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

Percentage Accurate: 100.0% → 100.0%

Time: 10.7s

Alternatives: 10

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. Final simplification 100.0%

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

Alternative 2: 86.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -680000:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq -1.7 \cdot 10^{-36}:\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1 - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -680000.0)
   (- 1.0 (/ x y))
   (if (<= y -1.7e-36)
     (/ y (+ y -1.0))
     (if (<= y 2.7e+14) (/ x (- 1.0 y)) (+ 1.0 (/ (- 1.0 x) y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -680000.0) {
		tmp = 1.0 - (x / y);
	} else if (y <= -1.7e-36) {
		tmp = y / (y + -1.0);
	} else if (y <= 2.7e+14) {
		tmp = x / (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 <= (-680000.0d0)) then
        tmp = 1.0d0 - (x / y)
    else if (y <= (-1.7d-36)) then
        tmp = y / (y + (-1.0d0))
    else if (y <= 2.7d+14) then
        tmp = x / (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 <= -680000.0) {
		tmp = 1.0 - (x / y);
	} else if (y <= -1.7e-36) {
		tmp = y / (y + -1.0);
	} else if (y <= 2.7e+14) {
		tmp = x / (1.0 - y);
	} else {
		tmp = 1.0 + ((1.0 - x) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -680000.0:
		tmp = 1.0 - (x / y)
	elif y <= -1.7e-36:
		tmp = y / (y + -1.0)
	elif y <= 2.7e+14:
		tmp = x / (1.0 - y)
	else:
		tmp = 1.0 + ((1.0 - x) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -680000.0)
		tmp = Float64(1.0 - Float64(x / y));
	elseif (y <= -1.7e-36)
		tmp = Float64(y / Float64(y + -1.0));
	elseif (y <= 2.7e+14)
		tmp = Float64(x / 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 <= -680000.0)
		tmp = 1.0 - (x / y);
	elseif (y <= -1.7e-36)
		tmp = y / (y + -1.0);
	elseif (y <= 2.7e+14)
		tmp = x / (1.0 - y);
	else
		tmp = 1.0 + ((1.0 - x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -680000.0], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.7e-36], N[(y / N[(y + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.7e+14], N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -6.8e5

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 83.2%

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

      2. pow2 83.2%

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

   4. Applied egg-rr 83.2%

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

   5. Taylor expanded in y around -inf 98.2%

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

      1. mul-1-neg 98.2%

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

      2. unsub-neg 98.2%

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

      3. sub-neg 98.2%

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

      4. metadata-eval 98.2%

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

   7. Simplified 98.2%

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

   8. Taylor expanded in x around inf 98.2%

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

   if -6.8e5 < y < -1.7000000000000001e-36

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 91.6%

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

      1. metadata-eval 91.6%

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

      2. times-frac 91.6%

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

      3. *-lft-identity 91.6%

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

      4. neg-mul-1 91.6%

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

      5. neg-sub0 91.6%

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

      6. associate--r- 91.6%

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

      7. metadata-eval 91.6%

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

   5. Simplified 91.6%

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

   if -1.7000000000000001e-36 < y < 2.7e14

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 82.6%

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

   if 2.7e14 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 100.0%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. div-sub 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -680000:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq -1.7 \cdot 10^{-36}:\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1 - x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+79}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -45:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq -1.32 \cdot 10^{-36}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5e+79)
   1.0
   (if (<= y -45.0)
     (/ (- x) y)
     (if (<= y -1.32e-36) (- y) (if (<= y 1.0) x 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5e+79) {
		tmp = 1.0;
	} else if (y <= -45.0) {
		tmp = -x / y;
	} else if (y <= -1.32e-36) {
		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 <= (-5d+79)) then
        tmp = 1.0d0
    else if (y <= (-45.0d0)) then
        tmp = -x / y
    else if (y <= (-1.32d-36)) 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 <= -5e+79) {
		tmp = 1.0;
	} else if (y <= -45.0) {
		tmp = -x / y;
	} else if (y <= -1.32e-36) {
		tmp = -y;
	} else if (y <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5e+79:
		tmp = 1.0
	elif y <= -45.0:
		tmp = -x / y
	elif y <= -1.32e-36:
		tmp = -y
	elif y <= 1.0:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5e+79)
		tmp = 1.0;
	elseif (y <= -45.0)
		tmp = Float64(Float64(-x) / y);
	elseif (y <= -1.32e-36)
		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 <= -5e+79)
		tmp = 1.0;
	elseif (y <= -45.0)
		tmp = -x / y;
	elseif (y <= -1.32e-36)
		tmp = -y;
	elseif (y <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5e+79], 1.0, If[LessEqual[y, -45.0], N[((-x) / y), $MachinePrecision], If[LessEqual[y, -1.32e-36], (-y), If[LessEqual[y, 1.0], x, 1.0]]]]

