Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, C

Percentage Accurate: 100.0% → 100.0%

Time: 5.3s

Alternatives: 11

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{x - y}{1 - y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (- x y) (- 1.0 y)))

C

double code(double x, double y) {
	return (x - y) / (1.0 - y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x - y) / (1.0d0 - y)
end function

Java

public static double code(double x, double y) {
	return (x - y) / (1.0 - y);
}

Python

def code(x, y):
	return (x - y) / (1.0 - y)

Julia

function code(x, y)
	return Float64(Float64(x - y) / Float64(1.0 - y))
end

MATLAB

function tmp = code(x, y)
	tmp = (x - y) / (1.0 - y);
end

Wolfram

code[x_, y_] := N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x - y}{1 - y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (- x y) (- 1.0 y)))

C

double code(double x, double y) {
	return (x - y) / (1.0 - y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x - y) / (1.0d0 - y)
end function

Java

public static double code(double x, double y) {
	return (x - y) / (1.0 - y);
}

Python

def code(x, y):
	return (x - y) / (1.0 - y)

Julia

function code(x, y)
	return Float64(Float64(x - y) / Float64(1.0 - y))
end

MATLAB

function tmp = code(x, y)
	tmp = (x - y) / (1.0 - y);
end

Wolfram

code[x_, y_] := N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x - y}{1 - y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (- x y) (- 1.0 y)))

C

double code(double x, double y) {
	return (x - y) / (1.0 - y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x - y) / (1.0d0 - y)
end function

Java

public static double code(double x, double y) {
	return (x - y) / (1.0 - y);
}

Python

def code(x, y):
	return (x - y) / (1.0 - y)

Julia

function code(x, y)
	return Float64(Float64(x - y) / Float64(1.0 - y))
end

MATLAB

function tmp = code(x, y)
	tmp = (x - y) / (1.0 - y);
end

Wolfram

code[x_, y_] := N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{x - y}{1 - y} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 85.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{+61}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{elif}\;y \leq 0.00038:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.6e+61)
   1.0
   (if (<= y -1.8e-7)
     (/ x (- 1.0 y))
     (if (<= y 0.00038) (- x y) (/ y (+ y -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.6e+61) {
		tmp = 1.0;
	} else if (y <= -1.8e-7) {
		tmp = x / (1.0 - y);
	} else if (y <= 0.00038) {
		tmp = x - y;
	} else {
		tmp = y / (y + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-4.6d+61)) then
        tmp = 1.0d0
    else if (y <= (-1.8d-7)) then
        tmp = x / (1.0d0 - y)
    else if (y <= 0.00038d0) then
        tmp = x - y
    else
        tmp = y / (y + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.6e+61) {
		tmp = 1.0;
	} else if (y <= -1.8e-7) {
		tmp = x / (1.0 - y);
	} else if (y <= 0.00038) {
		tmp = x - y;
	} else {
		tmp = y / (y + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.6e+61:
		tmp = 1.0
	elif y <= -1.8e-7:
		tmp = x / (1.0 - y)
	elif y <= 0.00038:
		tmp = x - y
	else:
		tmp = y / (y + -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.6e+61)
		tmp = 1.0;
	elseif (y <= -1.8e-7)
		tmp = Float64(x / Float64(1.0 - y));
	elseif (y <= 0.00038)
		tmp = Float64(x - y);
	else
		tmp = Float64(y / Float64(y + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.6e+61)
		tmp = 1.0;
	elseif (y <= -1.8e-7)
		tmp = x / (1.0 - y);
	elseif (y <= 0.00038)
		tmp = x - y;
	else
		tmp = y / (y + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.6e+61], 1.0, If[LessEqual[y, -1.8e-7], N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00038], N[(x - y), $MachinePrecision], N[(y / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -4.5999999999999999e61

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 74.8%

      \[\leadsto \color{blue}{1} \]

   if -4.5999999999999999e61 < y < -1.79999999999999997e-7

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 75.6%

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

   if -1.79999999999999997e-7 < y < 3.8000000000000002e-4

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. mul-1-neg 99.5%

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

      3. mul-1-neg 99.5%

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

      4. sub-neg 99.5%

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

      5. sub-neg 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0 98.7%

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

      1. mul-1-neg 98.7%

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

      2. distribute-rgt-neg-in 98.7%

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

      3. distribute-neg-in 98.7%

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

      4. metadata-eval 98.7%

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

      5. sub-neg 98.7%

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

   8. Simplified 98.7%

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

   9. Taylor expanded in y around 0 98.5%

      \[\leadsto x + \color{blue}{-1 \cdot y} \]
   10. Step-by-step derivation

       1. neg-mul-1 98.5%

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

   11. Simplified 98.5%

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

   if 3.8000000000000002e-4 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 81.1%

