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

Percentage Accurate: 65.8% → 99.9%

Time: 9.7s

Alternatives: 12

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ 1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -235000 \lor \neg \left(y \leq 280000\right):\\ \;\;\;\;\left(x - \frac{x + -1}{y}\right) + \frac{x + -1}{{y}^{2}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -235000.0) (not (<= y 280000.0)))
   (+ (- x (/ (+ x -1.0) y)) (/ (+ x -1.0) (pow y 2.0)))
   (+ 1.0 (/ (* y (+ x -1.0)) (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -235000.0) || !(y <= 280000.0)) {
		tmp = (x - ((x + -1.0) / y)) + ((x + -1.0) / pow(y, 2.0));
	} else {
		tmp = 1.0 + ((y * (x + -1.0)) / (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 <= (-235000.0d0)) .or. (.not. (y <= 280000.0d0))) then
        tmp = (x - ((x + (-1.0d0)) / y)) + ((x + (-1.0d0)) / (y ** 2.0d0))
    else
        tmp = 1.0d0 + ((y * (x + (-1.0d0))) / (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -235000.0) || !(y <= 280000.0)) {
		tmp = (x - ((x + -1.0) / y)) + ((x + -1.0) / Math.pow(y, 2.0));
	} else {
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -235000.0) or not (y <= 280000.0):
		tmp = (x - ((x + -1.0) / y)) + ((x + -1.0) / math.pow(y, 2.0))
	else:
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -235000.0) || !(y <= 280000.0))
		tmp = Float64(Float64(x - Float64(Float64(x + -1.0) / y)) + Float64(Float64(x + -1.0) / (y ^ 2.0)));
	else
		tmp = Float64(1.0 + Float64(Float64(y * Float64(x + -1.0)) / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -235000.0) || ~((y <= 280000.0)))
		tmp = (x - ((x + -1.0) / y)) + ((x + -1.0) / (y ^ 2.0));
	else
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -235000.0], N[Not[LessEqual[y, 280000.0]], $MachinePrecision]], N[(N[(x - N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] + N[(N[(x + -1.0), $MachinePrecision] / N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -235000 or 2.8e5 < y

   1. Initial program 30.7%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf 100.0%

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

      1. associate-+r+ 100.0%

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

      2. associate--l+ 100.0%

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

      3. mul-1-neg 100.0%

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

      4. unsub-neg 100.0%

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

      5. sub-neg 100.0%

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

      6. metadata-eval 100.0%

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

      7. div-sub 100.0%

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

      8. sub-neg 100.0%

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

      9. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   if -235000 < y < 2.8e5

   1. Initial program 99.9%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -235000 \lor \neg \left(y \leq 280000\right):\\ \;\;\;\;\left(x - \frac{x + -1}{y}\right) + \frac{x + -1}{{y}^{2}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{-1}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-159}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-107}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.16:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ -1.0 y))))
   (if (<= y -1.0)
     t_0
     (if (<= y 4.6e-159)
       (- 1.0 y)
       (if (<= y 1.7e-107) (* y x) (if (<= y 0.16) (- 1.0 y) t_0))))))

C

double code(double x, double y) {
	double t_0 = x - (-1.0 / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 4.6e-159) {
		tmp = 1.0 - y;
	} else if (y <= 1.7e-107) {
		tmp = y * x;
	} else if (y <= 0.16) {
		tmp = 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 = x - ((-1.0d0) / y)
    if (y <= (-1.0d0)) then
        tmp = t_0
    else if (y <= 4.6d-159) then
        tmp = 1.0d0 - y
    else if (y <= 1.7d-107) then
        tmp = y * x
    else if (y <= 0.16d0) then
        tmp = 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 = x - (-1.0 / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 4.6e-159) {
		tmp = 1.0 - y;
	} else if (y <= 1.7e-107) {
		tmp = y * x;
	} else if (y <= 0.16) {
		tmp = 1.0 - y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x - (-1.0 / y)
	tmp = 0
	if y <= -1.0:
		tmp = t_0
	elif y <= 4.6e-159:
		tmp = 1.0 - y
	elif y <= 1.7e-107:
		tmp = y * x
	elif y <= 0.16:
		tmp = 1.0 - y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x - Float64(-1.0 / y))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 4.6e-159)
		tmp = Float64(1.0 - y);
	elseif (y <= 1.7e-107)
		tmp = Float64(y * x);
	elseif (y <= 0.16)
		tmp = Float64(1.0 - y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x - (-1.0 / y);
	tmp = 0.0;
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 4.6e-159)
		tmp = 1.0 - y;
	elseif (y <= 1.7e-107)
		tmp = y * x;
	elseif (y <= 0.16)
		tmp = 1.0 - y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 4.6e-159], N[(1.0 - y), $MachinePrecision], If[LessEqual[y, 1.7e-107], N[(y * x), $MachinePrecision], If[LessEqual[y, 0.16], N[(1.0 - y), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 0.160000000000000003 < y

