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

Percentage Accurate: 99.9% → 99.9%

Time: 8.3s

Alternatives: 12

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: 84.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + y\right) \cdot 0.5\\ \mathbf{if}\;y \leq -1.05 \cdot 10^{-140}:\\ \;\;\;\;x - t\_0\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-168}:\\ \;\;\;\;x + \frac{\left|x\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (+ x y) 0.5)))
   (if (<= y -1.05e-140)
     (- x t_0)
     (if (<= y 3.5e-168) (+ x (/ (fabs x) 2.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = (x + y) * 0.5;
	double tmp;
	if (y <= -1.05e-140) {
		tmp = x - t_0;
	} else if (y <= 3.5e-168) {
		tmp = x + (fabs(x) / 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + y) * 0.5d0
    if (y <= (-1.05d-140)) then
        tmp = x - t_0
    else if (y <= 3.5d-168) then
        tmp = x + (abs(x) / 2.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x + y) * 0.5;
	double tmp;
	if (y <= -1.05e-140) {
		tmp = x - t_0;
	} else if (y <= 3.5e-168) {
		tmp = x + (Math.abs(x) / 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x + y) * 0.5
	tmp = 0
	if y <= -1.05e-140:
		tmp = x - t_0
	elif y <= 3.5e-168:
		tmp = x + (math.fabs(x) / 2.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x + y) * 0.5)
	tmp = 0.0
	if (y <= -1.05e-140)
		tmp = Float64(x - t_0);
	elseif (y <= 3.5e-168)
		tmp = Float64(x + Float64(abs(x) / 2.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x + y) * 0.5;
	tmp = 0.0;
	if (y <= -1.05e-140)
		tmp = x - t_0;
	elseif (y <= 3.5e-168)
		tmp = x + (abs(x) / 2.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[y, -1.05e-140], N[(x - t$95$0), $MachinePrecision], If[LessEqual[y, 3.5e-168], N[(x + N[(N[Abs[x], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.05000000000000009e-140

   1. Initial program 99.9%

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

      1. add-cube-cbrt 98.1%

         \[\leadsto x + \frac{\left|\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \sqrt[3]{y - x}}\right|}{2} \]

      2. fabs-mul 98.1%

         \[\leadsto x + \frac{\color{blue}{\left|\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{2} \]

      3. pow2 98.1%

         \[\leadsto x + \frac{\left|\color{blue}{{\left(\sqrt[3]{y - x}\right)}^{2}}\right| \cdot \left|\sqrt[3]{y - x}\right|}{2} \]

   4. Applied egg-rr 98.1%

      \[\leadsto x + \frac{\color{blue}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{2} \]

