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

Percentage Accurate: 99.9% → 99.9%

Time: 8.5s

Alternatives: 9

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: 83.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.1 \cdot 10^{-61}:\\ \;\;\;\;x + -0.5 \cdot \left(y - x\right)\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{-231}:\\ \;\;\;\;x + \frac{\left|x\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5.1e-61)
   (+ x (* -0.5 (- y x)))
   (if (<= y 5.1e-231) (+ x (/ (fabs x) 2.0)) (* 0.5 (+ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5.1e-61) {
		tmp = x + (-0.5 * (y - x));
	} else if (y <= 5.1e-231) {
		tmp = x + (fabs(x) / 2.0);
	} 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 <= (-5.1d-61)) then
        tmp = x + ((-0.5d0) * (y - x))
    else if (y <= 5.1d-231) then
        tmp = x + (abs(x) / 2.0d0)
    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 <= -5.1e-61) {
		tmp = x + (-0.5 * (y - x));
	} else if (y <= 5.1e-231) {
		tmp = x + (Math.abs(x) / 2.0);
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5.1e-61:
		tmp = x + (-0.5 * (y - x))
	elif y <= 5.1e-231:
		tmp = x + (math.fabs(x) / 2.0)
	else:
		tmp = 0.5 * (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5.1e-61)
		tmp = Float64(x + Float64(-0.5 * Float64(y - x)));
	elseif (y <= 5.1e-231)
		tmp = Float64(x + Float64(abs(x) / 2.0));
	else
		tmp = Float64(0.5 * Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -5.1e-61)
		tmp = x + (-0.5 * (y - x));
	elseif (y <= 5.1e-231)
		tmp = x + (abs(x) / 2.0);
	else
		tmp = 0.5 * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5.1e-61], N[(x + N[(-0.5 * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.1e-231], 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 < -5.09999999999999968e-61

   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 10.6%

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

      4. fabs-sqr 10.6%

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

      5. add-sqr-sqrt 13.8%

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

      6. frac-2neg 13.8%

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

      7. distribute-frac-neg 13.8%

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

      8. fmm-undef 13.8%

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

      9. *-un-lft-identity 13.8%

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

      10. metadata-eval 13.8%

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

   4. Applied egg-rr 13.8%

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

   6. Applied egg-rr 91.2%

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

      1. *-commutative 91.2%

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

      2. +-commutative 91.2%

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

      3. sub-neg 91.2%

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

   8. Simplified 91.2%

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

   if -5.09999999999999968e-61 < y < 5.1e-231

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 87.6%

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

      1. neg-mul-1 87.6%

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

   5. Simplified 87.6%

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

   if 5.1e-231 < 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 82.3%

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

      4. fabs-sqr 82.3%

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

      5. add-sqr-sqrt 85.9%

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

      6. frac-2neg 85.9%

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

      7. distribute-frac-neg 85.9%

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

      8. fmm-undef 85.9%

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

      9. *-un-lft-identity 85.9%

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

      10. metadata-eval 85.9%

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

   4. Applied egg-rr 85.9%

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

   5. Taylor expanded in x around 0 85.9%

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

      1. cancel-sign-sub-inv 85.9%

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

      2. metadata-eval 85.9%

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

      3. distribute-lft-in 85.9%

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

   7. Simplified 85.9%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.1 \cdot 10^{-61}:\\ \;\;\;\;x + -0.5 \cdot \left(y - x\right)\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{-231}:\\ \;\;\;\;x + \frac{\left|x\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 58.4% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.06 \cdot 10^{-96}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-91}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.06e-96) (* x 0.5) (if (<= x 2.4e-91) (* y 0.5) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.06e-96) {
		tmp = x * 0.5;
	} else if (x <= 2.4e-91) {
		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 <= (-1.06d-96)) then
        tmp = x * 0.5d0
    else if (x <= 2.4d-91) 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 <= -1.06e-96) {
		tmp = x * 0.5;
	} else if (x <= 2.4e-91) {
		tmp = y * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.06e-96:
		tmp = x * 0.5
	elif x <= 2.4e-91:
		tmp = y * 0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.06e-96)
		tmp = Float64(x * 0.5);
	elseif (x <= 2.4e-91)
		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 <= -1.06e-96)
		tmp = x * 0.5;
	elseif (x <= 2.4e-91)
		tmp = y * 0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.06e-96], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, 2.4e-91], N[(y * 0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05999999999999998e-96