   1. Initial program 32.2%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf 98.2%

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

      1. mul-1-neg 98.2%

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

      2. unsub-neg 98.2%

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

      3. sub-neg 98.2%

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

      4. metadata-eval 98.2%

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

   5. Simplified 98.2%

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

   6. Taylor expanded in x around 0 97.9%

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

   if -1 < y < 4.59999999999999957e-159 or 1.69999999999999997e-107 < y < 0.160000000000000003

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 71.7%

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

   4. Taylor expanded in y around 0 70.4%

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

      1. neg-mul-1 70.4%

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

      2. sub-neg 70.4%

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

   6. Simplified 70.4%

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

   if 4.59999999999999957e-159 < y < 1.69999999999999997e-107

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 76.4%

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

      1. associate-/l* 75.9%

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

      2. *-lft-identity 75.9%

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

      3. associate-*l/ 75.9%

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

      4. distribute-lft-in 75.9%

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

      5. *-rgt-identity 75.9%

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

      6. lft-mult-inverse 75.9%

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

      7. +-commutative 75.9%

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

   5. Simplified 75.9%

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

   6. Taylor expanded in y around 0 76.4%

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

      1. *-commutative 76.4%

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

   8. Simplified 76.4%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-159}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-107}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.16:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{-1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-159}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-107}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.39:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   x
   (if (<= y 4.6e-159)
     (- 1.0 y)
     (if (<= y 2.3e-107) (* y x) (if (<= y 0.39) (- 1.0 y) x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 4.6e-159) {
		tmp = 1.0 - y;
	} else if (y <= 2.3e-107) {
		tmp = y * x;
	} else if (y <= 0.39) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x
    else if (y <= 4.6d-159) then
        tmp = 1.0d0 - y
    else if (y <= 2.3d-107) then
        tmp = y * x
    else if (y <= 0.39d0) then
        tmp = 1.0d0 - y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 4.6e-159) {
		tmp = 1.0 - y;
	} else if (y <= 2.3e-107) {
		tmp = y * x;
	} else if (y <= 0.39) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 4.6e-159:
		tmp = 1.0 - y
	elif y <= 2.3e-107:
		tmp = y * x
	elif y <= 0.39:
		tmp = 1.0 - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 4.6e-159)
		tmp = Float64(1.0 - y);
	elseif (y <= 2.3e-107)
		tmp = Float64(y * x);
	elseif (y <= 0.39)
		tmp = Float64(1.0 - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 4.6e-159)
		tmp = 1.0 - y;
	elseif (y <= 2.3e-107)
		tmp = y * x;
	elseif (y <= 0.39)
		tmp = 1.0 - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 4.6e-159], N[(1.0 - y), $MachinePrecision], If[LessEqual[y, 2.3e-107], N[(y * x), $MachinePrecision], If[LessEqual[y, 0.39], N[(1.0 - y), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 0.39000000000000001 < y

   1. Initial program 32.2%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 70.0%

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

   if -1 < y < 4.59999999999999957e-159 or 2.30000000000000003e-107 < y < 0.39000000000000001