   6. Applied egg-rr 85.0%

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

   if -1.05000000000000009e-140 < y < 3.49999999999999982e-168

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 97.1%

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

      1. neg-mul-1 97.1%

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

   5. Simplified 97.1%

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

   if 3.49999999999999982e-168 < y

   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 85.1%

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

      4. fabs-sqr 85.1%

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

      5. add-sqr-sqrt 88.4%

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

      6. frac-2neg 88.4%

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

      7. distribute-frac-neg 88.4%

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

      8. fmm-undef 88.4%

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

      9. *-un-lft-identity 88.4%

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

      10. metadata-eval 88.4%

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

   4. Applied egg-rr 88.4%

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

   5. Taylor expanded in x around 0 88.4%

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

      1. cancel-sign-sub-inv 88.4%

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

      2. metadata-eval 88.4%

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

      3. distribute-lft-in 88.4%

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

      4. +-commutative 88.4%

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

   7. Simplified 88.4%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{-140}:\\ \;\;\;\;x - \left(x + y\right) \cdot 0.5\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-168}:\\ \;\;\;\;x + \frac{\left|x\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-107}:\\ \;\;\;\;x - \left(x + y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 11500000:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \left(x - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.8e-107)
   (- x (* (+ x y) 0.5))
   (if (<= x 11500000.0) (+ x (/ (fabs y) 2.0)) (+ x (* 0.5 (- x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.8e-107) {
		tmp = x - ((x + y) * 0.5);
	} else if (x <= 11500000.0) {
		tmp = x + (fabs(y) / 2.0);
	} else {
		tmp = x + (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 (x <= (-1.8d-107)) then
        tmp = x - ((x + y) * 0.5d0)
    else if (x <= 11500000.0d0) then
        tmp = x + (abs(y) / 2.0d0)
    else
        tmp = x + (0.5d0 * (x - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.8e-107) {
		tmp = x - ((x + y) * 0.5);
	} else if (x <= 11500000.0) {
		tmp = x + (Math.abs(y) / 2.0);
	} else {
		tmp = x + (0.5 * (x - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.8e-107:
		tmp = x - ((x + y) * 0.5)
	elif x <= 11500000.0:
		tmp = x + (math.fabs(y) / 2.0)
	else:
		tmp = x + (0.5 * (x - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.8e-107)
		tmp = Float64(x - Float64(Float64(x + y) * 0.5));
	elseif (x <= 11500000.0)
		tmp = Float64(x + Float64(abs(y) / 2.0));
	else
		tmp = Float64(x + Float64(0.5 * Float64(x - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.8e-107)
		tmp = x - ((x + y) * 0.5);
	elseif (x <= 11500000.0)
		tmp = x + (abs(y) / 2.0);
	else
		tmp = x + (0.5 * (x - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.8e-107], N[(x - N[(N[(x + y), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 11500000.0], N[(x + N[(N[Abs[y], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(x + N[(0.5 * N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.79999999999999988e-107

   1. Initial program 100.0%

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

      1. add-cube-cbrt 97.9%

         \[\leadsto x + \frac{\left|\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \sqrt[3]{y - x}}\right|}{2} \]

      2. fabs-mul 97.9%

         \[\leadsto x + \frac{\color{blue}{\left|\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{2} \]

      3. pow2 97.9%

         \[\leadsto x + \frac{\left|\color{blue}{{\left(\sqrt[3]{y - x}\right)}^{2}}\right| \cdot \left|\sqrt[3]{y - x}\right|}{2} \]

   4. Applied egg-rr 97.9%

      \[\leadsto x + \frac{\color{blue}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{2} \]

   6. Applied egg-rr 89.8%

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

   if -1.79999999999999988e-107 < x < 1.15e7

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 91.2%

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

   if 1.15e7 < x

   1. Initial program 99.8%

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

      1. add-cube-cbrt 98.9%

         \[\leadsto x + \frac{\left|\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \sqrt[3]{y - x}}\right|}{2} \]

      2. fabs-mul 98.9%

         \[\leadsto x + \frac{\color{blue}{\left|\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{2} \]

      3. pow2 98.9%

         \[\leadsto x + \frac{\left|\color{blue}{{\left(\sqrt[3]{y - x}\right)}^{2}}\right| \cdot \left|\sqrt[3]{y - x}\right|}{2} \]

   4. Applied egg-rr 98.9%

      \[\leadsto x + \frac{\color{blue}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{2} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 98.7%

         \[\leadsto x + \color{blue}{\sqrt{\frac{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}{2}} \cdot \sqrt{\frac{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}{2}}} \]

      2. sqrt-unprod 37.8%

         \[\leadsto x + \color{blue}{\sqrt{\frac{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}{2} \cdot \frac{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}{2}}} \]

      3. frac-2neg 37.8%

         \[\leadsto x + \sqrt{\color{blue}{\frac{-\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}{-2}} \cdot \frac{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}{2}} \]

      4. metadata-eval 37.8%

         \[\leadsto x + \sqrt{\frac{-\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}{\color{blue}{-2}} \cdot \frac{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}{2}} \]

      5. frac-2neg 37.8%

         \[\leadsto x + \sqrt{\frac{-\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}{-2} \cdot \color{blue}{\frac{-\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}{-2}}} \]

