Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3

Percentage Accurate: 99.9% → 99.9%

Time: 6.3s

Alternatives: 9

Speedup: 1.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. div-inv N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-fabs.f64 N/A

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

   7. lift--.f64 N/A

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

   8. fabs-sub N/A

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

   9. lower-fabs.f64 N/A

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

   10. lower--.f64 N/A

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

   11. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 74.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + \frac{\left|y - x\right|}{2} \leq 10^{-266}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - y, 0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ x (/ (fabs (- y x)) 2.0)) 1e-266) (* 0.5 x) (fma (- x y) 0.5 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x + (fabs((y - x)) / 2.0)) <= 1e-266) {
		tmp = 0.5 * x;
	} else {
		tmp = fma((x - y), 0.5, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x + Float64(abs(Float64(y - x)) / 2.0)) <= 1e-266)
		tmp = Float64(0.5 * x);
	else
		tmp = fma(Float64(x - y), 0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x + N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 1e-266], N[(0.5 * x), $MachinePrecision], N[(N[(x - y), $MachinePrecision] * 0.5 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (fabs.f64 (-.f64 y x)) #s(literal 2 binary64))) < 9.9999999999999998e-267

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. neg-mul-1 N/A

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

      9. distribute-lft-in N/A

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

      10. neg-mul-1 N/A

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

      11. lower-fabs.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 2.9

          \[\leadsto \left|\color{blue}{y - x}\right| \cdot 0.5 \]

   5. Applied rewrites 2.9%

      \[\leadsto \color{blue}{\left|y - x\right| \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 97.1%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 97.2%

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

   if 9.9999999999999998e-267 < (+.f64 x (/.f64 (fabs.f64 (-.f64 y x)) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. lift--.f64 N/A

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

      8. fabs-sub N/A

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

      9. lower-fabs.f64 N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-fabs.f64 N/A

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

      2. unpow1 N/A

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

      3. sqr-pow N/A

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

      4. fabs-sqr N/A

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

      5. sqr-pow N/A

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

      6. unpow1 66.3

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

   6. Applied rewrites 66.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - y, 0.5, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 50.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + \frac{\left|y - x\right|}{2} \leq 10^{-266}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ x (/ (fabs (- y x)) 2.0)) 1e-266) (* 0.5 x) (* -0.5 y)))

C

double code(double x, double y) {
	double tmp;
	if ((x + (fabs((y - x)) / 2.0)) <= 1e-266) {
		tmp = 0.5 * x;
	} 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 ((x + (abs((y - x)) / 2.0d0)) <= 1d-266) then
        tmp = 0.5d0 * x
    else
        tmp = (-0.5d0) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x + (Math.abs((y - x)) / 2.0)) <= 1e-266) {
		tmp = 0.5 * x;
	} else {
		tmp = -0.5 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x + (math.fabs((y - x)) / 2.0)) <= 1e-266:
		tmp = 0.5 * x
	else:
		tmp = -0.5 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x + Float64(abs(Float64(y - x)) / 2.0)) <= 1e-266)
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(-0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x + (abs((y - x)) / 2.0)) <= 1e-266)
		tmp = 0.5 * x;
	else
		tmp = -0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x + N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 1e-266], N[(0.5 * x), $MachinePrecision], N[(-0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (fabs.f64 (-.f64 y x)) #s(literal 2 binary64))) < 9.9999999999999998e-267

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. neg-mul-1 N/A

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

      9. distribute-lft-in N/A

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

      10. neg-mul-1 N/A

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

      11. lower-fabs.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 2.9

          \[\leadsto \left|\color{blue}{y - x}\right| \cdot 0.5 \]

   5. Applied rewrites 2.9%

      \[\leadsto \color{blue}{\left|y - x\right| \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 97.1%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 97.2%

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

   if 9.9999999999999998e-267 < (+.f64 x (/.f64 (fabs.f64 (-.f64 y x)) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. lift--.f64 N/A

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

      8. fabs-sub N/A

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

      9. lower-fabs.f64 N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-fabs.f64 N/A

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

      2. unpow1 N/A

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

      3. sqr-pow N/A

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

      4. fabs-sqr N/A

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

      5. sqr-pow N/A

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

      6. unpow1 66.3

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

   6. Applied rewrites 66.3%

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

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 34.4

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

   9. Applied rewrites 34.4%

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

Alternative 4: 83.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.22 \cdot 10^{-58}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(\left|-y\right|, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1.5, x, -0.5 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.22e-58)
   (* (- x y) 0.5)
   (if (<= x 2.5e-26) (fma (fabs (- y)) 0.5 x) (fma 1.5 x (* -0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.22e-58) {
		tmp = (x - y) * 0.5;
	} else if (x <= 2.5e-26) {
		tmp = fma(fabs(-y), 0.5, x);
	} else {
		tmp = fma(1.5, x, (-0.5 * y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.22e-58)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (x <= 2.5e-26)
		tmp = fma(abs(Float64(-y)), 0.5, x);
	else
		tmp = fma(1.5, x, Float64(-0.5 * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.22e-58], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 2.5e-26], N[(N[Abs[(-y)], $MachinePrecision] * 0.5 + x), $MachinePrecision], N[(1.5 * x + N[(-0.5 * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.2199999999999999e-58