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 71.7%

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

   4. Taylor expanded in y around 0 70.4%

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

      1. neg-mul-1 70.4%

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

      2. sub-neg 70.4%

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

   6. Simplified 70.4%

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

   if 4.59999999999999957e-159 < y < 2.30000000000000003e-107

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 76.4%

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

      1. associate-/l* 75.9%

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

      2. *-lft-identity 75.9%

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

      3. associate-*l/ 75.9%

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

      4. distribute-lft-in 75.9%

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

      5. *-rgt-identity 75.9%

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

      6. lft-mult-inverse 75.9%

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

      7. +-commutative 75.9%

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

   5. Simplified 75.9%

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

   6. Taylor expanded in y around 0 76.4%

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

      1. *-commutative 76.4%

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

   8. Simplified 76.4%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-159}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-107}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.39:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-159}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-107}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.17:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   x
   (if (<= y 4.6e-159)
     1.0
     (if (<= y 1.7e-107) (* y x) (if (<= y 0.17) 1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 4.6e-159) {
		tmp = 1.0;
	} else if (y <= 1.7e-107) {
		tmp = y * x;
	} else if (y <= 0.17) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x
    else if (y <= 4.6d-159) then
        tmp = 1.0d0
    else if (y <= 1.7d-107) then
        tmp = y * x
    else if (y <= 0.17d0) then
        tmp = 1.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 4.6e-159) {
		tmp = 1.0;
	} else if (y <= 1.7e-107) {
		tmp = y * x;
	} else if (y <= 0.17) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 4.6e-159:
		tmp = 1.0
	elif y <= 1.7e-107:
		tmp = y * x
	elif y <= 0.17:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 4.6e-159)
		tmp = 1.0;
	elseif (y <= 1.7e-107)
		tmp = Float64(y * x);
	elseif (y <= 0.17)
		tmp = 1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 4.6e-159)
		tmp = 1.0;
	elseif (y <= 1.7e-107)
		tmp = y * x;
	elseif (y <= 0.17)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 4.6e-159], 1.0, If[LessEqual[y, 1.7e-107], N[(y * x), $MachinePrecision], If[LessEqual[y, 0.17], 1.0, x]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 0.170000000000000012 < y

   1. Initial program 32.2%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 70.0%

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

   if -1 < y < 4.59999999999999957e-159 or 1.69999999999999997e-107 < y < 0.170000000000000012

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 69.5%

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

   if 4.59999999999999957e-159 < y < 1.69999999999999997e-107

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 76.4%

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

      1. associate-/l* 75.9%

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

      2. *-lft-identity 75.9%

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

      3. associate-*l/ 75.9%

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

      4. distribute-lft-in 75.9%

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

      5. *-rgt-identity 75.9%

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

      6. lft-mult-inverse 75.9%

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

      7. +-commutative 75.9%

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

   5. Simplified 75.9%

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

   6. Taylor expanded in y around 0 76.4%

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

      1. *-commutative 76.4%

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

   8. Simplified 76.4%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-159}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-107}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.17:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -55000000000 \lor \neg \left(y \leq 17000000000\right):\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -55000000000.0) (not (<= y 17000000000.0)))
   (- x (/ -1.0 y))
   (+ 1.0 (/ (* y (+ x -1.0)) (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -55000000000.0) || !(y <= 17000000000.0)) {
		tmp = x - (-1.0 / y);
	} else {
		tmp = 1.0 + ((y * (x + -1.0)) / (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 <= (-55000000000.0d0)) .or. (.not. (y <= 17000000000.0d0))) then
        tmp = x - ((-1.0d0) / y)
    else
        tmp = 1.0d0 + ((y * (x + (-1.0d0))) / (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -55000000000.0) || !(y <= 17000000000.0)) {
		tmp = x - (-1.0 / y);
	} else {
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -55000000000.0) or not (y <= 17000000000.0):
		tmp = x - (-1.0 / y)
	else:
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -55000000000.0) || !(y <= 17000000000.0))
		tmp = Float64(x - Float64(-1.0 / y));
	else
		tmp = Float64(1.0 + Float64(Float64(y * Float64(x + -1.0)) / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -55000000000.0) || ~((y <= 17000000000.0)))
		tmp = x - (-1.0 / y);
	else
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -55000000000.0], N[Not[LessEqual[y, 17000000000.0]], $MachinePrecision]], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.5e10 or 1.7e10 < y

   1. Initial program 30.2%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. unsub-neg 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around 0 99.9%

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

   if -5.5e10 < y < 1.7e10

   1. Initial program 99.9%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -55000000000 \lor \neg \left(y \leq 17000000000\right):\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 35.4%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf 97.9%

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

      1. mul-1-neg 97.9%

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

      2. unsub-neg 97.9%

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

      3. sub-neg 97.9%

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

      4. metadata-eval 97.9%

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

   5. Simplified 97.9%

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

   6. Taylor expanded in x around 0 97.9%

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

   if -1 < y < 1

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 98.4%

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

      1. sub-neg 98.4%

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

      2. metadata-eval 98.4%

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

      3. +-commutative 98.4%

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

      4. metadata-eval 98.4%

         \[\leadsto 1 + \left(y \cdot \left(\color{blue}{\left(0 - 1\right)} + x\right) + {y}^{2} \cdot \left(1 - x\right)\right) \]