Derivation

1. Split input into 4 regimes

2. if y < -5e79 or 1 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 79.4%

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

   if -5e79 < y < -45

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 72.8%

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

   4. Taylor expanded in y around inf 66.9%

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

      1. associate-*r/ 66.9%

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

      2. neg-mul-1 66.9%

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

   6. Simplified 66.9%

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

   if -45 < y < -1.31999999999999993e-36

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 91.6%

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

      1. metadata-eval 91.6%

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

      2. times-frac 91.6%

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

      3. *-lft-identity 91.6%

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

      4. neg-mul-1 91.6%

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

      5. neg-sub0 91.6%

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

      6. associate--r- 91.6%

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

      7. metadata-eval 91.6%

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

   5. Simplified 91.6%

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

   6. Taylor expanded in y around 0 74.4%

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

      1. neg-mul-1 74.4%

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

   8. Simplified 74.4%

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

   if -1.31999999999999993e-36 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 82.2%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+79}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -45:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq -1.32 \cdot 10^{-36}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.65 \cdot 10^{+81}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -0.185:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq -1.02 \cdot 10^{-35}:\\ \;\;\;\;y \cdot \left(-1 - y\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.65e+81)
   1.0
   (if (<= y -0.185)
     (/ (- x) y)
     (if (<= y -1.02e-35) (* y (- -1.0 y)) (if (<= y 1.0) x 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.65e+81) {
		tmp = 1.0;
	} else if (y <= -0.185) {
		tmp = -x / y;
	} else if (y <= -1.02e-35) {
		tmp = y * (-1.0 - 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 <= (-2.65d+81)) then
        tmp = 1.0d0
    else if (y <= (-0.185d0)) then
        tmp = -x / y
    else if (y <= (-1.02d-35)) then
        tmp = y * ((-1.0d0) - 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 <= -2.65e+81) {
		tmp = 1.0;
	} else if (y <= -0.185) {
		tmp = -x / y;
	} else if (y <= -1.02e-35) {
		tmp = y * (-1.0 - y);
	} else if (y <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.65e+81:
		tmp = 1.0
	elif y <= -0.185:
		tmp = -x / y
	elif y <= -1.02e-35:
		tmp = y * (-1.0 - y)
	elif y <= 1.0:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.65e+81)
		tmp = 1.0;
	elseif (y <= -0.185)
		tmp = Float64(Float64(-x) / y);
	elseif (y <= -1.02e-35)
		tmp = Float64(y * Float64(-1.0 - 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 <= -2.65e+81)
		tmp = 1.0;
	elseif (y <= -0.185)
		tmp = -x / y;
	elseif (y <= -1.02e-35)
		tmp = y * (-1.0 - y);
	elseif (y <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.65e+81], 1.0, If[LessEqual[y, -0.185], N[((-x) / y), $MachinePrecision], If[LessEqual[y, -1.02e-35], N[(y * N[(-1.0 - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], x, 1.0]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.65000000000000014e81 or 1 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 79.4%

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

   if -2.65000000000000014e81 < y < -0.185

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 69.3%

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

   4. Taylor expanded in y around inf 63.7%

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

      1. associate-*r/ 63.7%

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

      2. neg-mul-1 63.7%

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

   6. Simplified 63.7%

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

   if -0.185 < y < -1.01999999999999995e-35

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 90.8%

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

      1. metadata-eval 90.8%

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

      2. times-frac 90.8%

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

      3. *-lft-identity 90.8%

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

      4. neg-mul-1 90.8%

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

      5. neg-sub0 90.8%

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

      6. associate--r- 90.8%

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

      7. metadata-eval 90.8%

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

   5. Simplified 90.8%

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

   6. Taylor expanded in y around 0 84.6%

      \[\leadsto \color{blue}{-1 \cdot y + -1 \cdot {y}^{2}} \]
   7. Step-by-step derivation

