Graphics.Rendering.Chart.Axis.Types:hBufferRect from Chart-1.5.3

Percentage Accurate: 99.9% → 100.0%

Time: 7.7s

Alternatives: 5

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ x + \frac{x - y}{2} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x (/ (- x y) 2.0)))

C

double code(double x, double y) {
	return x + ((x - y) / 2.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + ((x - y) / 2.0d0)
end function

Java

public static double code(double x, double y) {
	return x + ((x - y) / 2.0);
}

Python

def code(x, y):
	return x + ((x - y) / 2.0)

Julia

function code(x, y)
	return Float64(x + Float64(Float64(x - y) / 2.0))
end

MATLAB

function tmp = code(x, y)
	tmp = x + ((x - y) / 2.0);
end

Wolfram

code[x_, y_] := N[(x + N[(N[(x - y), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{x - y}{2} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x (/ (- x y) 2.0)))

C

double code(double x, double y) {
	return x + ((x - y) / 2.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + ((x - y) / 2.0d0)
end function

Java

public static double code(double x, double y) {
	return x + ((x - y) / 2.0);
}

Python

def code(x, y):
	return x + ((x - y) / 2.0)

Julia

function code(x, y)
	return Float64(x + Float64(Float64(x - y) / 2.0))
end

MATLAB

function tmp = code(x, y)
	tmp = x + ((x - y) / 2.0);
end

Wolfram

code[x_, y_] := N[(x + N[(N[(x - y), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, 1.5, \frac{y}{-2}\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma x 1.5 (/ y -2.0)))

C

double code(double x, double y) {
	return fma(x, 1.5, (y / -2.0));
}

Julia

function code(x, y)
	return fma(x, 1.5, Float64(y / -2.0))
end

Wolfram

code[x_, y_] := N[(x * 1.5 + N[(y / -2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. div-sub N/A

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

   2. sub-neg N/A

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

   3. associate-+r+ N/A

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

   4. +-commutative N/A

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

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{y}{2}\right)\right), \color{blue}{\left(x + \frac{x}{2}\right)}\right) \]

   6. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{y}{\mathsf{neg}\left(2\right)}\right), \left(\color{blue}{x} + \frac{x}{2}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(2\right)\right)\right), \left(\color{blue}{x} + \frac{x}{2}\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \frac{x}{2}\right)\right) \]

   9. remove-double-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)}{2}\right)\right) \]

   10. neg-mul-1 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \frac{-1 \cdot \left(\mathsf{neg}\left(x\right)\right)}{2}\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \frac{\left(\mathsf{neg}\left(x\right)\right) \cdot -1}{2}\right)\right) \]

   12. associate-/l* N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\frac{-1}{2}}\right)\right) \]

   13. cancel-sign-sub-inv N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x - \color{blue}{x \cdot \frac{-1}{2}}\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x - \frac{-1}{2} \cdot \color{blue}{x}\right)\right) \]

   15. cancel-sign-sub-inv N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot x}\right)\right) \]

   16. *-lft-identity N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(1 \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)} \cdot x\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x + \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{2}}\right)\right) \cdot x\right)\right) \]

   18. distribute-rgt-out N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right)}\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right)}\right)\right) \]

   20. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \mathsf{*.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{2}}\right)\right)\right)\right)\right) \]

   21. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \mathsf{*.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right)\right)\right) \]

   22. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \mathsf{*.f64}\left(x, \left(1 + \frac{1}{2}\right)\right)\right) \]

   23. metadata-eval 99.9%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \mathsf{*.f64}\left(x, \frac{3}{2}\right)\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{y}{-2} + x \cdot 1.5} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \frac{3}{2} + \color{blue}{\frac{y}{-2}} \]

   2. fma-define N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{3}{2}}, \frac{y}{-2}\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\frac{3}{2}}, \left(\frac{y}{-2}\right)\right) \]

   4. /-lowering-/.f64 100.0%

      \[\leadsto \mathsf{fma.f64}\left(x, \frac{3}{2}, \mathsf{/.f64}\left(y, -2\right)\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1.5, \frac{y}{-2}\right)} \]
7. Add Preprocessing

Alternative 2: 73.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+55}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+82}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.7e+55) (* x 1.5) (if (<= x 1.25e+82) (* y -0.5) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.7e+55) {
		tmp = x * 1.5;
	} else if (x <= 1.25e+82) {
		tmp = y * -0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-4.7d+55)) then
        tmp = x * 1.5d0
    else if (x <= 1.25d+82) then
        tmp = y * (-0.5d0)
    else
        tmp = x * 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4.7e+55) {
		tmp = x * 1.5;
	} else if (x <= 1.25e+82) {
		tmp = y * -0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.7e+55:
		tmp = x * 1.5
	elif x <= 1.25e+82:
		tmp = y * -0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.7e+55)
		tmp = Float64(x * 1.5);
	elseif (x <= 1.25e+82)
		tmp = Float64(y * -0.5);
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.7e+55)
		tmp = x * 1.5;
	elseif (x <= 1.25e+82)
		tmp = y * -0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.7e+55], N[(x * 1.5), $MachinePrecision], If[LessEqual[x, 1.25e+82], N[(y * -0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.7000000000000001e55 or 1.25000000000000004e82 < x