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

      1. neg-mul-1 81.1%

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

      2. distribute-neg-frac2 81.1%

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

      3. neg-sub0 81.1%

         \[\leadsto \frac{y}{\color{blue}{0 - \left(1 - y\right)}} \]

      4. associate--r- 81.1%

         \[\leadsto \frac{y}{\color{blue}{\left(0 - 1\right) + y}} \]

      5. metadata-eval 81.1%

         \[\leadsto \frac{y}{\color{blue}{-1} + y} \]

   5. Simplified 81.1%

      \[\leadsto \color{blue}{\frac{y}{-1 + y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{+61}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{elif}\;y \leq 0.00038:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+61}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7e+61)
   1.0
   (if (<= y -2.7e-6) (/ x (- 1.0 y)) (if (<= y 1.0) (- x y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7e+61) {
		tmp = 1.0;
	} else if (y <= -2.7e-6) {
		tmp = x / (1.0 - y);
	} else if (y <= 1.0) {
		tmp = x - y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-7d+61)) then
        tmp = 1.0d0
    else if (y <= (-2.7d-6)) then
        tmp = x / (1.0d0 - y)
    else if (y <= 1.0d0) then
        tmp = x - y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7e+61) {
		tmp = 1.0;
	} else if (y <= -2.7e-6) {
		tmp = x / (1.0 - y);
	} else if (y <= 1.0) {
		tmp = x - y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7e+61:
		tmp = 1.0
	elif y <= -2.7e-6:
		tmp = x / (1.0 - y)
	elif y <= 1.0:
		tmp = x - y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7e+61)
		tmp = 1.0;
	elseif (y <= -2.7e-6)
		tmp = Float64(x / Float64(1.0 - y));
	elseif (y <= 1.0)
		tmp = Float64(x - y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7e+61)
		tmp = 1.0;
	elseif (y <= -2.7e-6)
		tmp = x / (1.0 - y);
	elseif (y <= 1.0)
		tmp = x - y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7e+61], 1.0, If[LessEqual[y, -2.7e-6], N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(x - y), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.00000000000000036e61 or 1 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 77.1%

      \[\leadsto \color{blue}{1} \]

   if -7.00000000000000036e61 < y < -2.69999999999999998e-6

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 75.6%

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

   if -2.69999999999999998e-6 < y < 1

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. mul-1-neg 99.5%

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

      3. mul-1-neg 99.5%

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

      4. sub-neg 99.5%

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

      5. sub-neg 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0 98.7%

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

      1. mul-1-neg 98.7%

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

      2. distribute-rgt-neg-in 98.7%

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

      3. distribute-neg-in 98.7%

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

      4. metadata-eval 98.7%

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

      5. sub-neg 98.7%

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

   8. Simplified 98.7%

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

   9. Taylor expanded in y around 0 98.5%

      \[\leadsto x + \color{blue}{-1 \cdot y} \]
   10. Step-by-step derivation

       1. neg-mul-1 98.5%

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

   11. Simplified 98.5%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+61}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{+61}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -29.5:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.6e+61)
   1.0
   (if (<= y -29.5) (/ x (- y)) (if (<= y 1.0) (- x y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.6e+61) {
		tmp = 1.0;
	} else if (y <= -29.5) {
		tmp = x / -y;
	} else if (y <= 1.0) {
		tmp = x - y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-4.6d+61)) then
        tmp = 1.0d0
    else if (y <= (-29.5d0)) then
        tmp = x / -y
    else if (y <= 1.0d0) then
        tmp = x - y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.6e+61) {
		tmp = 1.0;
	} else if (y <= -29.5) {
		tmp = x / -y;
	} else if (y <= 1.0) {
		tmp = x - y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.6e+61:
		tmp = 1.0
	elif y <= -29.5:
		tmp = x / -y
	elif y <= 1.0:
		tmp = x - y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.6e+61)
		tmp = 1.0;
	elseif (y <= -29.5)
		tmp = Float64(x / Float64(-y));
	elseif (y <= 1.0)
		tmp = Float64(x - y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.6e+61)
		tmp = 1.0;
	elseif (y <= -29.5)
		tmp = x / -y;
	elseif (y <= 1.0)
		tmp = x - y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.6e+61], 1.0, If[LessEqual[y, -29.5], N[(x / (-y)), $MachinePrecision], If[LessEqual[y, 1.0], N[(x - y), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.5999999999999999e61 or 1 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 77.1%

