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

Percentage Accurate: 99.9% → 100.0%

Time: 8.4s

Alternatives: 7

Speedup: 1.8×

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, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative N/A

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

   2. div-sub N/A

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

   3. sub-neg N/A

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

   4. associate-+l+ N/A

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

   5. div-inv N/A

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

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

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

   7. metadata-eval N/A

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

   8. div-inv N/A

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

   9. distribute-rgt-neg-in N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

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

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

   15. metadata-eval N/A

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

   16. metadata-eval 99.8

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot y + \frac{3}{2} \cdot x} \]
6. Step-by-step derivation

   1. metadata-eval N/A

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

   2. distribute-rgt1-in N/A

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

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

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

   4. distribute-rgt1-in N/A

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

   5. metadata-eval N/A

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

   6. *-lowering-*.f64 99.8

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

7. Simplified 99.8%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

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

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

   4. *-lowering-*.f64 100.0

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

9. Applied egg-rr 100.0%

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

Alternative 2: 75.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(y, -0.5, x\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+49}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5.5e+86)
   (fma y -0.5 x)
   (if (<= y 1.35e+49) (* x 1.5) (fma y -0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5.5e+86) {
		tmp = fma(y, -0.5, x);
	} else if (y <= 1.35e+49) {
		tmp = x * 1.5;
	} else {
		tmp = fma(y, -0.5, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5.5e+86)
		tmp = fma(y, -0.5, x);
	elseif (y <= 1.35e+49)
		tmp = Float64(x * 1.5);
	else
		tmp = fma(y, -0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5.5e+86], N[(y * -0.5 + x), $MachinePrecision], If[LessEqual[y, 1.35e+49], N[(x * 1.5), $MachinePrecision], N[(y * -0.5 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.5000000000000002e86 or 1.35000000000000005e49 < y

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

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

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

      5. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - \left(x - y\right)}, \frac{1}{\mathsf{neg}\left(2\right)}, x\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(x + \left(\mathsf{neg}\left(y\right)\right)\right)}, \frac{1}{\mathsf{neg}\left(2\right)}, x\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + x\right)}, \frac{1}{\mathsf{neg}\left(2\right)}, x\right) \]

      8. associate--r+ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - x}, \frac{1}{\mathsf{neg}\left(2\right)}, x\right) \]

      9. neg-sub0 N/A

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

      10. remove-double-neg N/A

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

      11. --lowering--.f64 N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 99.9

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

   4. Applied egg-rr 99.9%

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

   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 82.7%

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

   if -5.5000000000000002e86 < y < 1.35000000000000005e49

   1. Initial program 99.7%

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

      1. +-commutative N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. associate-+l+ N/A

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

      5. div-inv N/A

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

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

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

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

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

      15. metadata-eval N/A

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

      16. metadata-eval 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 77.8

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

   7. Simplified 77.8%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(y, -0.5, x\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+49}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -0.5, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+86}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+48}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5.5e+86) (* -0.5 y) (if (<= y 1.55e+48) (* x 1.5) (* -0.5 y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5.5e+86) {
		tmp = -0.5 * y;
	} else if (y <= 1.55e+48) {
		tmp = x * 1.5;
	} else {
		tmp = -0.5 * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-5.5d+86)) then
        tmp = (-0.5d0) * y
    else if (y <= 1.55d+48) then
        tmp = x * 1.5d0
    else
        tmp = (-0.5d0) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -5.5e+86) {
		tmp = -0.5 * y;
	} else if (y <= 1.55e+48) {
		tmp = x * 1.5;
	} else {
		tmp = -0.5 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5.5e+86:
		tmp = -0.5 * y
	elif y <= 1.55e+48:
		tmp = x * 1.5
	else:
		tmp = -0.5 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5.5e+86)
		tmp = Float64(-0.5 * y);
	elseif (y <= 1.55e+48)
		tmp = Float64(x * 1.5);
	else
		tmp = Float64(-0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -5.5e+86)
		tmp = -0.5 * y;
	elseif (y <= 1.55e+48)
		tmp = x * 1.5;
	else
		tmp = -0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5.5e+86], N[(-0.5 * y), $MachinePrecision], If[LessEqual[y, 1.55e+48], N[(x * 1.5), $MachinePrecision], N[(-0.5 * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.5000000000000002e86 or 1.55000000000000003e48 < 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. +-rgt-identity N/A