      6. metadata-eval 37.8%

         \[\leadsto x + \sqrt{\frac{-\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}{-2} \cdot \frac{-\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}{\color{blue}{-2}}} \]

      7. frac-times 37.8%

         \[\leadsto x + \sqrt{\color{blue}{\frac{\left(-\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|\right) \cdot \left(-\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|\right)}{-2 \cdot -2}}} \]

   6. Applied egg-rr 85.6%

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

   7. Taylor expanded in y around 0 85.6%

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

      1. mul-1-neg 85.6%

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

      2. sub-neg 85.6%

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

   9. Simplified 85.6%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-107}:\\ \;\;\;\;x - \left(x + y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 11500000:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \left(x - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.9% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-106}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-259}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-82}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-35}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.25e-106)
   (* x 0.5)
   (if (<= x 1.9e-259)
     (* y 0.5)
     (if (<= x 1.3e-82) (* y -0.5) (if (<= x 3.8e-35) (* y 0.5) (* x 1.5))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.25e-106) {
		tmp = x * 0.5;
	} else if (x <= 1.9e-259) {
		tmp = y * 0.5;
	} else if (x <= 1.3e-82) {
		tmp = y * -0.5;
	} else if (x <= 3.8e-35) {
		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 <= (-1.25d-106)) then
        tmp = x * 0.5d0
    else if (x <= 1.9d-259) then
        tmp = y * 0.5d0
    else if (x <= 1.3d-82) then
        tmp = y * (-0.5d0)
    else if (x <= 3.8d-35) 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 <= -1.25e-106) {
		tmp = x * 0.5;
	} else if (x <= 1.9e-259) {
		tmp = y * 0.5;
	} else if (x <= 1.3e-82) {
		tmp = y * -0.5;
	} else if (x <= 3.8e-35) {
		tmp = y * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.25e-106:
		tmp = x * 0.5
	elif x <= 1.9e-259:
		tmp = y * 0.5
	elif x <= 1.3e-82:
		tmp = y * -0.5
	elif x <= 3.8e-35:
		tmp = y * 0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.25e-106)
		tmp = Float64(x * 0.5);
	elseif (x <= 1.9e-259)
		tmp = Float64(y * 0.5);
	elseif (x <= 1.3e-82)
		tmp = Float64(y * -0.5);
	elseif (x <= 3.8e-35)
		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 <= -1.25e-106)
		tmp = x * 0.5;
	elseif (x <= 1.9e-259)
		tmp = y * 0.5;
	elseif (x <= 1.3e-82)
		tmp = y * -0.5;
	elseif (x <= 3.8e-35)
		tmp = y * 0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.25e-106], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, 1.9e-259], N[(y * 0.5), $MachinePrecision], If[LessEqual[x, 1.3e-82], N[(y * -0.5), $MachinePrecision], If[LessEqual[x, 3.8e-35], N[(y * 0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.24999999999999996e-106

   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 86.1%

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

      4. fabs-sqr 86.1%

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

      5. add-sqr-sqrt 86.9%

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

      6. frac-2neg 86.9%

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

      7. distribute-frac-neg 86.9%

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

      8. fmm-undef 86.9%

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

      9. *-un-lft-identity 86.9%

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

      10. metadata-eval 86.9%

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

   4. Applied egg-rr 86.9%

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

   5. Taylor expanded in x around inf 76.9%

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

   if -1.24999999999999996e-106 < x < 1.9e-259 or 1.3e-82 < x < 3.8000000000000001e-35

   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 66.2%

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

      4. fabs-sqr 66.2%

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

      5. add-sqr-sqrt 67.6%

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

      6. frac-2neg 67.6%

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

      7. distribute-frac-neg 67.6%

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

      8. fmm-undef 67.6%

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

      9. *-un-lft-identity 67.6%

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

      10. metadata-eval 67.6%

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

   4. Applied egg-rr 67.6%

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

   5. Taylor expanded in x around 0 59.9%

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

   if 1.9e-259 < x < 1.3e-82

   1. Initial program 100.0%

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

      1. add-cube-cbrt 97.8%

         \[\leadsto x + \frac{\left|\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \sqrt[3]{y - x}}\right|}{2} \]

