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

Percentage Accurate: 99.9% → 99.9%

Time: 8.1s

Alternatives: 6

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: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, \frac{y \cdot 1.5}{y}, y \cdot -0.5\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. flip-- N/A

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

   2. div-inv N/A

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

   3. difference-of-squares N/A

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

   4. associate-*l* N/A

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

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

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

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

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

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

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

   8. --lowering--.f64 N/A

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

   9. /-lowering-/.f64 N/A

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

   10. +-lowering-+.f64 99.8%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in y around inf

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

   1. distribute-lft1-in N/A

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

   2. metadata-eval N/A

      \[\leadsto y \cdot \left(\frac{3}{2} \cdot \frac{x}{y} - \frac{1}{2}\right) \]

   3. *-commutative N/A

      \[\leadsto y \cdot \left(\frac{x}{y} \cdot \frac{3}{2} - \frac{1}{2}\right) \]

   4. metadata-eval N/A

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

   5. distribute-rgt-out-- N/A

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

   6. associate--r+ N/A

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

   7. +-commutative N/A

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

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

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

   9. +-commutative N/A

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

   10. associate--r+ N/A

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

   11. distribute-rgt-out-- N/A

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

   12. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{x}{y} \cdot \frac{3}{2} - \frac{1}{2}\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{3}{2} \cdot \frac{x}{y} - \frac{1}{2}\right)\right) \]

   14. fmm-def N/A

       \[\leadsto \mathsf{*.f64}\left(y, \left(\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\frac{x}{y}}, \mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(y, \left(\mathsf{fma}\left(\frac{3}{2}, \frac{x}{y}, \frac{-1}{2}\right)\right)\right) \]

   16. fma-define N/A

       \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{3}{2} \cdot \frac{x}{y} + \color{blue}{\frac{-1}{2}}\right)\right) \]

   17. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{\frac{3}{2} \cdot x}{y} + \frac{-1}{2}\right)\right) \]

   18. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{x \cdot \frac{3}{2}}{y} + \frac{-1}{2}\right)\right) \]

   19. metadata-eval N/A

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

   20. distribute-rgt-out-- N/A

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

   21. +-commutative N/A

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

7. Simplified 87.0%

   \[\leadsto \color{blue}{y \cdot \left(-0.5 + 1.5 \cdot \frac{x}{y}\right)} \]
8. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \frac{-1}{2} \cdot y + \color{blue}{\left(\frac{3}{2} \cdot \frac{x}{y}\right) \cdot y} \]

   2. +-commutative N/A

      \[\leadsto \left(\frac{3}{2} \cdot \frac{x}{y}\right) \cdot y + \color{blue}{\frac{-1}{2} \cdot y} \]

   3. *-commutative N/A

      \[\leadsto \left(\frac{x}{y} \cdot \frac{3}{2}\right) \cdot y + \frac{-1}{2} \cdot y \]

   4. associate-*l* N/A

      \[\leadsto \frac{x}{y} \cdot \left(\frac{3}{2} \cdot y\right) + \color{blue}{\frac{-1}{2}} \cdot y \]

   5. fma-define N/A

      \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \color{blue}{\frac{3}{2} \cdot y}, \frac{-1}{2} \cdot y\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(\left(\frac{x}{y}\right), \color{blue}{\left(\frac{3}{2} \cdot y\right)}, \left(\frac{-1}{2} \cdot y\right)\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(x, y\right), \left(\color{blue}{\frac{3}{2}} \cdot y\right), \left(\frac{-1}{2} \cdot y\right)\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{*.f64}\left(\frac{3}{2}, \color{blue}{y}\right), \left(\frac{-1}{2} \cdot y\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{*.f64}\left(\frac{3}{2}, y\right), \left(y \cdot \frac{-1}{2}\right)\right) \]

   10. *-lowering-*.f64 87.1%

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{*.f64}\left(\frac{3}{2}, y\right), \mathsf{*.f64}\left(y, \frac{-1}{2}\right)\right) \]

9. Applied egg-rr 87.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, 1.5 \cdot y, y \cdot -0.5\right)} \]
10. Step-by-step derivation

    1. associate-*l/ N/A

       \[\leadsto \frac{x \cdot \left(\frac{3}{2} \cdot y\right)}{y} + \color{blue}{y} \cdot \frac{-1}{2} \]

    2. associate-/l* N/A

       \[\leadsto x \cdot \frac{\frac{3}{2} \cdot y}{y} + \color{blue}{y} \cdot \frac{-1}{2} \]

    3. fma-define N/A

       \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{\frac{3}{2} \cdot y}{y}}, y \cdot \frac{-1}{2}\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\left(\frac{\frac{3}{2} \cdot y}{y}\right)}, \left(y \cdot \frac{-1}{2}\right)\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{3}{2} \cdot y\right), \color{blue}{y}\right), \left(y \cdot \frac{-1}{2}\right)\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{/.f64}\left(\left(y \cdot \frac{3}{2}\right), y\right), \left(y \cdot \frac{-1}{2}\right)\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \frac{3}{2}\right), y\right), \left(y \cdot \frac{-1}{2}\right)\right) \]