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. neg-mul-1 N/A

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

      9. distribute-lft-in N/A

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

      10. neg-mul-1 N/A

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

      11. lower-fabs.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 34.3

          \[\leadsto \left|\color{blue}{y - x}\right| \cdot 0.5 \]

   5. Applied rewrites 34.3%

      \[\leadsto \color{blue}{\left|y - x\right| \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 83.8%

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

   if -1.2199999999999999e-58 < x < 2.5000000000000001e-26

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. lift--.f64 N/A

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

      8. fabs-sub N/A

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

      9. lower-fabs.f64 N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 85.0

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

   7. Applied rewrites 85.0%

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

   if 2.5000000000000001e-26 < x

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. lift--.f64 N/A

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

      8. fabs-sub N/A

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

      9. lower-fabs.f64 N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-fabs.f64 N/A

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

      2. unpow1 N/A

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

      3. sqr-pow N/A

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

      4. fabs-sqr N/A

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

      5. sqr-pow N/A

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

      6. unpow1 87.7

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

   6. Applied rewrites 87.7%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 87.9

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

   9. Applied rewrites 87.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1.5, x, -0.5 \cdot y\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 83.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.22 \cdot 10^{-58}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(\left|-y\right|, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - y, 0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.22e-58)
   (* (- x y) 0.5)
   (if (<= x 2.5e-26) (fma (fabs (- y)) 0.5 x) (fma (- x y) 0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.22e-58) {
		tmp = (x - y) * 0.5;
	} else if (x <= 2.5e-26) {
		tmp = fma(fabs(-y), 0.5, x);
	} else {
		tmp = fma((x - y), 0.5, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.22e-58)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (x <= 2.5e-26)
		tmp = fma(abs(Float64(-y)), 0.5, x);
	else
		tmp = fma(Float64(x - y), 0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.22e-58], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 2.5e-26], N[(N[Abs[(-y)], $MachinePrecision] * 0.5 + x), $MachinePrecision], N[(N[(x - y), $MachinePrecision] * 0.5 + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.2199999999999999e-58

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. neg-mul-1 N/A

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

      9. distribute-lft-in N/A

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

      10. neg-mul-1 N/A

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

      11. lower-fabs.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 34.3

          \[\leadsto \left|\color{blue}{y - x}\right| \cdot 0.5 \]

   5. Applied rewrites 34.3%

      \[\leadsto \color{blue}{\left|y - x\right| \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 83.8%

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

   if -1.2199999999999999e-58 < x < 2.5000000000000001e-26

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. lift--.f64 N/A

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

      8. fabs-sub N/A

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

      9. lower-fabs.f64 N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 85.0

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

   7. Applied rewrites 85.0%

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

   if 2.5000000000000001e-26 < x

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. lift--.f64 N/A

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

      8. fabs-sub N/A

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

      9. lower-fabs.f64 N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-fabs.f64 N/A

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

      2. unpow1 N/A

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

      3. sqr-pow N/A

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

      4. fabs-sqr N/A

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

      5. sqr-pow N/A

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

      6. unpow1 87.7

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

   6. Applied rewrites 87.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - y, 0.5, x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 82.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.22 \cdot 10^{-58}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-26}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - y, 0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.22e-58)
   (* (- x y) 0.5)
   (if (<= x 2.5e-26) (* (fabs (- y x)) 0.5) (fma (- x y) 0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.22e-58) {
		tmp = (x - y) * 0.5;
	} else if (x <= 2.5e-26) {
		tmp = fabs((y - x)) * 0.5;
	} else {
		tmp = fma((x - y), 0.5, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.22e-58)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (x <= 2.5e-26)
		tmp = Float64(abs(Float64(y - x)) * 0.5);
	else
		tmp = fma(Float64(x - y), 0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.22e-58], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 2.5e-26], N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(x - y), $MachinePrecision] * 0.5 + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.2199999999999999e-58

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. neg-mul-1 N/A

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

      9. distribute-lft-in N/A

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

      10. neg-mul-1 N/A

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

      11. lower-fabs.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 34.3

          \[\leadsto \left|\color{blue}{y - x}\right| \cdot 0.5 \]

   5. Applied rewrites 34.3%

      \[\leadsto \color{blue}{\left|y - x\right| \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 83.8%

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

   if -1.2199999999999999e-58 < x < 2.5000000000000001e-26

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. neg-mul-1 N/A

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

      9. distribute-lft-in N/A

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

      10. neg-mul-1 N/A

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

      11. lower-fabs.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 83.7

          \[\leadsto \left|\color{blue}{y - x}\right| \cdot 0.5 \]

   5. Applied rewrites 83.7%

      \[\leadsto \color{blue}{\left|y - x\right| \cdot 0.5} \]

   if 2.5000000000000001e-26 < x

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. lift--.f64 N/A

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

      8. fabs-sub N/A

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

      9. lower-fabs.f64 N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-fabs.f64 N/A

