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

Percentage Accurate: 100.0% → 100.0%

Time: 5.9s

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. Add Preprocessing

Alternative 2: 86.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{1 - x}{y}\\ \mathbf{if}\;y \leq -3000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-139}:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{-115}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (- 1.0 x) y))))
   (if (<= y -3000.0)
     t_0
     (if (<= y 1.5e-139)
       (/ x (- 1.0 y))
       (if (<= y 5.7e-115) (- y) (if (<= y 1.0) (+ x (* x y)) t_0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + ((1.0 - x) / y);
	double tmp;
	if (y <= -3000.0) {
		tmp = t_0;
	} else if (y <= 1.5e-139) {
		tmp = x / (1.0 - y);
	} else if (y <= 5.7e-115) {
		tmp = -y;
	} else if (y <= 1.0) {
		tmp = x + (x * 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 + ((1.0d0 - x) / y)
    if (y <= (-3000.0d0)) then
        tmp = t_0
    else if (y <= 1.5d-139) then
        tmp = x / (1.0d0 - y)
    else if (y <= 5.7d-115) then
        tmp = -y
    else if (y <= 1.0d0) then
        tmp = x + (x * 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 + ((1.0 - x) / y);
	double tmp;
	if (y <= -3000.0) {
		tmp = t_0;
	} else if (y <= 1.5e-139) {
		tmp = x / (1.0 - y);
	} else if (y <= 5.7e-115) {
		tmp = -y;
	} else if (y <= 1.0) {
		tmp = x + (x * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + ((1.0 - x) / y)
	tmp = 0
	if y <= -3000.0:
		tmp = t_0
	elif y <= 1.5e-139:
		tmp = x / (1.0 - y)
	elif y <= 5.7e-115:
		tmp = -y
	elif y <= 1.0:
		tmp = x + (x * y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(Float64(1.0 - x) / y))
	tmp = 0.0
	if (y <= -3000.0)
		tmp = t_0;
	elseif (y <= 1.5e-139)
		tmp = Float64(x / Float64(1.0 - y));
	elseif (y <= 5.7e-115)
		tmp = Float64(-y);
	elseif (y <= 1.0)
		tmp = Float64(x + Float64(x * y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + ((1.0 - x) / y);
	tmp = 0.0;
	if (y <= -3000.0)
		tmp = t_0;
	elseif (y <= 1.5e-139)
		tmp = x / (1.0 - y);
	elseif (y <= 5.7e-115)
		tmp = -y;
	elseif (y <= 1.0)
		tmp = x + (x * y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3000.0], t$95$0, If[LessEqual[y, 1.5e-139], N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.7e-115], (-y), If[LessEqual[y, 1.0], N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3e3 or 1 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 98.7%

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

      1. +-commutative 98.7%

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

      2. mul-1-neg 98.7%

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

      3. sub-neg 98.7%

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

      4. div-sub 98.7%

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

   5. Simplified 98.7%

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

   if -3e3 < y < 1.5e-139

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 86.4%

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

   if 1.5e-139 < y < 5.7000000000000001e-115

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. distribute-neg-frac2 100.0%

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

      3. neg-sub0 100.0%

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

      4. associate--r- 100.0%

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

      5. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 100.0%

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

      1. neg-mul-1 100.0%

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

   8. Simplified 100.0%

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

   if 5.7000000000000001e-115 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 57.3%

