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

Percentage Accurate: 99.9% → 99.9%

Time: 13.5s

Alternatives: 13

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 2: 84.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-182}:\\ \;\;\;\;\mathsf{fma}\left(x, 1.5, y \cdot -0.5\right)\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{-67}:\\ \;\;\;\;x + \frac{\left|x\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6e-182)
   (fma x 1.5 (* y -0.5))
   (if (<= y 9.2e-67) (+ x (/ (fabs x) 2.0)) (* 0.5 (+ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6e-182) {
		tmp = fma(x, 1.5, (y * -0.5));
	} else if (y <= 9.2e-67) {
		tmp = x + (fabs(x) / 2.0);
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6e-182)
		tmp = fma(x, 1.5, Float64(y * -0.5));
	elseif (y <= 9.2e-67)
		tmp = Float64(x + Float64(abs(x) / 2.0));
	else
		tmp = Float64(0.5 * Float64(x + y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6e-182], N[(x * 1.5 + N[(y * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.2e-67], N[(x + N[(N[Abs[x], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.0000000000000003e-182

   1. Initial program 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. fma-define 99.9%

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

      3. add-sqr-sqrt 15.6%

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

      4. fabs-sqr 15.6%

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

      5. add-sqr-sqrt 20.0%

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

      6. frac-2neg 20.0%

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

      7. distribute-frac-neg 20.0%

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

      8. fmm-undef 20.0%

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

      9. *-un-lft-identity 20.0%

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

      10. metadata-eval 20.0%

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

   4. Applied egg-rr 20.0%

      \[\leadsto \color{blue}{x - \frac{y - x}{-2}} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 20.0%

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

      2. div-inv 20.0%

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

      3. add-sqr-sqrt 15.6%

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

      4. fabs-sqr 15.6%

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

      5. add-sqr-sqrt 99.9%

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

      6. fabs-sub 99.9%

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

      7. add-sqr-sqrt 83.7%

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

      8. fabs-sqr 83.7%

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

      9. add-sqr-sqrt 87.0%

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

      10. metadata-eval 87.0%

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

   6. Applied egg-rr 87.0%

      \[\leadsto x - \color{blue}{1 \cdot \left(\left(x - y\right) \cdot -0.5\right)} \]
   7. Step-by-step derivation

      1. *-lft-identity 87.0%

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

   8. Simplified 87.0%

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

   9. Taylor expanded in x around 0 87.0%

      \[\leadsto \color{blue}{1.5 \cdot x - 0.5 \cdot y} \]
   10. Step-by-step derivation

       1. *-commutative 87.0%

          \[\leadsto \color{blue}{x \cdot 1.5} - 0.5 \cdot y \]

       2. fmm-def 87.1%

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

       3. distribute-lft-neg-in 87.1%

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

       4. metadata-eval 87.1%

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

       5. *-commutative 87.1%

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

   11. Simplified 87.1%

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

   if -6.0000000000000003e-182 < y < 9.2000000000000002e-67

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 90.3%

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

      1. neg-mul-1 90.3%

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

   5. Simplified 90.3%

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

   if 9.2000000000000002e-67 < y

   1. Initial program 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. fma-define 99.9%

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

      3. add-sqr-sqrt 87.3%

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

      4. fabs-sqr 87.3%

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

      5. add-sqr-sqrt 90.1%

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

      6. frac-2neg 90.1%

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

      7. distribute-frac-neg 90.1%

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

      8. fmm-undef 90.1%

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

      9. *-un-lft-identity 90.1%

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

      10. metadata-eval 90.1%

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

   4. Applied egg-rr 90.1%

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

   5. Taylor expanded in x around 0 90.1%

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

      1. cancel-sign-sub-inv 90.1%

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

      2. metadata-eval 90.1%

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

      3. distribute-lft-out 90.1%

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

   7. Simplified 90.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(x + y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-182}:\\ \;\;\;\;\mathsf{fma}\left(x, 1.5, y \cdot -0.5\right)\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{-67}:\\ \;\;\;\;x + \frac{\left|x\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-111}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-35}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 1.5, y \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.9e-111)
   (* 0.5 (+ x y))
   (if (<= x 5.5e-35) (+ x (/ (fabs y) 2.0)) (fma x 1.5 (* y -0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.9e-111) {
		tmp = 0.5 * (x + y);
	} else if (x <= 5.5e-35) {
		tmp = x + (fabs(y) / 2.0);
	} else {
		tmp = fma(x, 1.5, (y * -0.5));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.9e-111)
		tmp = Float64(0.5 * Float64(x + y));
	elseif (x <= 5.5e-35)
		tmp = Float64(x + Float64(abs(y) / 2.0));
	else
		tmp = fma(x, 1.5, Float64(y * -0.5));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.9e-111], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.5e-35], N[(x + N[(N[Abs[y], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(x * 1.5 + N[(y * -0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.90000000000000011e-111