   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 77.3%

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

      4. fabs-sqr 77.3%

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

      5. add-sqr-sqrt 78.0%

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

      6. frac-2neg 78.0%

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

      7. distribute-frac-neg 78.0%

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

      8. fmm-undef 78.0%

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

      9. *-un-lft-identity 78.0%

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

      10. metadata-eval 78.0%

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

   4. Applied egg-rr 78.0%

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

   5. Taylor expanded in x around inf 58.9%

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

      1. *-commutative 58.9%

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

   7. Simplified 58.9%

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

   if -1.05999999999999998e-96 < x < 2.40000000000000011e-91

   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 51.5%

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

      4. fabs-sqr 51.5%

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

      5. add-sqr-sqrt 53.7%

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

      6. frac-2neg 53.7%

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

      7. distribute-frac-neg 53.7%

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

      8. fmm-undef 53.7%

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

      9. *-un-lft-identity 53.7%

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

      10. metadata-eval 53.7%

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

   4. Applied egg-rr 53.7%

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

   5. Taylor expanded in x around 0 47.3%

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

      1. *-commutative 47.3%

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

   7. Simplified 47.3%

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

   if 2.40000000000000011e-91 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 64.7%

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

      1. neg-mul-1 64.7%

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

   5. Simplified 64.7%

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

      1. add-sqr-sqrt 64.7%

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

      2. associate-/l* 64.7%

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

      3. fabs-neg 64.7%

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

      4. add-sqr-sqrt 64.7%

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

      5. fabs-sqr 64.7%

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

      6. add-sqr-sqrt 64.7%

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

      7. fabs-neg 64.7%

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

      8. add-sqr-sqrt 64.7%

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

      9. fabs-sqr 64.7%

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

      10. add-sqr-sqrt 64.7%

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

   7. Applied egg-rr 64.7%

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

      1. associate-*r/ 64.7%

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

      2. rem-square-sqrt 64.7%

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

   9. Simplified 64.7%

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

   10. Taylor expanded in x around 0 64.7%

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

       1. *-commutative 64.7%

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

   12. Simplified 64.7%

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

Alternative 4: 78.6% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.8 \cdot 10^{-233}:\\ \;\;\;\;x + -0.5 \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7.8e-233) (+ x (* -0.5 (- y x))) (* 0.5 (+ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7.8e-233) {
		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 <= 7.8d-233) 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 <= 7.8e-233) {
		tmp = x + (-0.5 * (y - x));
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7.8e-233:
		tmp = x + (-0.5 * (y - x))
	else:
		tmp = 0.5 * (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7.8e-233)
		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 <= 7.8e-233)
		tmp = x + (-0.5 * (y - x));
	else
		tmp = 0.5 * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7.8e-233], 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 < 7.8000000000000002e-233

   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 19.3%

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

      4. fabs-sqr 19.3%

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

      5. add-sqr-sqrt 25.2%

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

      6. frac-2neg 25.2%

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

      7. distribute-frac-neg 25.2%

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

      8. fmm-undef 25.2%

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

      9. *-un-lft-identity 25.2%

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

      10. metadata-eval 25.2%

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

   4. Applied egg-rr 25.2%

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

   6. Applied egg-rr 83.9%

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

      1. *-commutative 83.9%

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

      2. +-commutative 83.9%

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

      3. sub-neg 83.9%

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

   8. Simplified 83.9%

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

   if 7.8000000000000002e-233 < 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 82.3%

