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

Percentage Accurate: 99.9% → 99.9%

Time: 3.6s

Alternatives: 6

Speedup: 1.0×

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, 1.0× 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]

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 70.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-135}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6.8e-135) (* (fabs (- y x)) 0.5) (/ (+ x y) 2.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6.8e-135) {
		tmp = fabs((y - x)) * 0.5;
	} else {
		tmp = (x + y) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-6.8d-135)) then
        tmp = abs((y - x)) * 0.5d0
    else
        tmp = (x + y) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -6.8e-135) {
		tmp = Math.abs((y - x)) * 0.5;
	} else {
		tmp = (x + y) / 2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -6.8e-135:
		tmp = math.fabs((y - x)) * 0.5
	else:
		tmp = (x + y) / 2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6.8e-135)
		tmp = Float64(abs(Float64(y - x)) * 0.5);
	else
		tmp = Float64(Float64(x + y) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -6.8e-135)
		tmp = abs((y - x)) * 0.5;
	else
		tmp = (x + y) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6.8e-135], N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(x + y), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.79999999999999978e-135

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 64.4%

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

   if -6.79999999999999978e-135 < y

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. div-inv 99.9%

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

      3. fma-def 99.9%

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

      4. add-sqr-sqrt 70.3%

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

      5. fabs-sqr 70.3%

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

      6. add-sqr-sqrt 75.7%

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

      7. metadata-eval 75.7%

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

   4. Applied egg-rr 75.7%

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

      1. fma-udef 75.7%

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

      2. flip-+ 38.0%

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

      3. pow2 38.0%

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

      4. pow2 38.0%

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

   6. Applied egg-rr 38.0%

      \[\leadsto \color{blue}{\frac{{\left(\left(y - x\right) \cdot 0.5\right)}^{2} - {x}^{2}}{\left(y - x\right) \cdot 0.5 - x}} \]
   7. Step-by-step derivation

      1. unpow2 38.0%

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

      2. unpow2 38.0%

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

      3. difference-of-squares 39.7%

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

      4. fma-def 39.7%

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

      5. associate-/l* 75.7%

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

      6. fma-def 75.7%

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

      7. +-commutative 75.7%

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

      8. *-commutative 75.7%

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

      9. sub-neg 75.7%

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

      10. +-commutative 75.7%

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

      11. distribute-lft-in 75.7%

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

      12. distribute-rgt-neg-in 75.7%

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

      13. distribute-lft-neg-in 75.7%

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

      14. metadata-eval 75.7%

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

      15. associate-+r+ 75.7%

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

      16. distribute-rgt1-in 75.7%

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

      17. metadata-eval 75.7%

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

      18. distribute-lft-out 75.7%

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

      19. *-inverses 75.7%

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

   8. Simplified 75.7%

      \[\leadsto \color{blue}{\frac{x + y}{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-135}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 46.0% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{-124} \lor \neg \left(y \leq 4.6 \cdot 10^{-58}\right) \land y \leq 1.15 \cdot 10^{-20}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y 1.05e-124) (and (not (<= y 4.6e-58)) (<= y 1.15e-20)))
   (* x 0.5)
   (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= 1.05e-124) || (!(y <= 4.6e-58) && (y <= 1.15e-20))) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= 1.05d-124) .or. (.not. (y <= 4.6d-58)) .and. (y <= 1.15d-20)) then
        tmp = x * 0.5d0
    else
        tmp = y * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= 1.05e-124) || (!(y <= 4.6e-58) && (y <= 1.15e-20))) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= 1.05e-124) or (not (y <= 4.6e-58) and (y <= 1.15e-20)):
		tmp = x * 0.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= 1.05e-124) || (!(y <= 4.6e-58) && (y <= 1.15e-20)))
		tmp = Float64(x * 0.5);
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= 1.05e-124) || (~((y <= 4.6e-58)) && (y <= 1.15e-20)))
		tmp = x * 0.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, 1.05e-124], And[N[Not[LessEqual[y, 4.6e-58]], $MachinePrecision], LessEqual[y, 1.15e-20]]], N[(x * 0.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.05e-124 or 4.5999999999999998e-58 < y < 1.15e-20