      \[\leadsto \color{blue}{1} \]

   if -4.5999999999999999e61 < y < -29.5

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 75.6%

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

   4. Taylor expanded in y around inf 73.2%

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

      1. associate-*r/ 73.2%

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

      2. neg-mul-1 73.2%

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

   6. Simplified 73.2%

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

   if -29.5 < y < 1

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. mul-1-neg 99.5%

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

      3. mul-1-neg 99.5%

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

      4. sub-neg 99.5%

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

      5. sub-neg 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0 98.7%

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

      1. mul-1-neg 98.7%

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

      2. distribute-rgt-neg-in 98.7%

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

      3. distribute-neg-in 98.7%

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

      4. metadata-eval 98.7%

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

      5. sub-neg 98.7%

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

   8. Simplified 98.7%

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

   9. Taylor expanded in y around 0 98.5%

      \[\leadsto x + \color{blue}{-1 \cdot y} \]
   10. Step-by-step derivation

       1. neg-mul-1 98.5%

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

   11. Simplified 98.5%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{+61}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -29.5:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.84:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x + y \cdot \left(-1 - y\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1 - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -0.84)
   (- 1.0 (/ x y))
   (if (<= y 1.0) (+ x (* y (- -1.0 y))) (+ 1.0 (/ (- 1.0 x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -0.84) {
		tmp = 1.0 - (x / y);
	} else if (y <= 1.0) {
		tmp = x + (y * (-1.0 - y));
	} else {
		tmp = 1.0 + ((1.0 - 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 <= (-0.84d0)) then
        tmp = 1.0d0 - (x / y)
    else if (y <= 1.0d0) then
        tmp = x + (y * ((-1.0d0) - y))
    else
        tmp = 1.0d0 + ((1.0d0 - x) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -0.84) {
		tmp = 1.0 - (x / y);
	} else if (y <= 1.0) {
		tmp = x + (y * (-1.0 - y));
	} else {
		tmp = 1.0 + ((1.0 - x) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -0.84:
		tmp = 1.0 - (x / y)
	elif y <= 1.0:
		tmp = x + (y * (-1.0 - y))
	else:
		tmp = 1.0 + ((1.0 - x) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -0.84)
		tmp = Float64(1.0 - Float64(x / y));
	elseif (y <= 1.0)
		tmp = Float64(x + Float64(y * Float64(-1.0 - y)));
	else
		tmp = Float64(1.0 + Float64(Float64(1.0 - x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -0.84)
		tmp = 1.0 - (x / y);
	elseif (y <= 1.0)
		tmp = x + (y * (-1.0 - y));
	else
		tmp = 1.0 + ((1.0 - x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -0.84], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(x + N[(y * N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.839999999999999969

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 99.4%

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

      1. +-commutative 99.4%

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

      2. mul-1-neg 99.4%

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

      3. sub-neg 99.4%

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

      4. div-sub 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in x around inf 99.4%

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

      1. neg-mul-1 99.4%

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

      2. distribute-neg-frac2 99.4%

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

   8. Simplified 99.4%

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

   if -0.839999999999999969 < y < 1

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. mul-1-neg 99.5%

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

      3. mul-1-neg 99.5%

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

      4. sub-neg 99.5%

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

      5. sub-neg 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0 98.7%

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

      1. mul-1-neg 98.7%

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

      2. distribute-rgt-neg-in 98.7%

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

      3. distribute-neg-in 98.7%

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

      4. metadata-eval 98.7%

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

      5. sub-neg 98.7%

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

   8. Simplified 98.7%

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

   if 1 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 98.2%

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

      1. +-commutative 98.2%

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

      2. mul-1-neg 98.2%

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

      3. sub-neg 98.2%

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

      4. div-sub 98.2%

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

   5. Simplified 98.2%

      \[\leadsto \color{blue}{1 + \frac{1 - x}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.84:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x + y \cdot \left(-1 - y\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1 - x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1 - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   (- 1.0 (/ x y))
   (if (<= y 1.0) (- x y) (+ 1.0 (/ (- 1.0 x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0 - (x / y);
	} else if (y <= 1.0) {
		tmp = x - y;
	} else {
		tmp = 1.0 + ((1.0 - x) / y);
	}
	return tmp;
}