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

      4. fabs-sqr 82.3%

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

      5. add-sqr-sqrt 85.9%

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

      6. frac-2neg 85.9%

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

      7. distribute-frac-neg 85.9%

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

      8. fmm-undef 85.9%

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

      9. *-un-lft-identity 85.9%

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

      10. metadata-eval 85.9%

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

   4. Applied egg-rr 85.9%

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

   5. Taylor expanded in x around 0 85.9%

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

      1. cancel-sign-sub-inv 85.9%

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

      2. metadata-eval 85.9%

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

      3. distribute-lft-in 85.9%

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

   7. Simplified 85.9%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.8 \cdot 10^{-233}:\\ \;\;\;\;x + -0.5 \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.8% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-74}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{\frac{2}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.4e-74) (* 0.5 (+ x y)) (+ x (/ 1.0 (/ 2.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.4e-74) {
		tmp = 0.5 * (x + y);
	} else {
		tmp = x + (1.0 / (2.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 2.4d-74) then
        tmp = 0.5d0 * (x + y)
    else
        tmp = x + (1.0d0 / (2.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2.4e-74) {
		tmp = 0.5 * (x + y);
	} else {
		tmp = x + (1.0 / (2.0 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.4e-74:
		tmp = 0.5 * (x + y)
	else:
		tmp = x + (1.0 / (2.0 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.4e-74)
		tmp = Float64(0.5 * Float64(x + y));
	else
		tmp = Float64(x + Float64(1.0 / Float64(2.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.4e-74)
		tmp = 0.5 * (x + y);
	else
		tmp = x + (1.0 / (2.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.4e-74], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.3999999999999999e-74

   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 63.2%

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

      4. fabs-sqr 63.2%

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

      5. add-sqr-sqrt 64.8%

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

      6. frac-2neg 64.8%

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

      7. distribute-frac-neg 64.8%

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

      8. fmm-undef 64.8%

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

      9. *-un-lft-identity 64.8%

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

      10. metadata-eval 64.8%

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

   4. Applied egg-rr 64.8%

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

   5. Taylor expanded in x around 0 64.9%

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

      1. cancel-sign-sub-inv 64.9%

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

      2. metadata-eval 64.9%

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

      3. distribute-lft-in 64.9%

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

   7. Simplified 64.9%

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

   if 2.3999999999999999e-74 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 68.0%

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

      1. neg-mul-1 68.0%

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

   5. Simplified 68.0%

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

      1. clear-num 68.0%

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

      2. inv-pow 68.0%

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

      3. fabs-neg 68.0%

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

      4. add-sqr-sqrt 67.9%

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

      5. fabs-sqr 67.9%

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

      6. add-sqr-sqrt 68.0%

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

   7. Applied egg-rr 68.0%

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

      1. unpow-1 68.0%

         \[\leadsto x + \color{blue}{\frac{1}{\frac{2}{x}}} \]

   9. Simplified 68.0%

      \[\leadsto x + \color{blue}{\frac{1}{\frac{2}{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 66.8% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-75}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.8e-75) (* 0.5 (+ x y)) (* x 1.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.8e-75) {
		tmp = 0.5 * (x + y);
	} 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 <= 2.8d-75) then
        tmp = 0.5d0 * (x + y)
    else
        tmp = x * 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2.8e-75) {
		tmp = 0.5 * (x + y);
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.8e-75:
		tmp = 0.5 * (x + y)
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.8e-75)
		tmp = Float64(0.5 * Float64(x + y));
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.8e-75)
		tmp = 0.5 * (x + y);
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.8e-75], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.79999999999999998e-75