   1. Initial program 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. fma-define 100.0%

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

      3. add-sqr-sqrt 83.2%

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

      4. fabs-sqr 83.2%

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

      5. add-sqr-sqrt 84.0%

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

      6. frac-2neg 84.0%

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

      7. distribute-frac-neg 84.0%

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

      8. fmm-undef 84.0%

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

      9. *-un-lft-identity 84.0%

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

      10. metadata-eval 84.0%

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

   4. Applied egg-rr 84.0%

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

   5. Taylor expanded in x around 0 84.0%

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

      1. cancel-sign-sub-inv 84.0%

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

      2. metadata-eval 84.0%

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

      3. distribute-lft-out 84.0%

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

   7. Simplified 84.0%

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

   if -1.90000000000000011e-111 < x < 5.4999999999999997e-35

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 81.9%

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

   if 5.4999999999999997e-35 < x

   1. Initial program 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. fma-define 99.7%

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

      3. add-sqr-sqrt 14.4%

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

      4. fabs-sqr 14.4%

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

      5. add-sqr-sqrt 27.7%

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

      6. frac-2neg 27.7%

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

      7. distribute-frac-neg 27.7%

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

      8. fmm-undef 27.7%

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

      9. *-un-lft-identity 27.7%

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

      10. metadata-eval 27.7%

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

   4. Applied egg-rr 27.7%

      \[\leadsto \color{blue}{x - \frac{y - x}{-2}} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 27.7%

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

      2. div-inv 27.7%

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

      3. add-sqr-sqrt 14.4%

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

      4. fabs-sqr 14.4%

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

      5. add-sqr-sqrt 99.7%

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

      6. fabs-sub 99.7%

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

      7. add-sqr-sqrt 85.0%

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

      8. fabs-sqr 85.0%

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

      9. add-sqr-sqrt 85.3%

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

      10. metadata-eval 85.3%

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

   6. Applied egg-rr 85.3%

      \[\leadsto x - \color{blue}{1 \cdot \left(\left(x - y\right) \cdot -0.5\right)} \]
   7. Step-by-step derivation

      1. *-lft-identity 85.3%

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

   8. Simplified 85.3%

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

   9. Taylor expanded in x around 0 85.3%

      \[\leadsto \color{blue}{1.5 \cdot x - 0.5 \cdot y} \]
   10. Step-by-step derivation

       1. *-commutative 85.3%

          \[\leadsto \color{blue}{x \cdot 1.5} - 0.5 \cdot y \]

       2. fmm-def 85.6%

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

       3. distribute-lft-neg-in 85.6%

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

       4. metadata-eval 85.6%

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

       5. *-commutative 85.6%

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

   11. Simplified 85.6%

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

Alternative 4: 84.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-111}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-35}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5 - y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.8e-111)
   (* 0.5 (+ x y))
   (if (<= x 4.2e-35) (+ x (/ (fabs y) 2.0)) (- (* x 1.5) (* y 0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.8e-111) {
		tmp = 0.5 * (x + y);
	} else if (x <= 4.2e-35) {
		tmp = x + (fabs(y) / 2.0);
	} else {
		tmp = (x * 1.5) - (y * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.8d-111)) then
        tmp = 0.5d0 * (x + y)
    else if (x <= 4.2d-35) then
        tmp = x + (abs(y) / 2.0d0)
    else
        tmp = (x * 1.5d0) - (y * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.8e-111) {
		tmp = 0.5 * (x + y);
	} else if (x <= 4.2e-35) {
		tmp = x + (Math.abs(y) / 2.0);
	} else {
		tmp = (x * 1.5) - (y * 0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.8e-111:
		tmp = 0.5 * (x + y)
	elif x <= 4.2e-35:
		tmp = x + (math.fabs(y) / 2.0)
	else:
		tmp = (x * 1.5) - (y * 0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.8e-111)
		tmp = Float64(0.5 * Float64(x + y));
	elseif (x <= 4.2e-35)
		tmp = Float64(x + Float64(abs(y) / 2.0));
	else
		tmp = Float64(Float64(x * 1.5) - Float64(y * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.8e-111)
		tmp = 0.5 * (x + y);
	elseif (x <= 4.2e-35)
		tmp = x + (abs(y) / 2.0);
	else
		tmp = (x * 1.5) - (y * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.8e-111], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.2e-35], N[(x + N[(N[Abs[y], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * 1.5), $MachinePrecision] - N[(y * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.79999999999999995e-111

