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

Percentage Accurate: 100.0% → 100.0%

Time: 8.1s

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

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

5. Simplified 100.0%

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

   1. +-rgt-identity N/A

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

   2. *-commutative N/A

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

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

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

   4. +-commutative N/A

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

   5. +-lowering-+.f64 100.0

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

7. Applied egg-rr 100.0%

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

Alternative 2: 50.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + \frac{y - x}{2} \leq -1 \cdot 10^{-308}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ x (/ (- y x) 2.0)) -1e-308) (* x 0.5) (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if ((x + ((y - x) / 2.0)) <= -1e-308) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	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-308)) then
        tmp = x * 0.5d0
    else
        tmp = y * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x + ((y - x) / 2.0)) <= -1e-308) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x + ((y - x) / 2.0)) <= -1e-308:
		tmp = x * 0.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x + Float64(Float64(y - x) / 2.0)) <= -1e-308)
		tmp = Float64(x * 0.5);
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x + ((y - x) / 2.0)) <= -1e-308)
		tmp = x * 0.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x + N[(N[(y - x), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], -1e-308], 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))) < -9.9999999999999991e-309

   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 44.1

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

   5. Simplified 44.1%

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

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 44.1

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

   7. Applied egg-rr 44.1%

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

   if -9.9999999999999991e-309 < (+.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 46.5

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

   5. Simplified 46.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, y, 0\right)} \]
   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 46.5

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

   7. Applied egg-rr 46.5%

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

Alternative 3: 51.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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 48.8

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

5. Simplified 48.8%

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

   1. +-rgt-identity N/A

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

   2. *-lowering-*.f64 48.8

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

7. Applied egg-rr 48.8%

   \[\leadsto \color{blue}{x \cdot 0.5} \]
8. 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 2024196 
(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)))