    8. *-lowering-*.f64 99.9%

       \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \frac{3}{2}\right), y\right), \mathsf{*.f64}\left(y, \frac{-1}{2}\right)\right) \]

11. Applied egg-rr 99.9%

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{y \cdot 1.5}{y}, y \cdot -0.5\right)} \]
12. Add Preprocessing

Alternative 2: 76.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + y \cdot -0.5\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{+36}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+52}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (* y -0.5))))
   (if (<= y -6.2e+36) t_0 (if (<= y 2.4e+52) (* x 1.5) t_0))))

C

double code(double x, double y) {
	double t_0 = x + (y * -0.5);
	double tmp;
	if (y <= -6.2e+36) {
		tmp = t_0;
	} else if (y <= 2.4e+52) {
		tmp = x * 1.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (y * (-0.5d0))
    if (y <= (-6.2d+36)) then
        tmp = t_0
    else if (y <= 2.4d+52) then
        tmp = x * 1.5d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x + (y * -0.5);
	double tmp;
	if (y <= -6.2e+36) {
		tmp = t_0;
	} else if (y <= 2.4e+52) {
		tmp = x * 1.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + (y * -0.5)
	tmp = 0
	if y <= -6.2e+36:
		tmp = t_0
	elif y <= 2.4e+52:
		tmp = x * 1.5
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(y * -0.5))
	tmp = 0.0
	if (y <= -6.2e+36)
		tmp = t_0;
	elseif (y <= 2.4e+52)
		tmp = Float64(x * 1.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + (y * -0.5);
	tmp = 0.0;
	if (y <= -6.2e+36)
		tmp = t_0;
	elseif (y <= 2.4e+52)
		tmp = x * 1.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(y * -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.2e+36], t$95$0, If[LessEqual[y, 2.4e+52], N[(x * 1.5), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.1999999999999999e36 or 2.4e52 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 84.2%

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

   5. Simplified 84.2%

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

   if -6.1999999999999999e36 < y < 2.4e52

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 79.3%

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

   5. Simplified 79.3%

      \[\leadsto \color{blue}{1.5 \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+36}:\\ \;\;\;\;x + y \cdot -0.5\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+52}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+30}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+52}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.55e+30) (* y -0.5) (if (<= y 4e+52) (* x 1.5) (* y -0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.55e+30) {
		tmp = y * -0.5;
	} else if (y <= 4e+52) {
		tmp = x * 1.5;
	} 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 (y <= (-1.55d+30)) then
        tmp = y * (-0.5d0)
    else if (y <= 4d+52) then
        tmp = x * 1.5d0
    else
        tmp = y * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.55e+30) {
		tmp = y * -0.5;
	} else if (y <= 4e+52) {
		tmp = x * 1.5;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.55e+30:
		tmp = y * -0.5
	elif y <= 4e+52:
		tmp = x * 1.5
	else:
		tmp = y * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.55e+30)
		tmp = Float64(y * -0.5);
	elseif (y <= 4e+52)
		tmp = Float64(x * 1.5);
	else
		tmp = Float64(y * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.55e+30)
		tmp = y * -0.5;
	elseif (y <= 4e+52)
		tmp = x * 1.5;
	else
		tmp = y * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.55e+30], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 4e+52], N[(x * 1.5), $MachinePrecision], N[(y * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.5499999999999999e30 or 4e52 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 81.3%

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

   5. Simplified 81.3%

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

   if -1.5499999999999999e30 < y < 4e52

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 79.3%

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

   5. Simplified 79.3%

      \[\leadsto \color{blue}{1.5 \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+30}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+52}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{+168}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -2.15e+168) x (* y -0.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.15e+168) {
		tmp = x;
	} 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 <= (-2.15d+168)) then
        tmp = x
    else
        tmp = y * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.15e+168) {
		tmp = x;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.15e+168:
		tmp = x
	else:
		tmp = y * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.15e+168)
		tmp = x;
	else
		tmp = Float64(y * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.15e+168)
		tmp = x;
	else
		tmp = y * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.15e+168], x, N[(y * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1499999999999999e168

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 22.2%

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

   5. Simplified 22.2%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 19.4%

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

   if -2.1499999999999999e168 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 56.2%

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

   5. Simplified 56.2%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{+168}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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 6: 11.7% accurate, 7.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 57.9%

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

5. Simplified 57.9%

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

6. Taylor expanded in x around inf

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

8. Simplified 11.8%

   \[\leadsto \color{blue}{x} \]
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 2024152 
(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)))