      2. fabs-mul 97.8%

         \[\leadsto x + \frac{\color{blue}{\left|\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{2} \]

      3. pow2 97.8%

         \[\leadsto x + \frac{\left|\color{blue}{{\left(\sqrt[3]{y - x}\right)}^{2}}\right| \cdot \left|\sqrt[3]{y - x}\right|}{2} \]

   4. Applied egg-rr 97.8%

      \[\leadsto x + \frac{\color{blue}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{2} \]

   6. Applied egg-rr 64.0%

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

   7. Taylor expanded in x around 0 63.2%

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

      1. *-commutative 63.2%

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

   9. Simplified 63.2%

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

   if 3.8000000000000001e-35 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 79.6%

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

      1. neg-mul-1 79.6%

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

   5. Simplified 79.6%

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

      1. add-sqr-sqrt 79.5%

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

      2. associate-/l* 79.5%

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

      3. fabs-neg 79.5%

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

      4. add-sqr-sqrt 79.5%

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

      5. fabs-sqr 79.5%

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

      6. add-sqr-sqrt 79.5%

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

      7. fabs-neg 79.5%

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

      8. add-sqr-sqrt 79.5%

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

      9. fabs-sqr 79.5%

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

      10. add-sqr-sqrt 79.5%

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

   7. Applied egg-rr 79.5%

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

      1. associate-*r/ 79.5%

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

      2. rem-square-sqrt 79.6%

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

   9. Simplified 79.6%

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

   10. Taylor expanded in x around 0 79.6%

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

       1. *-commutative 79.6%

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

   12. Simplified 79.6%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-106}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-259}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-82}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-35}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.5% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + y\right) \cdot 0.5\\ \mathbf{if}\;y \leq 9.8 \cdot 10^{-225}:\\ \;\;\;\;x - t\_0\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-180}:\\ \;\;\;\;x + \frac{1}{\frac{2}{x}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (+ x y) 0.5)))
   (if (<= y 9.8e-225)
     (- x t_0)
     (if (<= y 7.5e-180) (+ x (/ 1.0 (/ 2.0 x))) t_0))))

C

double code(double x, double y) {
	double t_0 = (x + y) * 0.5;
	double tmp;
	if (y <= 9.8e-225) {
		tmp = x - t_0;
	} else if (y <= 7.5e-180) {
		tmp = x + (1.0 / (2.0 / x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + y) * 0.5d0
    if (y <= 9.8d-225) then
        tmp = x - t_0
    else if (y <= 7.5d-180) then
        tmp = x + (1.0d0 / (2.0d0 / x))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x + y) * 0.5;
	double tmp;
	if (y <= 9.8e-225) {
		tmp = x - t_0;
	} else if (y <= 7.5e-180) {
		tmp = x + (1.0 / (2.0 / x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x + y) * 0.5
	tmp = 0
	if y <= 9.8e-225:
		tmp = x - t_0
	elif y <= 7.5e-180:
		tmp = x + (1.0 / (2.0 / x))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x + y) * 0.5)
	tmp = 0.0
	if (y <= 9.8e-225)
		tmp = Float64(x - t_0);
	elseif (y <= 7.5e-180)
		tmp = Float64(x + Float64(1.0 / Float64(2.0 / x)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x + y) * 0.5;
	tmp = 0.0;
	if (y <= 9.8e-225)
		tmp = x - t_0;
	elseif (y <= 7.5e-180)
		tmp = x + (1.0 / (2.0 / x));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[y, 9.8e-225], N[(x - t$95$0), $MachinePrecision], If[LessEqual[y, 7.5e-180], N[(x + N[(1.0 / N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if y < 9.79999999999999942e-225

   1. Initial program 100.0%

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

      1. add-cube-cbrt 98.2%

         \[\leadsto x + \frac{\left|\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \sqrt[3]{y - x}}\right|}{2} \]