      1. mul-1-neg 84.6%

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

      2. unpow2 84.6%

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

      3. distribute-lft-neg-in 84.6%

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

      4. distribute-rgt-in 84.5%

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

      5. sub-neg 84.5%

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

   8. Simplified 84.5%

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

   if -1.01999999999999995e-35 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 82.2%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.65 \cdot 10^{+81}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -0.185:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq -1.02 \cdot 10^{-35}:\\ \;\;\;\;y \cdot \left(-1 - y\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ \mathbf{if}\;y \leq -0.105:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -1.75 \cdot 10^{-36}:\\ \;\;\;\;y \cdot \left(-1 - y\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))))
   (if (<= y -0.105)
     t_0
     (if (<= y -1.75e-36) (* y (- -1.0 y)) (if (<= y 1.0) x t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -0.105) {
		tmp = t_0;
	} else if (y <= -1.75e-36) {
		tmp = y * (-1.0 - y);
	} else if (y <= 1.0) {
		tmp = 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 = 1.0d0 - (x / y)
    if (y <= (-0.105d0)) then
        tmp = t_0
    else if (y <= (-1.75d-36)) then
        tmp = y * ((-1.0d0) - y)
    else if (y <= 1.0d0) then
        tmp = x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -0.105) {
		tmp = t_0;
	} else if (y <= -1.75e-36) {
		tmp = y * (-1.0 - y);
	} else if (y <= 1.0) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (x / y)
	tmp = 0
	if y <= -0.105:
		tmp = t_0
	elif y <= -1.75e-36:
		tmp = y * (-1.0 - y)
	elif y <= 1.0:
		tmp = x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(x / y))
	tmp = 0.0
	if (y <= -0.105)
		tmp = t_0;
	elseif (y <= -1.75e-36)
		tmp = Float64(y * Float64(-1.0 - y));
	elseif (y <= 1.0)
		tmp = x;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (x / y);
	tmp = 0.0;
	if (y <= -0.105)
		tmp = t_0;
	elseif (y <= -1.75e-36)
		tmp = y * (-1.0 - y);
	elseif (y <= 1.0)
		tmp = x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.105], t$95$0, If[LessEqual[y, -1.75e-36], N[(y * N[(-1.0 - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], x, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.104999999999999996 or 1 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 87.6%

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

      2. pow2 87.6%

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

   4. Applied egg-rr 87.6%

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

   5. Taylor expanded in y around -inf 98.2%

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

      1. mul-1-neg 98.2%

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

      2. unsub-neg 98.2%

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

      3. sub-neg 98.2%

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

      4. metadata-eval 98.2%

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

   7. Simplified 98.2%

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

   8. Taylor expanded in x around inf 98.1%

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

   if -0.104999999999999996 < y < -1.75e-36

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 90.8%

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

      1. metadata-eval 90.8%

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

      2. times-frac 90.8%

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

      3. *-lft-identity 90.8%

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

      4. neg-mul-1 90.8%

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

      5. neg-sub0 90.8%

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

      6. associate--r- 90.8%

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

      7. metadata-eval 90.8%

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

   5. Simplified 90.8%

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

   6. Taylor expanded in y around 0 84.6%

      \[\leadsto \color{blue}{-1 \cdot y + -1 \cdot {y}^{2}} \]
   7. Step-by-step derivation