   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 63.2%

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

      4. fabs-sqr 63.2%

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

      5. add-sqr-sqrt 64.8%

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

      6. frac-2neg 64.8%

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

      7. distribute-frac-neg 64.8%

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

      8. fmm-undef 64.8%

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

      9. *-un-lft-identity 64.8%

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

      10. metadata-eval 64.8%

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

   4. Applied egg-rr 64.8%

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

   5. Taylor expanded in x around 0 64.9%

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

      1. cancel-sign-sub-inv 64.9%

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

      2. metadata-eval 64.9%

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

      3. distribute-lft-in 64.9%

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

   7. Simplified 64.9%

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

   if 2.79999999999999998e-75 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 68.0%

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

      1. neg-mul-1 68.0%

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

   5. Simplified 68.0%

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

      1. add-sqr-sqrt 67.9%

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

      2. associate-/l* 67.9%

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

      3. fabs-neg 67.9%

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

      4. add-sqr-sqrt 67.9%

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

      5. fabs-sqr 67.9%

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

      6. add-sqr-sqrt 67.9%

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

      7. fabs-neg 67.9%

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

      8. add-sqr-sqrt 67.9%

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

      9. fabs-sqr 67.9%

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

      10. add-sqr-sqrt 67.9%

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

   7. Applied egg-rr 67.9%

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

      1. associate-*r/ 67.9%

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

      2. rem-square-sqrt 68.0%

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

   9. Simplified 68.0%

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

   10. Taylor expanded in x around 0 68.0%

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

       1. *-commutative 68.0%

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

   12. Simplified 68.0%

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

Alternative 7: 50.6% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -1e-309) (* x 0.5) (* x 1.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1e-309) {
		tmp = x * 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 <= (-1d-309)) then
        tmp = x * 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 <= -1e-309) {
		tmp = x * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1e-309:
		tmp = x * 0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1e-309)
		tmp = Float64(x * 0.5);
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1e-309)
		tmp = x * 0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1e-309], N[(x * 0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.000000000000002e-309

   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 71.5%

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

      4. fabs-sqr 71.5%

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

      5. add-sqr-sqrt 72.3%

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

      6. frac-2neg 72.3%

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

      7. distribute-frac-neg 72.3%

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

      8. fmm-undef 72.3%

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

      9. *-un-lft-identity 72.3%

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

      10. metadata-eval 72.3%

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

   4. Applied egg-rr 72.3%

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

   5. Taylor expanded in x around inf 44.3%

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

      1. *-commutative 44.3%

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

   7. Simplified 44.3%

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

   if -1.000000000000002e-309 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 49.8%

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

      1. neg-mul-1 49.8%

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

   5. Simplified 49.8%

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

      1. add-sqr-sqrt 49.7%

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

      2. associate-/l* 49.7%

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

      3. fabs-neg 49.7%

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

      4. add-sqr-sqrt 49.7%

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

      5. fabs-sqr 49.7%

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

      6. add-sqr-sqrt 49.7%

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

      7. fabs-neg 49.7%

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

      8. add-sqr-sqrt 49.7%

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

      9. fabs-sqr 49.7%

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

      10. add-sqr-sqrt 49.7%

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

   7. Applied egg-rr 49.7%

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

      1. associate-*r/ 49.7%

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

      2. rem-square-sqrt 49.8%

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

   9. Simplified 49.8%

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

   10. Taylor expanded in x around 0 49.8%

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

       1. *-commutative 49.8%

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

   12. Simplified 49.8%

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

Alternative 8: 30.6% 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 48.1%

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

   4. fabs-sqr 48.1%

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

   5. add-sqr-sqrt 53.0%

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

   6. frac-2neg 53.0%

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

   7. distribute-frac-neg 53.0%

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

   8. fmm-undef 53.0%

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

   9. *-un-lft-identity 53.0%

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

   10. metadata-eval 53.0%

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

4. Applied egg-rr 53.0%

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

5. Taylor expanded in x around inf 27.3%

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

   1. *-commutative 27.3%

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

7. Simplified 27.3%

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

Alternative 9: 11.4% 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 10.7%

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

Reproduce

herbie shell --seed 2024131 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))