   1. Initial program 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. fma-define 100.0%

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

      3. add-sqr-sqrt 83.2%

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

      4. fabs-sqr 83.2%

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

      5. add-sqr-sqrt 84.0%

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

      6. frac-2neg 84.0%

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

      7. distribute-frac-neg 84.0%

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

      8. fmm-undef 84.0%

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

      9. *-un-lft-identity 84.0%

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

      10. metadata-eval 84.0%

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

   4. Applied egg-rr 84.0%

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

   5. Taylor expanded in x around 0 84.0%

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

      1. cancel-sign-sub-inv 84.0%

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

      2. metadata-eval 84.0%

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

      3. distribute-lft-out 84.0%

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

   7. Simplified 84.0%

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

   if -2.79999999999999995e-111 < x < 4.2e-35

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 81.9%

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

   if 4.2e-35 < x

   1. Initial program 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. fma-define 99.7%

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

      3. add-sqr-sqrt 14.4%

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

      4. fabs-sqr 14.4%

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

      5. add-sqr-sqrt 27.7%

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

      6. frac-2neg 27.7%

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

      7. distribute-frac-neg 27.7%

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

      8. fmm-undef 27.7%

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

      9. *-un-lft-identity 27.7%

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

      10. metadata-eval 27.7%

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

   4. Applied egg-rr 27.7%

      \[\leadsto \color{blue}{x - \frac{y - x}{-2}} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 27.7%

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

      2. div-inv 27.7%

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

      3. add-sqr-sqrt 14.4%

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

      4. fabs-sqr 14.4%

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

      5. add-sqr-sqrt 99.7%

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

      6. fabs-sub 99.7%

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

      7. add-sqr-sqrt 85.0%

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

      8. fabs-sqr 85.0%

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

      9. add-sqr-sqrt 85.3%

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

      10. metadata-eval 85.3%

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

   6. Applied egg-rr 85.3%

      \[\leadsto x - \color{blue}{1 \cdot \left(\left(x - y\right) \cdot -0.5\right)} \]
   7. Step-by-step derivation

      1. *-lft-identity 85.3%

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

   8. Simplified 85.3%

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

   9. Taylor expanded in x around 0 85.3%

      \[\leadsto \color{blue}{1.5 \cdot x - 0.5 \cdot y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-111}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-35}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5 - y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.9% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{-164}:\\ \;\;\;\;x - y \cdot 0.5\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-213}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.15e-164)
   (- x (* y 0.5))
   (if (<= y 9e-213) (* x 1.5) (* 0.5 (+ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.15e-164) {
		tmp = x - (y * 0.5);
	} else if (y <= 9e-213) {
		tmp = x * 1.5;
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3.15d-164)) then
        tmp = x - (y * 0.5d0)
    else if (y <= 9d-213) then
        tmp = x * 1.5d0
    else
        tmp = 0.5d0 * (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.15e-164) {
		tmp = x - (y * 0.5);
	} else if (y <= 9e-213) {
		tmp = x * 1.5;
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.15e-164:
		tmp = x - (y * 0.5)
	elif y <= 9e-213:
		tmp = x * 1.5
	else:
		tmp = 0.5 * (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.15e-164)
		tmp = Float64(x - Float64(y * 0.5));
	elseif (y <= 9e-213)
		tmp = Float64(x * 1.5);
	else
		tmp = Float64(0.5 * Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.15e-164)
		tmp = x - (y * 0.5);
	elseif (y <= 9e-213)
		tmp = x * 1.5;
	else
		tmp = 0.5 * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.15e-164], N[(x - N[(y * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9e-213], N[(x * 1.5), $MachinePrecision], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.15000000000000004e-164

   1. Initial program 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. fma-define 99.9%

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

      3. add-sqr-sqrt 16.7%

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

      4. fabs-sqr 16.7%

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

      5. add-sqr-sqrt 20.5%

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

      6. frac-2neg 20.5%

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

      7. distribute-frac-neg 20.5%

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

      8. fmm-undef 20.5%

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

      9. *-un-lft-identity 20.5%

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

      10. metadata-eval 20.5%

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

   4. Applied egg-rr 20.5%

      \[\leadsto \color{blue}{x - \frac{y - x}{-2}} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 20.5%

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

      2. div-inv 20.5%

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

      3. add-sqr-sqrt 16.7%

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

      4. fabs-sqr 16.7%

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

      5. add-sqr-sqrt 99.9%

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

      6. fabs-sub 99.9%

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

      7. add-sqr-sqrt 82.5%

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

      8. fabs-sqr 82.5%

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

      9. add-sqr-sqrt 86.0%

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

      10. metadata-eval 86.0%

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

   6. Applied egg-rr 86.0%

      \[\leadsto x - \color{blue}{1 \cdot \left(\left(x - y\right) \cdot -0.5\right)} \]
   7. Step-by-step derivation