      5. associate--r- 98.4%

         \[\leadsto 1 + \left(y \cdot \color{blue}{\left(0 - \left(1 - x\right)\right)} + {y}^{2} \cdot \left(1 - x\right)\right) \]

      6. neg-sub0 98.4%

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

      7. distribute-rgt-neg-in 98.4%

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

      8. *-commutative 98.4%

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

      9. neg-mul-1 98.4%

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

      10. unpow2 98.4%

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

      11. associate-*l* 98.4%

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

      12. *-commutative 98.4%

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

      13. distribute-rgt-in 98.3%

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

      14. +-commutative 98.3%

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

      15. *-commutative 98.3%

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

      16. associate-*l* 98.3%

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

   5. Simplified 98.3%

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

   if 1 < y

   1. Initial program 27.4%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf 98.6%

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

      1. mul-1-neg 98.6%

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

      2. unsub-neg 98.6%

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

      3. sub-neg 98.6%

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

      4. metadata-eval 98.6%

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

   5. Simplified 98.6%

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

4. Final simplification 98.2%

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

Alternative 7: 98.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 35.4%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf 97.9%

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

      1. mul-1-neg 97.9%

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

      2. unsub-neg 97.9%

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

      3. sub-neg 97.9%

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

      4. metadata-eval 97.9%

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

   5. Simplified 97.9%

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

   6. Taylor expanded in x around 0 97.9%

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

   if -1 < y < 1

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l* 99.8%

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

      2. +-commutative 99.8%

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

      3. clear-num 99.7%

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

      4. inv-pow 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

      2. associate-/l/ 99.9%

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

      3. *-commutative 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around inf 97.9%

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

      1. associate-*r/ 97.9%

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

      2. neg-mul-1 97.9%

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

      3. distribute-neg-in 97.9%

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

      4. metadata-eval 97.9%

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

      5. unsub-neg 97.9%

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

      6. *-commutative 97.9%

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

   9. Simplified 97.9%

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

   10. Taylor expanded in y around 0 97.4%

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

       1. *-commutative 97.4%

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

       2. *-commutative 97.4%

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

       3. unpow2 97.4%

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

       4. associate-*l* 97.4%

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

       5. distribute-rgt-in 97.4%

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

       6. *-commutative 97.4%

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

   12. Simplified 97.4%

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

   if 1 < y

   1. Initial program 27.4%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf 98.6%

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

      1. mul-1-neg 98.6%

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

      2. unsub-neg 98.6%

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

      3. sub-neg 98.6%

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

      4. metadata-eval 98.6%

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

   5. Simplified 98.6%

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

4. Final simplification 97.8%

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

Alternative 8: 98.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.78\right):\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 0.78)))
   (- x (/ -1.0 y))
   (+ 1.0 (* y (+ x -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.78)) {
		tmp = x - (-1.0 / y);
	} else {
		tmp = 1.0 + (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 <= 0.78d0))) then
        tmp = x - ((-1.0d0) / y)
    else
        tmp = 1.0d0 + (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 <= 0.78)) {
		tmp = x - (-1.0 / y);
	} else {
		tmp = 1.0 + (y * (x + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 0.78):
		tmp = x - (-1.0 / y)
	else:
		tmp = 1.0 + (y * (x + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 0.78))
		tmp = Float64(x - Float64(-1.0 / y));
	else
		tmp = Float64(1.0 + 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 <= 0.78)))
		tmp = x - (-1.0 / y);
	else
		tmp = 1.0 + (y * (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 0.78]], $MachinePrecision]], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.78000000000000003 < y

   1. Initial program 32.2%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf 98.2%

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

      1. mul-1-neg 98.2%

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

      2. unsub-neg 98.2%

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

      3. sub-neg 98.2%

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

      4. metadata-eval 98.2%

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

   5. Simplified 98.2%

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

   6. Taylor expanded in x around 0 97.9%

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

   if -1 < y < 0.78000000000000003

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 96.8%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.78\right):\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(x + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 35.4%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf 97.9%