      2. fabs-mul 98.2%

         \[\leadsto x + \frac{\color{blue}{\left|\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{2} \]

      3. pow2 98.2%

         \[\leadsto x + \frac{\left|\color{blue}{{\left(\sqrt[3]{y - x}\right)}^{2}}\right| \cdot \left|\sqrt[3]{y - x}\right|}{2} \]

   4. Applied egg-rr 98.2%

      \[\leadsto x + \frac{\color{blue}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{2} \]

   6. Applied egg-rr 80.5%

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

   if 9.79999999999999942e-225 < y < 7.50000000000000015e-180

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 92.5%

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

      1. neg-mul-1 92.5%

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

   5. Simplified 92.5%

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

      1. clear-num 92.5%

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

      2. inv-pow 92.5%

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

      3. fabs-neg 92.5%

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

      4. add-sqr-sqrt 77.0%

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

      5. fabs-sqr 77.0%

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

      6. add-sqr-sqrt 80.0%

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

   7. Applied egg-rr 80.0%

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

      1. unpow-1 80.0%

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

   9. Simplified 80.0%

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

   if 7.50000000000000015e-180 < 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 84.5%

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

      4. fabs-sqr 84.5%

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

      5. add-sqr-sqrt 87.9%

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

      6. frac-2neg 87.9%

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

      7. distribute-frac-neg 87.9%

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

      8. fmm-undef 87.9%

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

      9. *-un-lft-identity 87.9%

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

      10. metadata-eval 87.9%

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

   4. Applied egg-rr 87.9%

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

   5. Taylor expanded in x around 0 87.9%

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

      1. cancel-sign-sub-inv 87.9%

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

      2. metadata-eval 87.9%

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

      3. distribute-lft-in 87.9%

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

      4. +-commutative 87.9%

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

   7. Simplified 87.9%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.8 \cdot 10^{-225}:\\ \;\;\;\;x - \left(x + y\right) \cdot 0.5\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-180}:\\ \;\;\;\;x + \frac{1}{\frac{2}{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.2% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-106}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-22}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7e-106) (* x 0.5) (if (<= x 3.7e-22) (* y 0.5) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7e-106) {
		tmp = x * 0.5;
	} else if (x <= 3.7e-22) {
		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 <= (-7d-106)) then
        tmp = x * 0.5d0
    else if (x <= 3.7d-22) 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 <= -7e-106) {
		tmp = x * 0.5;
	} else if (x <= 3.7e-22) {
		tmp = y * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7e-106:
		tmp = x * 0.5
	elif x <= 3.7e-22:
		tmp = y * 0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7e-106)
		tmp = Float64(x * 0.5);
	elseif (x <= 3.7e-22)
		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 <= -7e-106)
		tmp = x * 0.5;
	elseif (x <= 3.7e-22)
		tmp = y * 0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7e-106], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, 3.7e-22], N[(y * 0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7e-106