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

   4. Taylor expanded in y around 0 57.3%

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

      1. *-commutative 57.3%

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

   6. Simplified 57.3%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3000:\\ \;\;\;\;1 + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-139}:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{-115}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1 - x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ \mathbf{if}\;y \leq -0.00024:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-138}:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-115}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))))
   (if (<= y -0.00024)
     t_0
     (if (<= y 4.5e-138)
       (/ x (- 1.0 y))
       (if (<= y 5.5e-115) (- y) (if (<= y 1.0) (+ x (* x y)) t_0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -0.00024) {
		tmp = t_0;
	} else if (y <= 4.5e-138) {
		tmp = x / (1.0 - y);
	} else if (y <= 5.5e-115) {
		tmp = -y;
	} else if (y <= 1.0) {
		tmp = x + (x * 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 <= (-0.00024d0)) then
        tmp = t_0
    else if (y <= 4.5d-138) then
        tmp = x / (1.0d0 - y)
    else if (y <= 5.5d-115) then
        tmp = -y
    else if (y <= 1.0d0) then
        tmp = x + (x * 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 <= -0.00024) {
		tmp = t_0;
	} else if (y <= 4.5e-138) {
		tmp = x / (1.0 - y);
	} else if (y <= 5.5e-115) {
		tmp = -y;
	} else if (y <= 1.0) {
		tmp = x + (x * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (x / y)
	tmp = 0
	if y <= -0.00024:
		tmp = t_0
	elif y <= 4.5e-138:
		tmp = x / (1.0 - y)
	elif y <= 5.5e-115:
		tmp = -y
	elif y <= 1.0:
		tmp = x + (x * 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 <= -0.00024)
		tmp = t_0;
	elseif (y <= 4.5e-138)
		tmp = Float64(x / Float64(1.0 - y));
	elseif (y <= 5.5e-115)
		tmp = Float64(-y);
	elseif (y <= 1.0)
		tmp = Float64(x + Float64(x * 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 <= -0.00024)
		tmp = t_0;
	elseif (y <= 4.5e-138)
		tmp = x / (1.0 - y);
	elseif (y <= 5.5e-115)
		tmp = -y;
	elseif (y <= 1.0)
		tmp = x + (x * 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, -0.00024], t$95$0, If[LessEqual[y, 4.5e-138], N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.5e-115], (-y), If[LessEqual[y, 1.0], N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.40000000000000006e-4 or 1 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 98.0%

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

      1. +-commutative 98.0%

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

      2. mul-1-neg 98.0%

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

      3. sub-neg 98.0%

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

      4. div-sub 98.0%

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

   5. Simplified 98.0%

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

   6. Taylor expanded in x around inf 97.2%

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

      1. neg-mul-1 97.2%

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

      2. distribute-neg-frac2 97.2%

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

   8. Simplified 97.2%

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

   9. Taylor expanded in x around 0 97.2%

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

       1. mul-1-neg 97.2%

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

       2. sub-neg 97.2%

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

   11. Simplified 97.2%

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

   if -2.40000000000000006e-4 < y < 4.50000000000000008e-138

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 87.2%

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

   if 4.50000000000000008e-138 < y < 5.50000000000000028e-115

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. distribute-neg-frac2 100.0%

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

      3. neg-sub0 100.0%

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

      4. associate--r- 100.0%

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

      5. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 100.0%

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

      1. neg-mul-1 100.0%

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

   8. Simplified 100.0%

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

   if 5.50000000000000028e-115 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 57.3%

