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

Percentage Accurate: 100.0% → 100.0%

Time: 2.6s

Alternatives: 3

Speedup: 1.4×

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, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. +-commutative 100.0%

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

   2. div-sub 100.0%

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

   3. sub-neg 100.0%

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

   4. distribute-frac-neg 100.0%

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

   5. associate-+l+ 100.0%

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

   6. remove-double-neg 100.0%

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

   7. sub-neg 100.0%

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

   8. neg-mul-1 100.0%

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

   9. *-commutative 100.0%

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

   10. associate-/l* 100.0%

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

   11. *-commutative 100.0%

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

   12. neg-mul-1 100.0%

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

   13. distribute-rgt-out-- 100.0%

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

   14. metadata-eval 100.0%

       \[\leadsto \frac{y}{2} + x \cdot \left(\color{blue}{-0.5} - -1\right) \]

   15. metadata-eval 100.0%

       \[\leadsto \frac{y}{2} + x \cdot \color{blue}{0.5} \]

   16. metadata-eval 100.0%

       \[\leadsto \frac{y}{2} + x \cdot \color{blue}{\left(--0.5\right)} \]

   17. metadata-eval 100.0%

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

   18. cancel-sign-sub 100.0%

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

3. Simplified 100.0%

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

Alternative 2: 62.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-94}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -1.1e-94) (* x 0.5) (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.1e-94) {
		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 <= (-1.1d-94)) 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 <= -1.1e-94) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.1e-94:
		tmp = x * 0.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.1e-94)
		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 <= -1.1e-94)
		tmp = x * 0.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.1e-94], N[(x * 0.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.10000000000000001e-94

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. div-sub 100.0%

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

      3. sub-neg 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. associate-+l+ 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-commutative 100.0%

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

      10. associate-/l* 100.0%

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

      11. *-commutative 100.0%

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

      12. neg-mul-1 100.0%

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

      13. distribute-rgt-out-- 100.0%

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

      14. metadata-eval 100.0%

          \[\leadsto \frac{y}{2} + x \cdot \left(\color{blue}{-0.5} - -1\right) \]

      15. metadata-eval 100.0%

          \[\leadsto \frac{y}{2} + x \cdot \color{blue}{0.5} \]

      16. metadata-eval 100.0%

          \[\leadsto \frac{y}{2} + x \cdot \color{blue}{\left(--0.5\right)} \]

      17. metadata-eval 100.0%

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

      18. cancel-sign-sub 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 59.8%

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

   if -1.10000000000000001e-94 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. div-sub 100.0%

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

      3. sub-neg 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. associate-+l+ 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-commutative 100.0%

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

      10. associate-/l* 100.0%

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

      11. *-commutative 100.0%

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

      12. neg-mul-1 100.0%

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

      13. distribute-rgt-out-- 100.0%

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

      14. metadata-eval 100.0%

          \[\leadsto \frac{y}{2} + x \cdot \left(\color{blue}{-0.5} - -1\right) \]

      15. metadata-eval 100.0%

          \[\leadsto \frac{y}{2} + x \cdot \color{blue}{0.5} \]

      16. metadata-eval 100.0%

          \[\leadsto \frac{y}{2} + x \cdot \color{blue}{\left(--0.5\right)} \]

      17. metadata-eval 100.0%

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

      18. cancel-sign-sub 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 61.6%

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

Alternative 3: 50.6% accurate, 2.3× 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. Step-by-step derivation

   1. +-commutative 100.0%

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

   2. div-sub 100.0%

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

   3. sub-neg 100.0%

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

   4. distribute-frac-neg 100.0%

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

   5. associate-+l+ 100.0%

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

   6. remove-double-neg 100.0%

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

   7. sub-neg 100.0%

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

   8. neg-mul-1 100.0%

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

   9. *-commutative 100.0%

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

   10. associate-/l* 100.0%

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

   11. *-commutative 100.0%

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

   12. neg-mul-1 100.0%

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

   13. distribute-rgt-out-- 100.0%

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

   14. metadata-eval 100.0%

       \[\leadsto \frac{y}{2} + x \cdot \left(\color{blue}{-0.5} - -1\right) \]

   15. metadata-eval 100.0%

       \[\leadsto \frac{y}{2} + x \cdot \color{blue}{0.5} \]

   16. metadata-eval 100.0%

       \[\leadsto \frac{y}{2} + x \cdot \color{blue}{\left(--0.5\right)} \]

   17. metadata-eval 100.0%

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

   18. cancel-sign-sub 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around inf 47.2%

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

Developer Target 1: 100.0% accurate, 1.4× 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 2024131 
(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)))