      1. mul-1-neg 84.6%

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

      2. unpow2 84.6%

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

      3. distribute-lft-neg-in 84.6%

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

      4. distribute-rgt-in 84.5%

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

      5. sub-neg 84.5%

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

   8. Simplified 84.5%

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

   if -1.75e-36 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 82.2%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.105:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq -1.75 \cdot 10^{-36}:\\ \;\;\;\;y \cdot \left(-1 - y\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ \mathbf{if}\;y \leq -0.076:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -1.62 \cdot 10^{-35}:\\ \;\;\;\;y \cdot \left(-1 - y\right)\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))))
   (if (<= y -0.076)
     t_0
     (if (<= y -1.62e-35)
       (* y (- -1.0 y))
       (if (<= y 2.7e+14) (/ x (- 1.0 y)) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -0.076) {
		tmp = t_0;
	} else if (y <= -1.62e-35) {
		tmp = y * (-1.0 - y);
	} else if (y <= 2.7e+14) {
		tmp = x / (1.0 - y);
	} 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 = 1.0d0 - (x / y)
    if (y <= (-0.076d0)) then
        tmp = t_0
    else if (y <= (-1.62d-35)) then
        tmp = y * ((-1.0d0) - y)
    else if (y <= 2.7d+14) then
        tmp = x / (1.0d0 - y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -0.076) {
		tmp = t_0;
	} else if (y <= -1.62e-35) {
		tmp = y * (-1.0 - y);
	} else if (y <= 2.7e+14) {
		tmp = x / (1.0 - y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (x / y)
	tmp = 0
	if y <= -0.076:
		tmp = t_0
	elif y <= -1.62e-35:
		tmp = y * (-1.0 - y)
	elif y <= 2.7e+14:
		tmp = x / (1.0 - y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(x / y))
	tmp = 0.0
	if (y <= -0.076)
		tmp = t_0;
	elseif (y <= -1.62e-35)
		tmp = Float64(y * Float64(-1.0 - y));
	elseif (y <= 2.7e+14)
		tmp = Float64(x / Float64(1.0 - y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (x / y);
	tmp = 0.0;
	if (y <= -0.076)
		tmp = t_0;
	elseif (y <= -1.62e-35)
		tmp = y * (-1.0 - y);
	elseif (y <= 2.7e+14)
		tmp = x / (1.0 - y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.076], t$95$0, If[LessEqual[y, -1.62e-35], N[(y * N[(-1.0 - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.7e+14], N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0759999999999999981 or 2.7e14 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 89.1%

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

      2. pow2 89.1%

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

   4. Applied egg-rr 89.1%

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

   5. Taylor expanded in y around -inf 98.5%

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

      1. mul-1-neg 98.5%

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

      2. unsub-neg 98.5%

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

      3. sub-neg 98.5%

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

      4. metadata-eval 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in x around inf 98.4%

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

   if -0.0759999999999999981 < y < -1.62000000000000011e-35

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 90.8%

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

      1. metadata-eval 90.8%

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

      2. times-frac 90.8%

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

      3. *-lft-identity 90.8%

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

      4. neg-mul-1 90.8%

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

      5. neg-sub0 90.8%

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

      6. associate--r- 90.8%

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

      7. metadata-eval 90.8%

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

   5. Simplified 90.8%

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

   6. Taylor expanded in y around 0 84.6%

      \[\leadsto \color{blue}{-1 \cdot y + -1 \cdot {y}^{2}} \]
   7. Step-by-step derivation

      1. mul-1-neg 84.6%

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

      2. unpow2 84.6%

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

      3. distribute-lft-neg-in 84.6%

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

      4. distribute-rgt-in 84.5%

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

      5. sub-neg 84.5%

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

   8. Simplified 84.5%

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

   if -1.62000000000000011e-35 < y < 2.7e14

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 82.6%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.076:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq -1.62 \cdot 10^{-35}:\\ \;\;\;\;y \cdot \left(-1 - y\right)\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ \mathbf{if}\;y \leq -900000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -7.8 \cdot 10^{-36}:\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))))
   (if (<= y -900000.0)
     t_0
     (if (<= y -7.8e-36)
       (/ y (+ y -1.0))
       (if (<= y 2.7e+14) (/ x (- 1.0 y)) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -900000.0) {
		tmp = t_0;
	} else if (y <= -7.8e-36) {
		tmp = y / (y + -1.0);
	} else if (y <= 2.7e+14) {
		tmp = x / (1.0 - y);
	} 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 = 1.0d0 - (x / y)
    if (y <= (-900000.0d0)) then
        tmp = t_0
    else if (y <= (-7.8d-36)) then
        tmp = y / (y + (-1.0d0))
    else if (y <= 2.7d+14) then
        tmp = x / (1.0d0 - y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -900000.0) {
		tmp = t_0;
	} else if (y <= -7.8e-36) {
		tmp = y / (y + -1.0);
	} else if (y <= 2.7e+14) {
		tmp = x / (1.0 - y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (x / y)
	tmp = 0
	if y <= -900000.0:
		tmp = t_0
	elif y <= -7.8e-36:
		tmp = y / (y + -1.0)
	elif y <= 2.7e+14:
		tmp = x / (1.0 - y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(x / y))
	tmp = 0.0
	if (y <= -900000.0)
		tmp = t_0;
	elseif (y <= -7.8e-36)
		tmp = Float64(y / Float64(y + -1.0));
	elseif (y <= 2.7e+14)
		tmp = Float64(x / Float64(1.0 - y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (x / y);
	tmp = 0.0;
	if (y <= -900000.0)
		tmp = t_0;
	elseif (y <= -7.8e-36)
		tmp = y / (y + -1.0);
	elseif (y <= 2.7e+14)
		tmp = x / (1.0 - y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -900000.0], t$95$0, If[LessEqual[y, -7.8e-36], N[(y / N[(y + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.7e+14], N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9e5 or 2.7e14 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 89.0%