Fortran

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

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0 - (x / y);
	} else if (y <= 1.0) {
		tmp = x - y;
	} else {
		tmp = 1.0 + ((1.0 - x) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = 1.0 - (x / y)
	elif y <= 1.0:
		tmp = x - y
	else:
		tmp = 1.0 + ((1.0 - x) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(1.0 - Float64(x / y));
	elseif (y <= 1.0)
		tmp = Float64(x - y);
	else
		tmp = Float64(1.0 + Float64(Float64(1.0 - x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = 1.0 - (x / y);
	elseif (y <= 1.0)
		tmp = x - y;
	else
		tmp = 1.0 + ((1.0 - x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(x - y), $MachinePrecision], N[(1.0 + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 99.4%

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

      1. +-commutative 99.4%

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

      2. mul-1-neg 99.4%

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

      3. sub-neg 99.4%

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

      4. div-sub 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in x around inf 99.4%

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

      1. neg-mul-1 99.4%

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

      2. distribute-neg-frac2 99.4%

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

   8. Simplified 99.4%

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

   if -1 < y < 1

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. mul-1-neg 99.5%

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

      3. mul-1-neg 99.5%

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

      4. sub-neg 99.5%

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

      5. sub-neg 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0 98.7%

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

      1. mul-1-neg 98.7%

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

      2. distribute-rgt-neg-in 98.7%

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

      3. distribute-neg-in 98.7%

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

      4. metadata-eval 98.7%

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

      5. sub-neg 98.7%

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

   8. Simplified 98.7%

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

   9. Taylor expanded in y around 0 98.5%

      \[\leadsto x + \color{blue}{-1 \cdot y} \]
   10. Step-by-step derivation

       1. neg-mul-1 98.5%

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

   11. Simplified 98.5%

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

   if 1 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 98.2%

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

      1. +-commutative 98.2%

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

      2. mul-1-neg 98.2%

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

      3. sub-neg 98.2%

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

      4. div-sub 98.2%

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

   5. Simplified 98.2%

      \[\leadsto \color{blue}{1 + \frac{1 - x}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1 - x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+19}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -2.2 \cdot 10^{-101}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.1e+19) 1.0 (if (<= y -2.2e-101) (- y) (if (<= y 1.0) x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.1e+19) {
		tmp = 1.0;
	} else if (y <= -2.2e-101) {
		tmp = -y;
	} else if (y <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3.1d+19)) then
        tmp = 1.0d0
    else if (y <= (-2.2d-101)) then
        tmp = -y
    else if (y <= 1.0d0) then
        tmp = x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.1e+19) {
		tmp = 1.0;
	} else if (y <= -2.2e-101) {
		tmp = -y;
	} else if (y <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.1e+19:
		tmp = 1.0
	elif y <= -2.2e-101:
		tmp = -y
	elif y <= 1.0:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.1e+19)
		tmp = 1.0;
	elseif (y <= -2.2e-101)
		tmp = Float64(-y);
	elseif (y <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.1e+19)
		tmp = 1.0;
	elseif (y <= -2.2e-101)
		tmp = -y;
	elseif (y <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.1e+19], 1.0, If[LessEqual[y, -2.2e-101], (-y), If[LessEqual[y, 1.0], x, 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.1e19 or 1 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 74.3%

      \[\leadsto \color{blue}{1} \]

   if -3.1e19 < y < -2.1999999999999999e-101

   1. Initial program 99.9%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 58.4%

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

      1. neg-mul-1 58.4%

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

      2. distribute-neg-frac2 58.4%

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

      3. neg-sub0 58.4%

         \[\leadsto \frac{y}{\color{blue}{0 - \left(1 - y\right)}} \]

      4. associate--r- 58.4%

         \[\leadsto \frac{y}{\color{blue}{\left(0 - 1\right) + y}} \]

      5. metadata-eval 58.4%

         \[\leadsto \frac{y}{\color{blue}{-1} + y} \]

   5. Simplified 58.4%

      \[\leadsto \color{blue}{\frac{y}{-1 + y}} \]