   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 86.1%

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

      4. fabs-sqr 86.1%

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

      5. add-sqr-sqrt 86.9%

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

      6. frac-2neg 86.9%

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

      7. distribute-frac-neg 86.9%

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

      8. fmm-undef 86.9%

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

      9. *-un-lft-identity 86.9%

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

      10. metadata-eval 86.9%

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

   4. Applied egg-rr 86.9%

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

   5. Taylor expanded in x around inf 76.9%

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

   if -7e-106 < x < 3.7e-22

   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 52.3%

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

      4. fabs-sqr 52.3%

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

      5. add-sqr-sqrt 54.1%

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

      6. frac-2neg 54.1%

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

      7. distribute-frac-neg 54.1%

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

      8. fmm-undef 54.1%

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

      9. *-un-lft-identity 54.1%

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

      10. metadata-eval 54.1%

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

   4. Applied egg-rr 54.1%

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

   5. Taylor expanded in x around 0 49.0%

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

   if 3.7e-22 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 79.6%

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

      1. neg-mul-1 79.6%

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

   5. Simplified 79.6%

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

      1. add-sqr-sqrt 79.5%

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

      2. associate-/l* 79.5%

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

      3. fabs-neg 79.5%

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

      4. add-sqr-sqrt 79.5%

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

      5. fabs-sqr 79.5%

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

      6. add-sqr-sqrt 79.5%

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

      7. fabs-neg 79.5%

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

      8. add-sqr-sqrt 79.5%

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

      9. fabs-sqr 79.5%

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

      10. add-sqr-sqrt 79.5%

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

   7. Applied egg-rr 79.5%

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

      1. associate-*r/ 79.5%

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

      2. rem-square-sqrt 79.6%

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

   9. Simplified 79.6%

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

   10. Taylor expanded in x around 0 79.6%

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

       1. *-commutative 79.6%

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

   12. Simplified 79.6%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-106}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-22}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.4% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.25 \cdot 10^{-106}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+178}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.25e-106) (* x 0.5) (if (<= x 1.35e+178) (* y 0.5) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.25e-106) {
		tmp = x * 0.5;
	} else if (x <= 1.35e+178) {
		tmp = y * 0.5;
	} else {
		tmp = 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 <= (-3.25d-106)) then
        tmp = x * 0.5d0
    else if (x <= 1.35d+178) then
        tmp = y * 0.5d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.25e-106) {
		tmp = x * 0.5;
	} else if (x <= 1.35e+178) {
		tmp = y * 0.5;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.25e-106:
		tmp = x * 0.5
	elif x <= 1.35e+178:
		tmp = y * 0.5
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.25e-106)
		tmp = Float64(x * 0.5);
	elseif (x <= 1.35e+178)
		tmp = Float64(y * 0.5);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.25e-106)
		tmp = x * 0.5;
	elseif (x <= 1.35e+178)
		tmp = y * 0.5;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.25e-106], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, 1.35e+178], N[(y * 0.5), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if x < -3.2499999999999998e-106

   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 86.1%

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

      4. fabs-sqr 86.1%

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

      5. add-sqr-sqrt 86.9%

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

      6. frac-2neg 86.9%

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

      7. distribute-frac-neg 86.9%

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

      8. fmm-undef 86.9%

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

      9. *-un-lft-identity 86.9%

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

      10. metadata-eval 86.9%

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

   4. Applied egg-rr 86.9%

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

   5. Taylor expanded in x around inf 76.9%

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

   if -3.2499999999999998e-106 < x < 1.35000000000000009e178

   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 44.1%

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

      4. fabs-sqr 44.1%

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

      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. Taylor expanded in x around 0 41.9%

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

   if 1.35000000000000009e178 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 19.5%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.25 \cdot 10^{-106}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+178}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.5% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{-141}:\\ \;\;\;\;x + 0.5 \cdot \left(x - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.1e-141) (+ x (* 0.5 (- x y))) (* (+ x y) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.1e-141) {
		tmp = x + (0.5 * (x - y));
	} else {
		tmp = (x + 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-141)) then
        tmp = x + (0.5d0 * (x - y))
    else
        tmp = (x + y) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.1e-141) {
		tmp = x + (0.5 * (x - y));
	} else {
		tmp = (x + y) * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.1e-141:
		tmp = x + (0.5 * (x - y))
	else:
		tmp = (x + y) * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.1e-141)
		tmp = Float64(x + Float64(0.5 * Float64(x - y)));
	else
		tmp = Float64(Float64(x + y) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.1e-141)
		tmp = x + (0.5 * (x - y));
	else
		tmp = (x + y) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.1e-141], N[(x + N[(0.5 * N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + y), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.10000000000000027e-141

   1. Initial program 99.9%

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

      1. add-cube-cbrt 98.1%

         \[\leadsto x + \frac{\left|\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \sqrt[3]{y - x}}\right|}{2} \]

      2. fabs-mul 98.1%

         \[\leadsto x + \frac{\color{blue}{\left|\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{2} \]