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

      1. mul-1-neg 97.9%

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

      2. unsub-neg 97.9%

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

      3. sub-neg 97.9%

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

      4. metadata-eval 97.9%

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

   5. Simplified 97.9%

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

   6. Taylor expanded in x around 0 97.9%

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

   if -1 < y < 1

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 96.8%

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

   if 1 < y

   1. Initial program 27.4%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf 98.6%

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

      1. mul-1-neg 98.6%

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

      2. unsub-neg 98.6%

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

      3. sub-neg 98.6%

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

      4. metadata-eval 98.6%

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

   5. Simplified 98.6%

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

4. Final simplification 97.6%

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

Alternative 10: 98.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = x - (-1.0 / y);
	} else {
		tmp = 1.0 + (y * x);
	}
	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 = x - ((-1.0d0) / y)
    else
        tmp = 1.0d0 + (y * x)
    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 = x - (-1.0 / y);
	} else {
		tmp = 1.0 + (y * x);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 32.2%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf 98.2%

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

      1. mul-1-neg 98.2%

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

      2. unsub-neg 98.2%

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

      3. sub-neg 98.2%

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

      4. metadata-eval 98.2%

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

   5. Simplified 98.2%

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

   6. Taylor expanded in x around 0 97.9%

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

   if -1 < y < 1

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l* 99.8%

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

      2. +-commutative 99.8%

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

      3. clear-num 99.7%

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

      4. inv-pow 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

      2. associate-/l/ 99.9%

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

      3. *-commutative 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around inf 97.9%

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

      1. associate-*r/ 97.9%

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

      2. neg-mul-1 97.9%

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

      3. distribute-neg-in 97.9%

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

      4. metadata-eval 97.9%

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

      5. unsub-neg 97.9%

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

      6. *-commutative 97.9%

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

   9. Simplified 97.9%

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

   10. Taylor expanded in y around 0 96.1%

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

       1. mul-1-neg 96.1%

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

       2. *-commutative 96.1%

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

       3. distribute-rgt-neg-in 96.1%

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

   12. Simplified 96.1%

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

4. Final simplification 97.0%

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

Alternative 11: 73.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1.0) x (if (<= y 2.5) 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 2.5) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x
    else if (y <= 2.5d0) then
        tmp = 1.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 2.5) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 2.5:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 2.5)
		tmp = 1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 2.5)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 2.5], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 2.5 < y

   1. Initial program 32.2%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 70.0%

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

   if -1 < y < 2.5

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 65.6%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 39.1% accurate, 11.0× 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 63.7%

   \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
2. Add Preprocessing

3. Taylor expanded in y around 0 32.5%

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

4. Final simplification 32.5%

   \[\leadsto 1 \]
5. Add Preprocessing

Developer target: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{if}\;y < -3693.8482788297247:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 6799310503.41891:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 y) (- (/ x y) x))))
   (if (< y -3693.8482788297247)
     t_0
     (if (< y 6799310503.41891) (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))) t_0))))

C

double code(double x, double y) {
	double t_0 = (1.0 / y) - ((x / y) - x);
	double tmp;
	if (y < -3693.8482788297247) {
		tmp = t_0;
	} else if (y < 6799310503.41891) {
		tmp = 1.0 - (((1.0 - x) * 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 / y) - ((x / y) - x)
    if (y < (-3693.8482788297247d0)) then
        tmp = t_0
    else if (y < 6799310503.41891d0) then
        tmp = 1.0d0 - (((1.0d0 - x) * 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 / y) - ((x / y) - x);
	double tmp;
	if (y < -3693.8482788297247) {
		tmp = t_0;
	} else if (y < 6799310503.41891) {
		tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.0 / y) - ((x / y) - x)
	tmp = 0
	if y < -3693.8482788297247:
		tmp = t_0
	elif y < 6799310503.41891:
		tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 / y) - Float64(Float64(x / y) - x))
	tmp = 0.0
	if (y < -3693.8482788297247)
		tmp = t_0;
	elseif (y < 6799310503.41891)
		tmp = Float64(1.0 - Float64(Float64(Float64(1.0 - x) * y) / Float64(y + 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (1.0 / y) - ((x / y) - x);
	tmp = 0.0;
	if (y < -3693.8482788297247)
		tmp = t_0;
	elseif (y < 6799310503.41891)
		tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 / y), $MachinePrecision] - N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[Less[y, -3693.8482788297247], t$95$0, If[Less[y, 6799310503.41891], N[(1.0 - N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024034 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1.0 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))) (- (/ 1.0 y) (- (/ x y) x))))

  (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))