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

   4. Taylor expanded in y around 0 57.3%

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

      1. *-commutative 57.3%

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

   6. Simplified 57.3%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00024:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-138}:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-115}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ t_1 := x + x \cdot y\\ \mathbf{if}\;y \leq -0.00024:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-138}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-114}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))) (t_1 (+ x (* x y))))
   (if (<= y -0.00024)
     t_0
     (if (<= y 5.5e-138)
       t_1
       (if (<= y 4.2e-114) (- y) (if (<= y 1.0) t_1 t_0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (x / y);
	double t_1 = x + (x * y);
	double tmp;
	if (y <= -0.00024) {
		tmp = t_0;
	} else if (y <= 5.5e-138) {
		tmp = t_1;
	} else if (y <= 4.2e-114) {
		tmp = -y;
	} else if (y <= 1.0) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (x / y)
    t_1 = x + (x * y)
    if (y <= (-0.00024d0)) then
        tmp = t_0
    else if (y <= 5.5d-138) then
        tmp = t_1
    else if (y <= 4.2d-114) then
        tmp = -y
    else if (y <= 1.0d0) then
        tmp = t_1
    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 t_1 = x + (x * y);
	double tmp;
	if (y <= -0.00024) {
		tmp = t_0;
	} else if (y <= 5.5e-138) {
		tmp = t_1;
	} else if (y <= 4.2e-114) {
		tmp = -y;
	} else if (y <= 1.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (x / y)
	t_1 = x + (x * y)
	tmp = 0
	if y <= -0.00024:
		tmp = t_0
	elif y <= 5.5e-138:
		tmp = t_1
	elif y <= 4.2e-114:
		tmp = -y
	elif y <= 1.0:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(x / y))
	t_1 = Float64(x + Float64(x * y))
	tmp = 0.0
	if (y <= -0.00024)
		tmp = t_0;
	elseif (y <= 5.5e-138)
		tmp = t_1;
	elseif (y <= 4.2e-114)
		tmp = Float64(-y);
	elseif (y <= 1.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (x / y);
	t_1 = x + (x * y);
	tmp = 0.0;
	if (y <= -0.00024)
		tmp = t_0;
	elseif (y <= 5.5e-138)
		tmp = t_1;
	elseif (y <= 4.2e-114)
		tmp = -y;
	elseif (y <= 1.0)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.00024], t$95$0, If[LessEqual[y, 5.5e-138], t$95$1, If[LessEqual[y, 4.2e-114], (-y), If[LessEqual[y, 1.0], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.40000000000000006e-4 or 1 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 98.0%

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

      1. +-commutative 98.0%

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

      2. mul-1-neg 98.0%

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

      3. sub-neg 98.0%

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

      4. div-sub 98.0%

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

   5. Simplified 98.0%

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

   6. Taylor expanded in x around inf 97.2%

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

      1. neg-mul-1 97.2%

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

      2. distribute-neg-frac2 97.2%

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

   8. Simplified 97.2%

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

   9. Taylor expanded in x around 0 97.2%

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

       1. mul-1-neg 97.2%

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

       2. sub-neg 97.2%

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

   11. Simplified 97.2%

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

   if -2.40000000000000006e-4 < y < 5.5000000000000003e-138 or 4.19999999999999985e-114 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 82.7%

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

   4. Taylor expanded in y around 0 82.4%

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

      1. *-commutative 82.4%

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

   6. Simplified 82.4%

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

   if 5.5000000000000003e-138 < y < 4.19999999999999985e-114

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. distribute-neg-frac2 100.0%

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

      3. neg-sub0 100.0%

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

      4. associate--r- 100.0%

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

      5. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 100.0%

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

      1. neg-mul-1 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00024:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-138}:\\ \;\;\;\;x + x \cdot y\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-114}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + x \cdot y\\ \mathbf{if}\;y \leq -0.00024:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-138}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-115}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (* x y))))
   (if (<= y -0.00024)
     1.0
     (if (<= y 5.5e-138)
       t_0
       (if (<= y 5.8e-115) (- y) (if (<= y 1.0) t_0 1.0))))))

C

double code(double x, double y) {
	double t_0 = x + (x * y);
	double tmp;
	if (y <= -0.00024) {
		tmp = 1.0;
	} else if (y <= 5.5e-138) {
		tmp = t_0;
	} else if (y <= 5.8e-115) {
		tmp = -y;
	} else if (y <= 1.0) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = x + (x * y)
    if (y <= (-0.00024d0)) then
        tmp = 1.0d0
    else if (y <= 5.5d-138) then
        tmp = t_0
    else if (y <= 5.8d-115) then
        tmp = -y
    else if (y <= 1.0d0) then
        tmp = t_0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x + (x * y);
	double tmp;
	if (y <= -0.00024) {
		tmp = 1.0;
	} else if (y <= 5.5e-138) {
		tmp = t_0;
	} else if (y <= 5.8e-115) {
		tmp = -y;
	} else if (y <= 1.0) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + (x * y)
	tmp = 0
	if y <= -0.00024:
		tmp = 1.0
	elif y <= 5.5e-138:
		tmp = t_0
	elif y <= 5.8e-115:
		tmp = -y
	elif y <= 1.0:
		tmp = t_0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(x * y))
	tmp = 0.0
	if (y <= -0.00024)
		tmp = 1.0;
	elseif (y <= 5.5e-138)
		tmp = t_0;
	elseif (y <= 5.8e-115)
		tmp = Float64(-y);
	elseif (y <= 1.0)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + (x * y);
	tmp = 0.0;
	if (y <= -0.00024)
		tmp = 1.0;
	elseif (y <= 5.5e-138)
		tmp = t_0;
	elseif (y <= 5.8e-115)
		tmp = -y;
	elseif (y <= 1.0)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.00024], 1.0, If[LessEqual[y, 5.5e-138], t$95$0, If[LessEqual[y, 5.8e-115], (-y), If[LessEqual[y, 1.0], t$95$0, 1.0]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.40000000000000006e-4 or 1 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 75.9%