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

      2. unpow1 N/A

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

      3. sqr-pow N/A

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

      4. fabs-sqr N/A

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

      5. sqr-pow N/A

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

      6. unpow1 87.7

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

   6. Applied rewrites 87.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - y, 0.5, x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 58.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-15}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-161}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3e-15) (* 0.5 x) (if (<= x 1.05e-161) (* -0.5 y) (* 1.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3e-15) {
		tmp = 0.5 * x;
	} else if (x <= 1.05e-161) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3d-15)) then
        tmp = 0.5d0 * x
    else if (x <= 1.05d-161) then
        tmp = (-0.5d0) * y
    else
        tmp = 1.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3e-15) {
		tmp = 0.5 * x;
	} else if (x <= 1.05e-161) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3e-15:
		tmp = 0.5 * x
	elif x <= 1.05e-161:
		tmp = -0.5 * y
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3e-15)
		tmp = Float64(0.5 * x);
	elseif (x <= 1.05e-161)
		tmp = Float64(-0.5 * y);
	else
		tmp = Float64(1.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3e-15)
		tmp = 0.5 * x;
	elseif (x <= 1.05e-161)
		tmp = -0.5 * y;
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3e-15], N[(0.5 * x), $MachinePrecision], If[LessEqual[x, 1.05e-161], N[(-0.5 * y), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3e-15

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. neg-mul-1 N/A

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

      9. distribute-lft-in N/A

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

      10. neg-mul-1 N/A

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

      11. lower-fabs.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 28.6

          \[\leadsto \left|\color{blue}{y - x}\right| \cdot 0.5 \]

   5. Applied rewrites 28.6%

      \[\leadsto \color{blue}{\left|y - x\right| \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 83.6%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 71.3%

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

   if -3e-15 < x < 1.05e-161

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. lift--.f64 N/A

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

      8. fabs-sub N/A

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

      9. lower-fabs.f64 N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fabs.f64 N/A

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

      2. unpow1 N/A

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

      3. sqr-pow N/A

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

      4. fabs-sqr N/A

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

      5. sqr-pow N/A

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

      6. unpow1 50.3

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

   6. Applied rewrites 50.3%

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

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 43.9

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

   9. Applied rewrites 43.9%

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

   if 1.05e-161 < x

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. lift--.f64 N/A

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

      8. fabs-sub N/A

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

      9. lower-fabs.f64 N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-fabs.f64 N/A

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

      2. unpow1 N/A

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

      3. sqr-pow N/A

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

      4. fabs-sqr N/A

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

      5. sqr-pow N/A

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

      6. unpow1 80.6

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

   6. Applied rewrites 80.6%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 65.9

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

   9. Applied rewrites 65.9%

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

Alternative 8: 66.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.95 \cdot 10^{-113}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.95e-113) (* (- x y) 0.5) (* 1.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.95e-113) {
		tmp = (x - y) * 0.5;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.95d-113) then
        tmp = (x - y) * 0.5d0
    else
        tmp = 1.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.95e-113) {
		tmp = (x - y) * 0.5;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.95e-113:
		tmp = (x - y) * 0.5
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.95e-113)
		tmp = Float64(Float64(x - y) * 0.5);
	else
		tmp = Float64(1.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.95e-113)
		tmp = (x - y) * 0.5;
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.95e-113], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.9499999999999999e-113

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. neg-mul-1 N/A

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

      9. distribute-lft-in N/A

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

      10. neg-mul-1 N/A

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

      11. lower-fabs.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 62.7

          \[\leadsto \left|\color{blue}{y - x}\right| \cdot 0.5 \]

   5. Applied rewrites 62.7%

      \[\leadsto \color{blue}{\left|y - x\right| \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 65.2%

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

   if 1.9499999999999999e-113 < x

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. lift--.f64 N/A

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

      8. fabs-sub N/A

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

      9. lower-fabs.f64 N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-fabs.f64 N/A

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

      2. unpow1 N/A

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

      3. sqr-pow N/A

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

      4. fabs-sqr N/A

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

      5. sqr-pow N/A

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

      6. unpow1 81.6

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

   6. Applied rewrites 81.6%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 69.3

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

   9. Applied rewrites 69.3%

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

Alternative 9: 31.2% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. sub-neg N/A

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

   3. mul-1-neg N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. remove-double-neg N/A

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

   7. mul-1-neg N/A

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

   8. neg-mul-1 N/A

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

   9. distribute-lft-in N/A

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

   10. neg-mul-1 N/A

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

   11. lower-fabs.f64 N/A

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

   12. +-commutative N/A

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

   13. distribute-neg-in N/A

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

   14. mul-1-neg N/A

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

   15. remove-double-neg N/A

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

   16. sub-neg N/A

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

   17. lower--.f64 56.2

       \[\leadsto \left|\color{blue}{y - x}\right| \cdot 0.5 \]

5. Applied rewrites 56.2%

   \[\leadsto \color{blue}{\left|y - x\right| \cdot 0.5} \]
6. Step-by-step derivation

7. Applied rewrites 52.5%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 28.7%

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

Reproduce

herbie shell --seed 2024318 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))