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

      2. pow2 89.0%

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

   4. Applied egg-rr 89.0%

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

   5. Taylor expanded in y around -inf 99.1%

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

      1. mul-1-neg 99.1%

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

      2. unsub-neg 99.1%

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

      3. sub-neg 99.1%

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

      4. metadata-eval 99.1%

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

   7. Simplified 99.1%

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

   8. Taylor expanded in x around inf 99.1%

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

   if -9e5 < y < -7.8000000000000001e-36

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 91.6%

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

      1. metadata-eval 91.6%

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

      2. times-frac 91.6%

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

      3. *-lft-identity 91.6%

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

      4. neg-mul-1 91.6%

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

      5. neg-sub0 91.6%

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

      6. associate--r- 91.6%

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

      7. metadata-eval 91.6%

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

   5. Simplified 91.6%

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

   if -7.8000000000000001e-36 < y < 2.7e14

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 82.6%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -900000:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq -7.8 \cdot 10^{-36}:\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -4.55 \cdot 10^{-41}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) 1.0 (if (<= y -4.55e-41) (- y) (if (<= y 1.0) x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= -4.55e-41) {
		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 <= (-1.0d0)) then
        tmp = 1.0d0
    else if (y <= (-4.55d-41)) 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 <= -1.0) {
		tmp = 1.0;
	} else if (y <= -4.55e-41) {
		tmp = -y;
	} else if (y <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = 1.0
	elif y <= -4.55e-41:
		tmp = -y
	elif y <= 1.0:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = 1.0;
	elseif (y <= -4.55e-41)
		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 <= -1.0)
		tmp = 1.0;
	elseif (y <= -4.55e-41)
		tmp = -y;
	elseif (y <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], 1.0, If[LessEqual[y, -4.55e-41], (-y), If[LessEqual[y, 1.0], x, 1.0]]]

Derivation

1. Split input into 3 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 71.8%

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

   if -1 < y < -4.55000000000000015e-41

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 90.8%

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

      1. metadata-eval 90.8%

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

      2. times-frac 90.8%

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

      3. *-lft-identity 90.8%

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

      4. neg-mul-1 90.8%

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

      5. neg-sub0 90.8%

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

      6. associate--r- 90.8%

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

      7. metadata-eval 90.8%

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

   5. Simplified 90.8%

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

   6. Taylor expanded in y around 0 79.7%

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

      1. neg-mul-1 79.7%

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

   8. Simplified 79.7%

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

   if -4.55000000000000015e-41 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 82.2%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -4.55 \cdot 10^{-41}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+15}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6.2e+15) 1.0 (if (<= y 1.0) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6.2e+15) {
		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 <= (-6.2d+15)) 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 <= -6.2e+15) {
		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 <= -6.2e+15:
		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 <= -6.2e+15)
		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 <= -6.2e+15)
		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, -6.2e+15], 1.0, If[LessEqual[y, 1.0], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -6.2e15 or 1 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 74.5%

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

   if -6.2e15 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 73.9%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+15}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 38.7% 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 36.3%

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

4. Final simplification 36.3%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024026 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, C"
  :precision binary64
  (/ (- x y) (- 1.0 y)))