      1. *-lft-identity 86.0%

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

   8. Simplified 86.0%

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

   9. Taylor expanded in x around 0 70.2%

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

   if -3.15000000000000004e-164 < y < 9.0000000000000002e-213

   1. Initial program 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. fma-define 99.7%

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

      3. add-sqr-sqrt 38.5%

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

      4. fabs-sqr 38.5%

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

      5. add-sqr-sqrt 48.7%

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

      6. frac-2neg 48.7%

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

      7. distribute-frac-neg 48.7%

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

      8. fmm-undef 48.7%

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

      9. *-un-lft-identity 48.7%

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

      10. metadata-eval 48.7%

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

   4. Applied egg-rr 48.7%

      \[\leadsto \color{blue}{x - \frac{y - x}{-2}} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 48.7%

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

      2. div-inv 48.7%

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

      3. add-sqr-sqrt 38.5%

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

      4. fabs-sqr 38.5%

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

      5. add-sqr-sqrt 99.7%

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

      6. fabs-sub 99.7%

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

      7. add-sqr-sqrt 60.7%

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

      8. fabs-sqr 60.7%

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

      9. add-sqr-sqrt 67.3%

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

      10. metadata-eval 67.3%

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

   6. Applied egg-rr 67.3%

      \[\leadsto x - \color{blue}{1 \cdot \left(\left(x - y\right) \cdot -0.5\right)} \]
   7. Step-by-step derivation

      1. *-lft-identity 67.3%

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

   8. Simplified 67.3%

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

   9. Taylor expanded in x around inf 60.1%

      \[\leadsto \color{blue}{1.5 \cdot x} \]
   10. Step-by-step derivation

       1. *-commutative 60.1%

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

   11. Simplified 60.1%

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

   if 9.0000000000000002e-213 < y

   1. Initial program 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. fma-define 99.9%

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

      3. add-sqr-sqrt 79.3%

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

      4. fabs-sqr 79.3%

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

      5. add-sqr-sqrt 83.4%

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

      6. frac-2neg 83.4%

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

      7. distribute-frac-neg 83.4%

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

      8. fmm-undef 83.4%

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

      9. *-un-lft-identity 83.4%

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

      10. metadata-eval 83.4%

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

   4. Applied egg-rr 83.4%

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

   5. Taylor expanded in x around 0 83.4%

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

      1. cancel-sign-sub-inv 83.4%

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

      2. metadata-eval 83.4%

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

      3. distribute-lft-out 83.4%

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

   7. Simplified 83.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(x + y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{-164}:\\ \;\;\;\;x - y \cdot 0.5\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-213}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.7% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.12 \cdot 10^{-152}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-213}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.12e-152)
   (* y -0.5)
   (if (<= y 5.8e-213) (* x 1.5) (* 0.5 (+ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.12e-152) {
		tmp = y * -0.5;
	} else if (y <= 5.8e-213) {
		tmp = x * 1.5;
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.12d-152)) then
        tmp = y * (-0.5d0)
    else if (y <= 5.8d-213) then
        tmp = x * 1.5d0
    else
        tmp = 0.5d0 * (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.12e-152) {
		tmp = y * -0.5;
	} else if (y <= 5.8e-213) {
		tmp = x * 1.5;
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.12e-152:
		tmp = y * -0.5
	elif y <= 5.8e-213:
		tmp = x * 1.5
	else:
		tmp = 0.5 * (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.12e-152)
		tmp = Float64(y * -0.5);
	elseif (y <= 5.8e-213)
		tmp = Float64(x * 1.5);
	else
		tmp = Float64(0.5 * Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.12e-152)
		tmp = y * -0.5;
	elseif (y <= 5.8e-213)
		tmp = x * 1.5;
	else
		tmp = 0.5 * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.12e-152], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 5.8e-213], N[(x * 1.5), $MachinePrecision], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.12e-152

   1. Initial program 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. fma-define 99.9%

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

      3. add-sqr-sqrt 14.7%

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

      4. fabs-sqr 14.7%

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

      5. add-sqr-sqrt 18.5%

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

      6. frac-2neg 18.5%

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

      7. distribute-frac-neg 18.5%

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

      8. fmm-undef 18.5%

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

      9. *-un-lft-identity 18.5%

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

      10. metadata-eval 18.5%

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

   4. Applied egg-rr 18.5%

      \[\leadsto \color{blue}{x - \frac{y - x}{-2}} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 18.5%

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

      2. div-inv 18.5%

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

      3. add-sqr-sqrt 14.7%

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

      4. fabs-sqr 14.7%

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

      5. add-sqr-sqrt 99.9%

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

      6. fabs-sub 99.9%

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

      7. add-sqr-sqrt 84.5%

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

      8. fabs-sqr 84.5%

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

      9. add-sqr-sqrt 87.7%

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

      10. metadata-eval 87.7%

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

   6. Applied egg-rr 87.7%

      \[\leadsto x - \color{blue}{1 \cdot \left(\left(x - y\right) \cdot -0.5\right)} \]
   7. Step-by-step derivation