   1. Initial program 99.7%

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{y}{2}\right)\right), \color{blue}{\left(x + \frac{x}{2}\right)}\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{y}{\mathsf{neg}\left(2\right)}\right), \left(\color{blue}{x} + \frac{x}{2}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(2\right)\right)\right), \left(\color{blue}{x} + \frac{x}{2}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \frac{x}{2}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)}{2}\right)\right) \]

      10. neg-mul-1 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \frac{-1 \cdot \left(\mathsf{neg}\left(x\right)\right)}{2}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \frac{\left(\mathsf{neg}\left(x\right)\right) \cdot -1}{2}\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\frac{-1}{2}}\right)\right) \]

      13. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x - \color{blue}{x \cdot \frac{-1}{2}}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x - \frac{-1}{2} \cdot \color{blue}{x}\right)\right) \]

      15. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot x}\right)\right) \]

      16. *-lft-identity N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(1 \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)} \cdot x\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x + \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{2}}\right)\right) \cdot x\right)\right) \]

      18. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right)}\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \mathsf{*.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{2}}\right)\right)\right)\right)\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \mathsf{*.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right)\right)\right) \]

      22. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \mathsf{*.f64}\left(x, \left(1 + \frac{1}{2}\right)\right)\right) \]

      23. metadata-eval 99.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \mathsf{*.f64}\left(x, \frac{3}{2}\right)\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{y}{-2} + x \cdot 1.5} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{3}{2} \cdot x} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 82.7%

         \[\leadsto \mathsf{*.f64}\left(\frac{3}{2}, \color{blue}{x}\right) \]

   7. Simplified 82.7%

      \[\leadsto \color{blue}{1.5 \cdot x} \]

   if -4.7000000000000001e55 < x < 1.25000000000000004e82

   1. Initial program 99.9%

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{y}{2}\right)\right), \color{blue}{\left(x + \frac{x}{2}\right)}\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{y}{\mathsf{neg}\left(2\right)}\right), \left(\color{blue}{x} + \frac{x}{2}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(2\right)\right)\right), \left(\color{blue}{x} + \frac{x}{2}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \frac{x}{2}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)}{2}\right)\right) \]

      10. neg-mul-1 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \frac{-1 \cdot \left(\mathsf{neg}\left(x\right)\right)}{2}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \frac{\left(\mathsf{neg}\left(x\right)\right) \cdot -1}{2}\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\frac{-1}{2}}\right)\right) \]

      13. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x - \color{blue}{x \cdot \frac{-1}{2}}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x - \frac{-1}{2} \cdot \color{blue}{x}\right)\right) \]

      15. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot x}\right)\right) \]

      16. *-lft-identity N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(1 \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)} \cdot x\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x + \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{2}}\right)\right) \cdot x\right)\right) \]

      18. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right)}\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \mathsf{*.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{2}}\right)\right)\right)\right)\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \mathsf{*.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right)\right)\right) \]

      22. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \mathsf{*.f64}\left(x, \left(1 + \frac{1}{2}\right)\right)\right) \]

      23. metadata-eval 99.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \mathsf{*.f64}\left(x, \frac{3}{2}\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{y}{-2} + x \cdot 1.5} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 76.2%

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{y}\right) \]

   7. Simplified 76.2%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+55}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+82}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{y}{-2} + x \cdot 1.5 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (/ y -2.0) (* x 1.5)))

C

double code(double x, double y) {
	return (y / -2.0) + (x * 1.5);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (y / (-2.0d0)) + (x * 1.5d0)
end function

Java

public static double code(double x, double y) {
	return (y / -2.0) + (x * 1.5);
}

Python

def code(x, y):
	return (y / -2.0) + (x * 1.5)

Julia

function code(x, y)
	return Float64(Float64(y / -2.0) + Float64(x * 1.5))
end

MATLAB

function tmp = code(x, y)
	tmp = (y / -2.0) + (x * 1.5);
end

Wolfram

code[x_, y_] := N[(N[(y / -2.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. div-sub N/A

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

   2. sub-neg N/A

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

   3. associate-+r+ N/A

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

   4. +-commutative N/A

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

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{y}{2}\right)\right), \color{blue}{\left(x + \frac{x}{2}\right)}\right) \]

   6. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{y}{\mathsf{neg}\left(2\right)}\right), \left(\color{blue}{x} + \frac{x}{2}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(2\right)\right)\right), \left(\color{blue}{x} + \frac{x}{2}\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \frac{x}{2}\right)\right) \]