   6. Taylor expanded in y around 0 57.7%

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

      1. neg-mul-1 57.7%

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

   8. Simplified 57.7%

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

   if -2.1999999999999999e-101 < y < 1

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 80.5%

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

Alternative 8: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.0))) (- 1.0 (/ x y)) (- x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x - y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = 1.0d0 - (x / y)
    else
        tmp = x - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 1.0):
		tmp = 1.0 - (x / y)
	else:
		tmp = x - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 1.0))
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = Float64(x - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 1.0)))
		tmp = 1.0 - (x / y);
	else
		tmp = x - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 98.8%

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

      1. +-commutative 98.8%

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

      2. mul-1-neg 98.8%

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

      3. sub-neg 98.8%

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

      4. div-sub 98.8%

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

   5. Simplified 98.8%

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

   6. Taylor expanded in x around inf 98.4%

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

      1. neg-mul-1 98.4%

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

      2. distribute-neg-frac2 98.4%

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

   8. Simplified 98.4%

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

   if -1 < y < 1

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. mul-1-neg 99.5%

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

      3. mul-1-neg 99.5%

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

      4. sub-neg 99.5%

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

      5. sub-neg 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0 98.7%

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

      1. mul-1-neg 98.7%

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

      2. distribute-rgt-neg-in 98.7%

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

      3. distribute-neg-in 98.7%

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

      4. metadata-eval 98.7%

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

      5. sub-neg 98.7%

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

   8. Simplified 98.7%

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

   9. Taylor expanded in y around 0 98.5%

      \[\leadsto x + \color{blue}{-1 \cdot y} \]
   10. Step-by-step derivation

       1. neg-mul-1 98.5%

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

   11. Simplified 98.5%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.15e+24) 1.0 (if (<= y 1.0) (- x y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.15e+24) {
		tmp = 1.0;
	} else if (y <= 1.0) {
		tmp = x - y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.15d+24)) then
        tmp = 1.0d0
    else if (y <= 1.0d0) then
        tmp = x - y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.15e+24) {
		tmp = 1.0;
	} else if (y <= 1.0) {
		tmp = x - y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.15e+24:
		tmp = 1.0
	elif y <= 1.0:
		tmp = x - y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.15e+24)
		tmp = 1.0;
	elseif (y <= 1.0)
		tmp = Float64(x - y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.15e+24)
		tmp = 1.0;
	elseif (y <= 1.0)
		tmp = x - y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.15e+24], 1.0, If[LessEqual[y, 1.0], N[(x - y), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1.15e24 or 1 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 75.0%

      \[\leadsto \color{blue}{1} \]

   if -1.15e24 < y < 1

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 96.4%

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

      1. mul-1-neg 96.4%

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

      2. mul-1-neg 96.4%

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

      3. mul-1-neg 96.4%

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

      4. sub-neg 96.4%

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

      5. sub-neg 96.4%

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

   5. Simplified 96.4%

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

   6. Taylor expanded in x around 0 95.7%

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

      1. mul-1-neg 95.7%

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

      2. distribute-rgt-neg-in 95.7%

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

      3. distribute-neg-in 95.7%

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

      4. metadata-eval 95.7%

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

      5. sub-neg 95.7%

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

   8. Simplified 95.7%

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

   9. Taylor expanded in y around 0 95.6%

      \[\leadsto x + \color{blue}{-1 \cdot y} \]
   10. Step-by-step derivation

       1. neg-mul-1 95.6%

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

   11. Simplified 95.6%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.15e+24) 1.0 (if (<= y 1.0) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.15e+24) {
		tmp = 1.0;
	} else if (y <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.15d+24)) then
        tmp = 1.0d0
    else if (y <= 1.0d0) then
        tmp = x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.15e+24) {
		tmp = 1.0;
	} else if (y <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.15e+24:
		tmp = 1.0
	elif y <= 1.0:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.15e+24)
		tmp = 1.0;
	elseif (y <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.15e+24)
		tmp = 1.0;
	elseif (y <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.15e+24], 1.0, If[LessEqual[y, 1.0], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1.15e24 or 1 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 75.0%

      \[\leadsto \color{blue}{1} \]

   if -1.15e24 < y < 1

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 70.6%

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

Alternative 11: 38.5% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

   \[\frac{x - y}{1 - y} \]
2. Add Preprocessing

3. Taylor expanded in y around inf 33.2%

   \[\leadsto \color{blue}{1} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024106 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, C"
  :precision binary64
  (/ (- x y) (- 1.0 y)))