      1. *-lft-identity 87.7%

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

   8. Simplified 87.7%

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

   9. Taylor expanded in x around 0 66.2%

      \[\leadsto \color{blue}{-0.5 \cdot y} \]
   10. Step-by-step derivation

       1. *-commutative 66.2%

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

   11. Simplified 66.2%

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

   if -1.12e-152 < y < 5.7999999999999999e-213

   1. Initial program 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. fma-define 99.7%

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

      3. add-sqr-sqrt 40.6%

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

      4. fabs-sqr 40.6%

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

      5. add-sqr-sqrt 50.5%

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

      6. frac-2neg 50.5%

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

      7. distribute-frac-neg 50.5%

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

      8. fmm-undef 50.5%

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

      9. *-un-lft-identity 50.5%

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

      10. metadata-eval 50.5%

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

   4. Applied egg-rr 50.5%

      \[\leadsto \color{blue}{x - \frac{y - x}{-2}} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 50.5%

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

      2. div-inv 50.5%

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

      3. add-sqr-sqrt 40.6%

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

      4. fabs-sqr 40.6%

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

      5. add-sqr-sqrt 99.7%

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

      6. fabs-sub 99.7%

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

      7. add-sqr-sqrt 58.6%

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

      8. fabs-sqr 58.6%

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

      9. add-sqr-sqrt 65.5%

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

      10. metadata-eval 65.5%

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

   6. Applied egg-rr 65.5%

      \[\leadsto x - \color{blue}{1 \cdot \left(\left(x - y\right) \cdot -0.5\right)} \]
   7. Step-by-step derivation

      1. *-lft-identity 65.5%

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

   8. Simplified 65.5%

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

   9. Taylor expanded in x around inf 58.6%

      \[\leadsto \color{blue}{1.5 \cdot x} \]
   10. Step-by-step derivation

       1. *-commutative 58.6%

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

   11. Simplified 58.6%

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

   if 5.7999999999999999e-213 < y

   1. Initial program 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. fma-define 99.9%

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

      3. add-sqr-sqrt 79.3%

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

      4. fabs-sqr 79.3%

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

      5. add-sqr-sqrt 83.4%

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

      6. frac-2neg 83.4%

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

      7. distribute-frac-neg 83.4%

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

      8. fmm-undef 83.4%

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

      9. *-un-lft-identity 83.4%

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

      10. metadata-eval 83.4%

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

   4. Applied egg-rr 83.4%

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

   5. Taylor expanded in x around 0 83.4%

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

      1. cancel-sign-sub-inv 83.4%

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

      2. metadata-eval 83.4%

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

      3. distribute-lft-out 83.4%

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

   7. Simplified 83.4%

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

Alternative 7: 59.4% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{-154}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-63}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.1e-154) (* y -0.5) (if (<= y 8.2e-63) (* x 1.5) (* y 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.1e-154) {
		tmp = y * -0.5;
	} else if (y <= 8.2e-63) {
		tmp = x * 1.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3.1d-154)) then
        tmp = y * (-0.5d0)
    else if (y <= 8.2d-63) then
        tmp = x * 1.5d0
    else
        tmp = y * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.1e-154) {
		tmp = y * -0.5;
	} else if (y <= 8.2e-63) {
		tmp = x * 1.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.1e-154:
		tmp = y * -0.5
	elif y <= 8.2e-63:
		tmp = x * 1.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.1e-154)
		tmp = Float64(y * -0.5);
	elseif (y <= 8.2e-63)
		tmp = Float64(x * 1.5);
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.1e-154)
		tmp = y * -0.5;
	elseif (y <= 8.2e-63)
		tmp = x * 1.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.1e-154], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 8.2e-63], N[(x * 1.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.09999999999999982e-154

   1. Initial program 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. fma-define 99.9%

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

      3. add-sqr-sqrt 14.7%

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

      4. fabs-sqr 14.7%

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

      5. add-sqr-sqrt 18.5%

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

      6. frac-2neg 18.5%

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

      7. distribute-frac-neg 18.5%

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

      8. fmm-undef 18.5%

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

      9. *-un-lft-identity 18.5%

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

      10. metadata-eval 18.5%

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

   4. Applied egg-rr 18.5%

      \[\leadsto \color{blue}{x - \frac{y - x}{-2}} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 18.5%

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

      2. div-inv 18.5%

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

      3. add-sqr-sqrt 14.7%

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

      4. fabs-sqr 14.7%

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

      5. add-sqr-sqrt 99.9%

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

      6. fabs-sub 99.9%

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

      7. add-sqr-sqrt 84.5%

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

      8. fabs-sqr 84.5%

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

      9. add-sqr-sqrt 87.7%

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

      10. metadata-eval 87.7%

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

   6. Applied egg-rr 87.7%

      \[\leadsto x - \color{blue}{1 \cdot \left(\left(x - y\right) \cdot -0.5\right)} \]
   7. Step-by-step derivation

