

      2. accelerator-lowering-fma.f64 43.6

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, y, 0\right)} \]

   5. Simplified 43.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, y, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 43.6

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

   7. Applied egg-rr 43.6%

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

Alternative 3: 51.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ x \cdot 0.5 \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (* x 0.5))

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * 0.5d0
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y):
	return x * 0.5

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(x * 0.5)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = x * 0.5;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(x * 0.5), $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. metadata-eval N/A

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

   2. associate-*r* N/A

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

   3. +-rgt-identity N/A

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{-1}{2} \cdot x\right) + 0} \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{-1}{2}\right) \cdot x} + 0 \]

   5. metadata-eval N/A

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot x + 0 \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{x \cdot \frac{1}{2}} + 0 \]

   7. accelerator-lowering-fma.f64 55.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, 0\right)} \]

5. Simplified 55.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. *-lowering-*.f64 55.3

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

7. Applied egg-rr 55.3%

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

Alternative 4: 11.2% accurate, 18.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ x \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 x)

C

assert(x < y);
double code(double x, double y) {
	return x;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x
end function

Java

assert x < y;
public static double code(double x, double y) {
	return x;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return x

Julia

x, y = sort([x, y])
function code(x, y)
	return x
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = x;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := x

Derivation

1. Initial program 100.0%

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

   1. +-commutative N/A

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

   2. div-inv N/A

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

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

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

   4. --lowering--.f64 N/A

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

   5. metadata-eval 100.0

      \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{0.5}, x\right) \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 0.5, x\right)} \]

5. Taylor expanded in y around inf

   \[\leadsto \mathsf{fma}\left(\color{blue}{y}, \frac{1}{2}, x\right) \]
6. Step-by-step derivation

7. Simplified 54.2%

   \[\leadsto \mathsf{fma}\left(\color{blue}{y}, 0.5, x\right) \]

8. Taylor expanded in y around 0

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

10. Simplified 11.8%

    \[\leadsto \color{blue}{x} \]
11. 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 2024195 
(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)))