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

      2. accelerator-lowering-fma.f64 79.4

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

   5. Simplified 79.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, y, 0\right)} \]
   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 79.4

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

   7. Applied egg-rr 79.4%

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

   if -5.5000000000000002e86 < y < 1.55000000000000003e48

   1. Initial program 99.7%

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

      1. +-commutative N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. associate-+l+ N/A

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

      5. div-inv N/A

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

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

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

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

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

      15. metadata-eval N/A

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

      16. metadata-eval 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 77.8

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

   7. Simplified 77.8%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+86}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+48}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative N/A

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

   2. div-sub N/A

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

   3. sub-neg N/A

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

   4. associate-+l+ N/A

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

   5. div-inv N/A

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

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

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

   7. metadata-eval N/A

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

   8. div-inv N/A

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

   9. distribute-rgt-neg-in N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

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

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

   15. metadata-eval N/A

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

   16. metadata-eval 99.8

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot y + \frac{3}{2} \cdot x} \]
6. Step-by-step derivation

   1. metadata-eval N/A

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

   2. distribute-rgt1-in N/A

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

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

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

   4. distribute-rgt1-in N/A

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

   5. metadata-eval N/A

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

   6. *-lowering-*.f64 99.8

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 5: 99.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative N/A

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

   2. frac-2neg N/A

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

   3. div-inv N/A

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

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

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

   5. neg-sub0 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{0 - \left(x - y\right)}, \frac{1}{\mathsf{neg}\left(2\right)}, x\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(x + \left(\mathsf{neg}\left(y\right)\right)\right)}, \frac{1}{\mathsf{neg}\left(2\right)}, x\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + x\right)}, \frac{1}{\mathsf{neg}\left(2\right)}, x\right) \]

   8. associate--r+ N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - x}, \frac{1}{\mathsf{neg}\left(2\right)}, x\right) \]

   9. neg-sub0 N/A

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

   10. remove-double-neg N/A

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

   11. --lowering--.f64 N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval 99.8

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

4. Applied egg-rr 99.8%

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

Alternative 6: 51.2% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative N/A

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

   2. div-sub N/A

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

   3. sub-neg N/A

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

   4. associate-+l+ N/A

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

   5. div-inv N/A

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

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

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

   7. metadata-eval N/A

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

   8. div-inv N/A

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

   9. distribute-rgt-neg-in N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

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

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

   15. metadata-eval N/A

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

   16. metadata-eval 99.8

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in x around inf

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

   1. *-lowering-*.f64 55.2

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

7. Simplified 55.2%

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

8. Final simplification 55.2%

   \[\leadsto x \cdot 1.5 \]
9. Add Preprocessing

Alternative 7: 11.8% accurate, 18.0× 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 99.8%

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

   1. +-commutative N/A

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

   2. frac-2neg N/A

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

   3. div-inv N/A

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

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

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

   5. neg-sub0 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{0 - \left(x - y\right)}, \frac{1}{\mathsf{neg}\left(2\right)}, x\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(x + \left(\mathsf{neg}\left(y\right)\right)\right)}, \frac{1}{\mathsf{neg}\left(2\right)}, x\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + x\right)}, \frac{1}{\mathsf{neg}\left(2\right)}, x\right) \]

   8. associate--r+ N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - x}, \frac{1}{\mathsf{neg}\left(2\right)}, x\right) \]

   9. neg-sub0 N/A

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

   10. remove-double-neg N/A

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

   11. --lowering--.f64 N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval 99.8

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

4. Applied egg-rr 99.8%

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

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 54.8%

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

8. Taylor expanded in y around 0

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

10. Simplified 12.5%

    \[\leadsto \color{blue}{x} \]
11. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.3× 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 2024195 
(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)))