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. div-inv 99.9%

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

      3. fma-def 99.9%

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

      4. add-sqr-sqrt 32.7%

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

      5. fabs-sqr 32.7%

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

      6. add-sqr-sqrt 39.1%

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

      7. metadata-eval 39.1%

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

   4. Applied egg-rr 39.1%

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

   5. Taylor expanded in y around 0 36.9%

      \[\leadsto \color{blue}{x + -0.5 \cdot x} \]
   6. Step-by-step derivation

      1. distribute-rgt1-in 36.9%

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

      2. metadata-eval 36.9%

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

      3. *-commutative 36.9%

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

   7. Simplified 36.9%

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

   if 1.05e-124 < y < 4.5999999999999998e-58 or 1.15e-20 < y

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. div-inv 99.9%

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

      3. fma-def 99.9%

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

      4. add-sqr-sqrt 86.0%

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

      5. fabs-sqr 86.0%

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

      6. add-sqr-sqrt 89.1%

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

      7. metadata-eval 89.1%

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

   4. Applied egg-rr 89.1%

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

   5. Taylor expanded in y around inf 78.0%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{-124} \lor \neg \left(y \leq 4.6 \cdot 10^{-58}\right) \land y \leq 1.15 \cdot 10^{-20}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 32.7% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{-190}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.9e-190) x (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.9e-190) {
		tmp = x;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.9d-190) then
        tmp = x
    else
        tmp = y * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.9e-190) {
		tmp = x;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.9e-190:
		tmp = x
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.9e-190)
		tmp = x;
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.9e-190)
		tmp = x;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.9e-190], x, N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.8999999999999999e-190

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 13.0%

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

   if 1.8999999999999999e-190 < y

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. div-inv 99.9%

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

      3. fma-def 99.9%

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

      4. add-sqr-sqrt 83.6%

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

      5. fabs-sqr 83.6%

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

      6. add-sqr-sqrt 87.1%

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

      7. metadata-eval 87.1%

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

   4. Applied egg-rr 87.1%

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

   5. Taylor expanded in y around inf 66.9%

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

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{-190}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.5% accurate, 21.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. +-commutative 99.9%

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

   2. div-inv 99.9%

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

   3. fma-def 99.9%

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

   4. add-sqr-sqrt 50.0%

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

   5. fabs-sqr 50.0%

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

   6. add-sqr-sqrt 55.3%

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

   7. metadata-eval 55.3%

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

4. Applied egg-rr 55.3%

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

   1. fma-udef 55.3%

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

   2. flip-+ 27.5%

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

   3. pow2 27.5%

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

   4. pow2 27.5%

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

6. Applied egg-rr 27.5%

   \[\leadsto \color{blue}{\frac{{\left(\left(y - x\right) \cdot 0.5\right)}^{2} - {x}^{2}}{\left(y - x\right) \cdot 0.5 - x}} \]
7. Step-by-step derivation

   1. unpow2 27.5%

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

   2. unpow2 27.5%

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

   3. difference-of-squares 29.2%

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

   4. fma-def 29.2%

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

   5. associate-/l* 55.3%

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

   6. fma-def 55.3%

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

   7. +-commutative 55.3%

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

   8. *-commutative 55.3%

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

   9. sub-neg 55.3%

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

   10. +-commutative 55.3%

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

   11. distribute-lft-in 55.3%

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

   12. distribute-rgt-neg-in 55.3%

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

   13. distribute-lft-neg-in 55.3%

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

   14. metadata-eval 55.3%

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

   15. associate-+r+ 55.3%

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

   16. distribute-rgt1-in 55.3%

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

   17. metadata-eval 55.3%

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

   18. distribute-lft-out 55.3%

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

   19. *-inverses 55.3%

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

8. Simplified 55.3%

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

9. Final simplification 55.3%

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

Alternative 6: 11.3% accurate, 107.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around inf 11.2%

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

4. Final simplification 11.2%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))