      3. pow2 98.1%

         \[\leadsto x + \frac{\left|\color{blue}{{\left(\sqrt[3]{y - x}\right)}^{2}}\right| \cdot \left|\sqrt[3]{y - x}\right|}{2} \]

   4. Applied egg-rr 98.1%

      \[\leadsto x + \frac{\color{blue}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{2} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 98.0%

         \[\leadsto x + \color{blue}{\sqrt{\frac{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}{2}} \cdot \sqrt{\frac{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}{2}}} \]

      2. sqrt-unprod 53.6%

         \[\leadsto x + \color{blue}{\sqrt{\frac{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}{2} \cdot \frac{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}{2}}} \]

      3. frac-2neg 53.6%

         \[\leadsto x + \sqrt{\color{blue}{\frac{-\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}{-2}} \cdot \frac{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}{2}} \]

      4. metadata-eval 53.6%

         \[\leadsto x + \sqrt{\frac{-\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}{\color{blue}{-2}} \cdot \frac{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}{2}} \]

      5. frac-2neg 53.6%

         \[\leadsto x + \sqrt{\frac{-\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}{-2} \cdot \color{blue}{\frac{-\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}{-2}}} \]

      6. metadata-eval 53.6%

         \[\leadsto x + \sqrt{\frac{-\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}{-2} \cdot \frac{-\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}{\color{blue}{-2}}} \]

      7. frac-times 53.6%

         \[\leadsto x + \sqrt{\color{blue}{\frac{\left(-\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|\right) \cdot \left(-\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|\right)}{-2 \cdot -2}}} \]

   6. Applied egg-rr 81.7%

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

   7. Taylor expanded in y around 0 81.7%

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

      1. mul-1-neg 81.7%

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

      2. sub-neg 81.7%

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

   9. Simplified 81.7%

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

   if -3.10000000000000027e-141 < 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 74.1%

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

      4. fabs-sqr 74.1%

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

      5. add-sqr-sqrt 79.1%

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

      6. frac-2neg 79.1%

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

      7. distribute-frac-neg 79.1%

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

      8. fmm-undef 79.1%

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

      9. *-un-lft-identity 79.1%

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

      10. metadata-eval 79.1%

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

   4. Applied egg-rr 79.1%

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

   5. Taylor expanded in x around 0 79.1%

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

      1. cancel-sign-sub-inv 79.1%

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

      2. metadata-eval 79.1%

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

      3. distribute-lft-in 79.1%

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

      4. +-commutative 79.1%

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

   7. Simplified 79.1%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{-141}:\\ \;\;\;\;x + 0.5 \cdot \left(x - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.0% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.8 \cdot 10^{-104}:\\ \;\;\;\;x - y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9.8e-104) (- x (* y 0.5)) (* (+ x y) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9.8e-104) {
		tmp = x - (y * 0.5);
	} else {
		tmp = (x + 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 <= (-9.8d-104)) then
        tmp = x - (y * 0.5d0)
    else
        tmp = (x + y) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -9.8e-104) {
		tmp = x - (y * 0.5);
	} else {
		tmp = (x + y) * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -9.8e-104:
		tmp = x - (y * 0.5)
	else:
		tmp = (x + y) * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9.8e-104)
		tmp = Float64(x - Float64(y * 0.5));
	else
		tmp = Float64(Float64(x + y) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -9.8e-104)
		tmp = x - (y * 0.5);
	else
		tmp = (x + y) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9.8e-104], N[(x - N[(y * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(x + y), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.8000000000000006e-104

   1. Initial program 100.0%

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

      1. add-cube-cbrt 98.2%

         \[\leadsto x + \frac{\left|\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \sqrt[3]{y - x}}\right|}{2} \]

      2. fabs-mul 98.2%

         \[\leadsto x + \frac{\color{blue}{\left|\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{2} \]

      3. pow2 98.2%

         \[\leadsto x + \frac{\left|\color{blue}{{\left(\sqrt[3]{y - x}\right)}^{2}}\right| \cdot \left|\sqrt[3]{y - x}\right|}{2} \]