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

   if -2.40000000000000006e-4 < y < 5.5000000000000003e-138 or 5.7999999999999996e-115 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 82.7%

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

   4. Taylor expanded in y around 0 82.4%

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

      1. *-commutative 82.4%

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

   6. Simplified 82.4%

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

   if 5.5000000000000003e-138 < y < 5.7999999999999996e-115

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. distribute-neg-frac2 100.0%

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

      3. neg-sub0 100.0%

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

      4. associate--r- 100.0%

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

      5. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 100.0%

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

      1. neg-mul-1 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00024:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-138}:\\ \;\;\;\;x + x \cdot y\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-115}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.00024:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-138}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-113}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -0.00024)
   1.0
   (if (<= y 5.5e-138) x (if (<= y 3.3e-113) (- y) (if (<= y 1.0) x 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -0.00024) {
		tmp = 1.0;
	} else if (y <= 5.5e-138) {
		tmp = x;
	} else if (y <= 3.3e-113) {
		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 <= (-0.00024d0)) then
        tmp = 1.0d0
    else if (y <= 5.5d-138) then
        tmp = x
    else if (y <= 3.3d-113) 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 <= -0.00024) {
		tmp = 1.0;
	} else if (y <= 5.5e-138) {
		tmp = x;
	} else if (y <= 3.3e-113) {
		tmp = -y;
	} else if (y <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -0.00024:
		tmp = 1.0
	elif y <= 5.5e-138:
		tmp = x
	elif y <= 3.3e-113:
		tmp = -y
	elif y <= 1.0:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -0.00024)
		tmp = 1.0;
	elseif (y <= 5.5e-138)
		tmp = x;
	elseif (y <= 3.3e-113)
		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 <= -0.00024)
		tmp = 1.0;
	elseif (y <= 5.5e-138)
		tmp = x;
	elseif (y <= 3.3e-113)
		tmp = -y;
	elseif (y <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -0.00024], 1.0, If[LessEqual[y, 5.5e-138], x, If[LessEqual[y, 3.3e-113], (-y), If[LessEqual[y, 1.0], x, 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.40000000000000006e-4 or 1 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 75.9%

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

   if -2.40000000000000006e-4 < y < 5.5000000000000003e-138 or 3.3000000000000002e-113 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 80.8%

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

   if 5.5000000000000003e-138 < y < 3.3000000000000002e-113

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. distribute-neg-frac2 100.0%