      1. *-lft-identity 87.7%

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

   8. Simplified 87.7%

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

   9. Taylor expanded in x around 0 66.2%

      \[\leadsto \color{blue}{-0.5 \cdot y} \]
   10. Step-by-step derivation

       1. *-commutative 66.2%

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

   11. Simplified 66.2%

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

   if -3.09999999999999982e-154 < y < 8.1999999999999995e-63

   1. Initial program 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. fma-define 99.8%

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

      3. add-sqr-sqrt 46.1%

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

      4. fabs-sqr 46.1%

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

      5. add-sqr-sqrt 55.3%

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

      6. frac-2neg 55.3%

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

      7. distribute-frac-neg 55.3%

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

      8. fmm-undef 55.3%

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

      9. *-un-lft-identity 55.3%

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

      10. metadata-eval 55.3%

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

   4. Applied egg-rr 55.3%

      \[\leadsto \color{blue}{x - \frac{y - x}{-2}} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 55.3%

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

      2. div-inv 55.3%

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

      3. add-sqr-sqrt 46.1%

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

      4. fabs-sqr 46.1%

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

      5. add-sqr-sqrt 99.8%

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

      6. fabs-sub 99.8%

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

      7. add-sqr-sqrt 53.2%

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

      8. fabs-sqr 53.2%

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

      9. add-sqr-sqrt 60.7%

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

      10. metadata-eval 60.7%

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

   6. Applied egg-rr 60.7%

      \[\leadsto x - \color{blue}{1 \cdot \left(\left(x - y\right) \cdot -0.5\right)} \]
   7. Step-by-step derivation

      1. *-lft-identity 60.7%

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

   8. Simplified 60.7%

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

   9. Taylor expanded in x around inf 55.8%

      \[\leadsto \color{blue}{1.5 \cdot x} \]
   10. Step-by-step derivation

       1. *-commutative 55.8%

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

   11. Simplified 55.8%

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

   if 8.1999999999999995e-63 < y

   1. Initial program 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. fma-define 99.9%

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

      3. add-sqr-sqrt 87.0%

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

      4. fabs-sqr 87.0%

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

      5. add-sqr-sqrt 89.8%

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

      6. frac-2neg 89.8%

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

      7. distribute-frac-neg 89.8%

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

      8. fmm-undef 89.8%

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

      9. *-un-lft-identity 89.8%

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

      10. metadata-eval 89.8%

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

   4. Applied egg-rr 89.8%

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

   5. Taylor expanded in x around 0 78.4%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{-154}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-63}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.6% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-32}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-130}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.2e-32) (* x 0.5) (if (<= x 3e-130) (* y 0.5) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.2e-32) {
		tmp = x * 0.5;
	} else if (x <= 3e-130) {
		tmp = y * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.2d-32)) then
        tmp = x * 0.5d0
    else if (x <= 3d-130) then
        tmp = y * 0.5d0
    else
        tmp = x * 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.2e-32) {
		tmp = x * 0.5;
	} else if (x <= 3e-130) {
		tmp = y * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.2e-32:
		tmp = x * 0.5
	elif x <= 3e-130:
		tmp = y * 0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.2e-32)
		tmp = Float64(x * 0.5);
	elseif (x <= 3e-130)
		tmp = Float64(y * 0.5);
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.2e-32)
		tmp = x * 0.5;
	elseif (x <= 3e-130)
		tmp = y * 0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.2e-32], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, 3e-130], N[(y * 0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.2e-32

   1. Initial program 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. fma-define 100.0%

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

      3. add-sqr-sqrt 83.4%

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

      4. fabs-sqr 83.4%

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

      5. add-sqr-sqrt 84.1%

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

      6. frac-2neg 84.1%

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

      7. distribute-frac-neg 84.1%

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

      8. fmm-undef 84.1%

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

      9. *-un-lft-identity 84.1%

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

      10. metadata-eval 84.1%

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

   4. Applied egg-rr 84.1%

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

   5. Taylor expanded in x around inf 65.7%

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

   if -2.2e-32 < x < 2.99999999999999986e-130

   1. Initial program 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. fma-define 99.9%

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

      3. add-sqr-sqrt 59.1%

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

      4. fabs-sqr 59.1%

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

      5. add-sqr-sqrt 61.1%

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

      6. frac-2neg 61.1%

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

      7. distribute-frac-neg 61.1%

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

      8. fmm-undef 61.1%

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

      9. *-un-lft-identity 61.1%

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

      10. metadata-eval 61.1%

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

   4. Applied egg-rr 61.1%

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

   5. Taylor expanded in x around 0 49.4%

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

   if 2.99999999999999986e-130 < x

   1. Initial program 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. fma-define 99.8%

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

      3. add-sqr-sqrt 19.2%

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

      4. fabs-sqr 19.2%

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

      5. add-sqr-sqrt 30.7%

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

      6. frac-2neg 30.7%

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

      7. distribute-frac-neg 30.7%

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

      8. fmm-undef 30.7%

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

      9. *-un-lft-identity 30.7%

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

      10. metadata-eval 30.7%

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

   4. Applied egg-rr 30.7%

      \[\leadsto \color{blue}{x - \frac{y - x}{-2}} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 30.7%