   4. Applied egg-rr 98.2%

      \[\leadsto x + \frac{\color{blue}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{2} \]

   6. Applied egg-rr 85.5%

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

   7. Taylor expanded in y around inf 73.5%

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

   if -9.8000000000000006e-104 < 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 72.2%

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

      4. fabs-sqr 72.2%

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

      5. add-sqr-sqrt 77.2%

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

      6. frac-2neg 77.2%

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

      7. distribute-frac-neg 77.2%

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

      8. fmm-undef 77.2%

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

      9. *-un-lft-identity 77.2%

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

      10. metadata-eval 77.2%

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

   4. Applied egg-rr 77.2%

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

   5. Taylor expanded in x around 0 77.2%

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

      1. cancel-sign-sub-inv 77.2%

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

      2. metadata-eval 77.2%

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

      3. distribute-lft-in 77.2%

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

      4. +-commutative 77.2%

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

   7. Simplified 77.2%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.8 \cdot 10^{-104}:\\ \;\;\;\;x - y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.1% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-47}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.5e-47) (* y -0.5) (* (+ x y) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.5e-47) {
		tmp = y * -0.5;
	} else {
		tmp = (x + 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 <= (-2.5d-47)) then
        tmp = y * (-0.5d0)
    else
        tmp = (x + y) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.5e-47) {
		tmp = y * -0.5;
	} else {
		tmp = (x + y) * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.5e-47:
		tmp = y * -0.5
	else:
		tmp = (x + y) * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.5e-47)
		tmp = Float64(y * -0.5);
	else
		tmp = Float64(Float64(x + y) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.5e-47)
		tmp = y * -0.5;
	else
		tmp = (x + y) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.5e-47], N[(y * -0.5), $MachinePrecision], N[(N[(x + y), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.50000000000000006e-47

   1. Initial program 100.0%

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

      1. add-cube-cbrt 98.3%

         \[\leadsto x + \frac{\left|\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \sqrt[3]{y - x}}\right|}{2} \]

      2. fabs-mul 98.3%

         \[\leadsto x + \frac{\color{blue}{\left|\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{2} \]

      3. pow2 98.3%

         \[\leadsto x + \frac{\left|\color{blue}{{\left(\sqrt[3]{y - x}\right)}^{2}}\right| \cdot \left|\sqrt[3]{y - x}\right|}{2} \]

   4. Applied egg-rr 98.3%

      \[\leadsto x + \frac{\color{blue}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{2} \]

   6. Applied egg-rr 84.6%

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

   7. Taylor expanded in x around 0 72.6%

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

      1. *-commutative 72.6%

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

   9. Simplified 72.6%

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

   if -2.50000000000000006e-47 < 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 70.4%

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

      4. fabs-sqr 70.4%

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

      5. add-sqr-sqrt 75.3%

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

      6. frac-2neg 75.3%

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

      7. distribute-frac-neg 75.3%

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

      8. fmm-undef 75.3%

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

      9. *-un-lft-identity 75.3%

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

      10. metadata-eval 75.3%

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

   4. Applied egg-rr 75.3%

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

   5. Taylor expanded in x around 0 75.3%

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

      1. cancel-sign-sub-inv 75.3%

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

      2. metadata-eval 75.3%

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

      3. distribute-lft-in 75.3%

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

      4. +-commutative 75.3%

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

   7. Simplified 75.3%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-47}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 30.4% 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 55.8%

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

   4. fabs-sqr 55.8%

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

   5. add-sqr-sqrt 60.4%

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

   6. frac-2neg 60.4%

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

   7. distribute-frac-neg 60.4%

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

   8. fmm-undef 60.4%

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

   9. *-un-lft-identity 60.4%

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

   10. metadata-eval 60.4%

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

4. Applied egg-rr 60.4%

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

5. Taylor expanded in x around inf 36.7%

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

6. Final simplification 36.7%

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

Alternative 12: 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 11.9%

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

Reproduce

herbie shell --seed 2024170 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))