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

      3. neg-sub0 100.0%

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

      4. associate--r- 100.0%

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

      5. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 100.0%

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

      1. neg-mul-1 100.0%

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

   8. Simplified 100.0%

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

Alternative 7: 84.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ \mathbf{if}\;y \leq -0.00024:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-138}:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{elif}\;y \leq 4000000:\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))))
   (if (<= y -0.00024)
     t_0
     (if (<= y 5.5e-138)
       (/ x (- 1.0 y))
       (if (<= y 4000000.0) (/ y (+ y -1.0)) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -0.00024) {
		tmp = t_0;
	} else if (y <= 5.5e-138) {
		tmp = x / (1.0 - y);
	} else if (y <= 4000000.0) {
		tmp = y / (y + -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x / y)
    if (y <= (-0.00024d0)) then
        tmp = t_0
    else if (y <= 5.5d-138) then
        tmp = x / (1.0d0 - y)
    else if (y <= 4000000.0d0) then
        tmp = y / (y + (-1.0d0))
    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 <= -0.00024) {
		tmp = t_0;
	} else if (y <= 5.5e-138) {
		tmp = x / (1.0 - y);
	} else if (y <= 4000000.0) {
		tmp = y / (y + -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (x / y)
	tmp = 0
	if y <= -0.00024:
		tmp = t_0
	elif y <= 5.5e-138:
		tmp = x / (1.0 - y)
	elif y <= 4000000.0:
		tmp = y / (y + -1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(x / y))
	tmp = 0.0
	if (y <= -0.00024)
		tmp = t_0;
	elseif (y <= 5.5e-138)
		tmp = Float64(x / Float64(1.0 - y));
	elseif (y <= 4000000.0)
		tmp = Float64(y / Float64(y + -1.0));
	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 <= -0.00024)
		tmp = t_0;
	elseif (y <= 5.5e-138)
		tmp = x / (1.0 - y);
	elseif (y <= 4000000.0)
		tmp = y / (y + -1.0);
	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, -0.00024], t$95$0, If[LessEqual[y, 5.5e-138], N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4000000.0], N[(y / N[(y + -1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.40000000000000006e-4 or 4e6 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 98.0%

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

      1. +-commutative 98.0%

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

      2. mul-1-neg 98.0%

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

      3. sub-neg 98.0%

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

      4. div-sub 98.0%

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

   5. Simplified 98.0%

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

   6. Taylor expanded in x around inf 97.5%

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

      1. neg-mul-1 97.5%

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

      2. distribute-neg-frac2 97.5%

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

   8. Simplified 97.5%

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

   9. Taylor expanded in x around 0 97.5%

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

       1. mul-1-neg 97.5%

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

       2. sub-neg 97.5%

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

   11. Simplified 97.5%

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

   if -2.40000000000000006e-4 < y < 5.5000000000000003e-138

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 87.2%

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

   if 5.5000000000000003e-138 < y < 4e6

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 61.1%

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

      1. neg-mul-1 61.1%

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

      2. distribute-neg-frac2 61.1%

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

      3. neg-sub0 61.1%

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

      4. associate--r- 61.1%

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

      5. metadata-eval 61.1%

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

   5. Simplified 61.1%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00024:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-138}:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{elif}\;y \leq 4000000:\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = 1.0 + ((1.0 - x) / y);
	} else {
		tmp = x + (y * (x + -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)) .or. (.not. (y <= 1.0d0))) then
        tmp = 1.0d0 + ((1.0d0 - x) / y)
    else
        tmp = x + (y * (x + (-1.0d0)))
    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 + ((1.0 - x) / y);
	} else {
		tmp = x + (y * (x + -1.0));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 1.0)))
		tmp = 1.0 + ((1.0 - x) / y);
	else
		tmp = x + (y * (x + -1.0));
	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[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $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 98.7%

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

      1. +-commutative 98.7%

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

      2. mul-1-neg 98.7%

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

      3. sub-neg 98.7%

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

      4. div-sub 98.7%

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

   5. Simplified 98.7%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.2%

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

      1. mul-1-neg 99.2%

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

      2. unsub-neg 99.2%

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

      3. mul-1-neg 99.2%

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

      4. sub-neg 99.2%

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

   5. Simplified 99.2%

      \[\leadsto \color{blue}{x - y \cdot \left(1 - x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

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

Alternative 9: 73.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.00024:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -0.00024) 1.0 (if (<= y 1.0) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -0.00024) {
		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 <= (-0.00024d0)) 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 <= -0.00024) {
		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 <= -0.00024:
		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 <= -0.00024)
		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 <= -0.00024)
		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, -0.00024], 1.0, If[LessEqual[y, 1.0], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -2.40000000000000006e-4 or 1 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 75.9%

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

   if -2.40000000000000006e-4 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 77.2%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. 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 40.5%

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

Reproduce

herbie shell --seed 2024091 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, C"
  :precision binary64
  (/ (- x y) (- 1.0 y)))