   9. remove-double-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)}{2}\right)\right) \]

   10. neg-mul-1 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \frac{-1 \cdot \left(\mathsf{neg}\left(x\right)\right)}{2}\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \frac{\left(\mathsf{neg}\left(x\right)\right) \cdot -1}{2}\right)\right) \]

   12. associate-/l* N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\frac{-1}{2}}\right)\right) \]

   13. cancel-sign-sub-inv N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x - \color{blue}{x \cdot \frac{-1}{2}}\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x - \frac{-1}{2} \cdot \color{blue}{x}\right)\right) \]

   15. cancel-sign-sub-inv N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot x}\right)\right) \]

   16. *-lft-identity N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(1 \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)} \cdot x\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x + \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{2}}\right)\right) \cdot x\right)\right) \]

   18. distribute-rgt-out N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right)}\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right)}\right)\right) \]

   20. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \mathsf{*.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{2}}\right)\right)\right)\right)\right) \]

   21. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \mathsf{*.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right)\right)\right) \]

   22. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \mathsf{*.f64}\left(x, \left(1 + \frac{1}{2}\right)\right)\right) \]

   23. metadata-eval 99.9%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \mathsf{*.f64}\left(x, \frac{3}{2}\right)\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{y}{-2} + x \cdot 1.5} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 4: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{x - y}{2} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x (/ (- x y) 2.0)))

C

double code(double x, double y) {
	return x + ((x - y) / 2.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + ((x - y) / 2.0d0)
end function

Java

public static double code(double x, double y) {
	return x + ((x - y) / 2.0);
}

Python

def code(x, y):
	return x + ((x - y) / 2.0)

Julia

function code(x, y)
	return Float64(x + Float64(Float64(x - y) / 2.0))
end

MATLAB

function tmp = code(x, y)
	tmp = x + ((x - y) / 2.0);
end

Wolfram

code[x_, y_] := N[(x + N[(N[(x - y), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

Alternative 5: 51.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. div-sub N/A

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

   2. sub-neg N/A

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

   3. associate-+r+ N/A

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

   4. +-commutative N/A

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

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{y}{2}\right)\right), \color{blue}{\left(x + \frac{x}{2}\right)}\right) \]

   6. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{y}{\mathsf{neg}\left(2\right)}\right), \left(\color{blue}{x} + \frac{x}{2}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(2\right)\right)\right), \left(\color{blue}{x} + \frac{x}{2}\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \frac{x}{2}\right)\right) \]

   9. remove-double-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)}{2}\right)\right) \]

   10. neg-mul-1 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \frac{-1 \cdot \left(\mathsf{neg}\left(x\right)\right)}{2}\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \frac{\left(\mathsf{neg}\left(x\right)\right) \cdot -1}{2}\right)\right) \]

   12. associate-/l* N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\frac{-1}{2}}\right)\right) \]

   13. cancel-sign-sub-inv N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x - \color{blue}{x \cdot \frac{-1}{2}}\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x - \frac{-1}{2} \cdot \color{blue}{x}\right)\right) \]

   15. cancel-sign-sub-inv N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot x}\right)\right) \]

   16. *-lft-identity N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(1 \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)} \cdot x\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x + \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{2}}\right)\right) \cdot x\right)\right) \]

   18. distribute-rgt-out N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right)}\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right)}\right)\right) \]

   20. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \mathsf{*.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{2}}\right)\right)\right)\right)\right) \]

   21. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \mathsf{*.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right)\right)\right) \]

   22. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \mathsf{*.f64}\left(x, \left(1 + \frac{1}{2}\right)\right)\right) \]

   23. metadata-eval 99.9%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \mathsf{*.f64}\left(x, \frac{3}{2}\right)\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{y}{-2} + x \cdot 1.5} \]
4. Add Preprocessing

5. Taylor expanded in y around inf

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

   1. *-lowering-*.f64 54.2%

      \[\leadsto \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{y}\right) \]

7. Simplified 54.2%

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

8. Final simplification 54.2%

   \[\leadsto y \cdot -0.5 \]
9. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1.5 \cdot x - 0.5 \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (* 1.5 x) (* 0.5 y)))

C

double code(double x, double y) {
	return (1.5 * x) - (0.5 * y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.5d0 * x) - (0.5d0 * y)
end function

Java

public static double code(double x, double y) {
	return (1.5 * x) - (0.5 * y);
}

Python

def code(x, y):
	return (1.5 * x) - (0.5 * y)

Julia

function code(x, y)
	return Float64(Float64(1.5 * x) - Float64(0.5 * y))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.5 * x) - (0.5 * y);
end

Wolfram

code[x_, y_] := N[(N[(1.5 * x), $MachinePrecision] - N[(0.5 * y), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024158 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Axis.Types:hBufferRect from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* 3/2 x) (* 1/2 y)))

  (+ x (/ (- x y) 2.0)))