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

      2. div-inv 30.7%

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

      3. add-sqr-sqrt 19.2%

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

      4. fabs-sqr 19.2%

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

      5. add-sqr-sqrt 99.8%

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

      6. fabs-sub 99.8%

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

      7. add-sqr-sqrt 80.0%

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

      8. fabs-sqr 80.0%

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

      9. add-sqr-sqrt 80.5%

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

      10. metadata-eval 80.5%

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

   6. Applied egg-rr 80.5%

      \[\leadsto x - \color{blue}{1 \cdot \left(\left(x - y\right) \cdot -0.5\right)} \]
   7. Step-by-step derivation

      1. *-lft-identity 80.5%

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

   8. Simplified 80.5%

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

   9. Taylor expanded in x around inf 63.2%

      \[\leadsto \color{blue}{1.5 \cdot x} \]
   10. Step-by-step derivation

       1. *-commutative 63.2%

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

   11. Simplified 63.2%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-32}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-130}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.5% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-213}:\\ \;\;\;\;x \cdot 1.5 - y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7e-213) (- (* x 1.5) (* y 0.5)) (* 0.5 (+ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7e-213) {
		tmp = (x * 1.5) - (y * 0.5);
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 7d-213) then
        tmp = (x * 1.5d0) - (y * 0.5d0)
    else
        tmp = 0.5d0 * (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 7e-213) {
		tmp = (x * 1.5) - (y * 0.5);
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7e-213:
		tmp = (x * 1.5) - (y * 0.5)
	else:
		tmp = 0.5 * (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7e-213)
		tmp = Float64(Float64(x * 1.5) - Float64(y * 0.5));
	else
		tmp = Float64(0.5 * Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7e-213)
		tmp = (x * 1.5) - (y * 0.5);
	else
		tmp = 0.5 * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7e-213], N[(N[(x * 1.5), $MachinePrecision] - N[(y * 0.5), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.00000000000000034e-213

   1. Initial program 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. fma-define 99.8%

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

      3. add-sqr-sqrt 25.3%

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

      4. fabs-sqr 25.3%

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

      5. add-sqr-sqrt 31.6%

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

      6. frac-2neg 31.6%

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

      7. distribute-frac-neg 31.6%

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

      8. fmm-undef 31.6%

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

      9. *-un-lft-identity 31.6%

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

      10. metadata-eval 31.6%

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

   4. Applied egg-rr 31.6%

      \[\leadsto \color{blue}{x - \frac{y - x}{-2}} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 31.6%

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

      2. div-inv 31.6%

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

      3. add-sqr-sqrt 25.3%

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

      4. fabs-sqr 25.3%

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

      5. add-sqr-sqrt 99.8%

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

      6. fabs-sub 99.8%

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

      7. add-sqr-sqrt 73.9%

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

      8. fabs-sqr 73.9%

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

      9. add-sqr-sqrt 78.7%

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

      10. metadata-eval 78.7%

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

   6. Applied egg-rr 78.7%

      \[\leadsto x - \color{blue}{1 \cdot \left(\left(x - y\right) \cdot -0.5\right)} \]
   7. Step-by-step derivation

      1. *-lft-identity 78.7%

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

   8. Simplified 78.7%

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

   9. Taylor expanded in x around 0 78.7%

      \[\leadsto \color{blue}{1.5 \cdot x - 0.5 \cdot y} \]

   if 7.00000000000000034e-213 < y

   1. Initial program 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. fma-define 99.9%

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

      3. add-sqr-sqrt 79.3%

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

      4. fabs-sqr 79.3%

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

      5. add-sqr-sqrt 83.4%

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

      6. frac-2neg 83.4%

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

      7. distribute-frac-neg 83.4%

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

      8. fmm-undef 83.4%

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

      9. *-un-lft-identity 83.4%

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

      10. metadata-eval 83.4%

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

   4. Applied egg-rr 83.4%

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

   5. Taylor expanded in x around 0 83.4%

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

      1. cancel-sign-sub-inv 83.4%

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

      2. metadata-eval 83.4%

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

      3. distribute-lft-out 83.4%

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

   7. Simplified 83.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(x + y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-213}:\\ \;\;\;\;x \cdot 1.5 - y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.5% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-213}:\\ \;\;\;\;x + -0.5 \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7e-213) (+ x (* -0.5 (- y x))) (* 0.5 (+ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7e-213) {
		tmp = x + (-0.5 * (y - x));
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 7d-213) then
        tmp = x + ((-0.5d0) * (y - x))
    else
        tmp = 0.5d0 * (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 7e-213) {
		tmp = x + (-0.5 * (y - x));
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7e-213:
		tmp = x + (-0.5 * (y - x))
	else:
		tmp = 0.5 * (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7e-213)
		tmp = Float64(x + Float64(-0.5 * Float64(y - x)));
	else
		tmp = Float64(0.5 * Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7e-213)
		tmp = x + (-0.5 * (y - x));
	else
		tmp = 0.5 * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7e-213], N[(x + N[(-0.5 * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.00000000000000034e-213

   1. Initial program 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. fma-define 99.8%

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

      3. add-sqr-sqrt 25.3%

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

      4. fabs-sqr 25.3%

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

      5. add-sqr-sqrt 31.6%

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

      6. frac-2neg 31.6%

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

      7. distribute-frac-neg 31.6%

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

      8. fmm-undef 31.6%

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

      9. *-un-lft-identity 31.6%

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

      10. metadata-eval 31.6%

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

   4. Applied egg-rr 31.6%

      \[\leadsto \color{blue}{x - \frac{y - x}{-2}} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 31.6%

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

      2. div-inv 31.6%

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

      3. add-sqr-sqrt 25.3%

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

      4. fabs-sqr 25.3%

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

      5. add-sqr-sqrt 99.8%

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

      6. fabs-sub 99.8%

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

      7. add-sqr-sqrt 73.9%

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

      8. fabs-sqr 73.9%

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

      9. add-sqr-sqrt 78.7%

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

      10. metadata-eval 78.7%

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

   6. Applied egg-rr 78.7%

      \[\leadsto x - \color{blue}{1 \cdot \left(\left(x - y\right) \cdot -0.5\right)} \]
   7. Step-by-step derivation

      1. *-lft-identity 78.7%

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

   8. Simplified 78.7%

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

   if 7.00000000000000034e-213 < y

   1. Initial program 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. fma-define 99.9%

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

      3. add-sqr-sqrt 79.3%

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

      4. fabs-sqr 79.3%

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

      5. add-sqr-sqrt 83.4%

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

      6. frac-2neg 83.4%

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

      7. distribute-frac-neg 83.4%

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

      8. fmm-undef 83.4%

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

      9. *-un-lft-identity 83.4%

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

      10. metadata-eval 83.4%

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

   4. Applied egg-rr 83.4%

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

   5. Taylor expanded in x around 0 83.4%

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

      1. cancel-sign-sub-inv 83.4%

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

      2. metadata-eval 83.4%

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

      3. distribute-lft-out 83.4%

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

   7. Simplified 83.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(x + y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-213}:\\ \;\;\;\;x + -0.5 \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.1% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{-63}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 9.2e-63) (* x 0.5) (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 9.2e-63) {
		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 <= 9.2d-63) 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 <= 9.2e-63) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 9.2e-63:
		tmp = x * 0.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 9.2e-63)
		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 <= 9.2e-63)
		tmp = x * 0.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 9.2e-63], N[(x * 0.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.2e-63

   1. Initial program 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. fma-define 99.8%

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

      3. add-sqr-sqrt 30.9%

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

      4. fabs-sqr 30.9%

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

      5. add-sqr-sqrt 37.5%

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

      6. frac-2neg 37.5%

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

      7. distribute-frac-neg 37.5%

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

      8. fmm-undef 37.5%

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

      9. *-un-lft-identity 37.5%

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

      10. metadata-eval 37.5%

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

   4. Applied egg-rr 37.5%

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

   5. Taylor expanded in x around inf 35.1%

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

   if 9.2e-63 < y

   1. Initial program 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. fma-define 99.9%

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

      3. add-sqr-sqrt 87.0%

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

      4. fabs-sqr 87.0%

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

      5. add-sqr-sqrt 89.8%

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

      6. frac-2neg 89.8%

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

      7. distribute-frac-neg 89.8%

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

      8. fmm-undef 89.8%

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

      9. *-un-lft-identity 89.8%

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

      10. metadata-eval 89.8%

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

   4. Applied egg-rr 89.8%

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

   5. Taylor expanded in x around 0 78.4%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{-63}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 31.6% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. fma-define 99.9%

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

   3. add-sqr-sqrt 50.4%

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

   4. fabs-sqr 50.4%

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

   5. add-sqr-sqrt 55.7%

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

   6. frac-2neg 55.7%

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

   7. distribute-frac-neg 55.7%

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

   8. fmm-undef 55.7%

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

   9. *-un-lft-identity 55.7%

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

   10. metadata-eval 55.7%

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

4. Applied egg-rr 55.7%

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

5. Taylor expanded in x around inf 27.5%

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

6. Final simplification 27.5%

   \[\leadsto x \cdot 0.5 \]
7. Add Preprocessing

Alternative 13: 11.6% 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 11.0%

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

Reproduce

herbie shell --seed 2024172 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))