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

Percentage Accurate: 65.6% → 99.9%

Time: 9.6s

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: 65.6% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -300000 \lor \neg \left(y \leq 340000\right):\\ \;\;\;\;x + \frac{\left(1 - x\right) + \frac{x + -1}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -300000.0) (not (<= y 340000.0)))
   (+ x (/ (+ (- 1.0 x) (/ (+ x -1.0) y)) y))
   (+ 1.0 (/ (* y (- 1.0 x)) (- -1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -300000.0) || !(y <= 340000.0)) {
		tmp = x + (((1.0 - x) + ((x + -1.0) / y)) / y);
	} else {
		tmp = 1.0 + ((y * (1.0 - 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 <= (-300000.0d0)) .or. (.not. (y <= 340000.0d0))) then
        tmp = x + (((1.0d0 - x) + ((x + (-1.0d0)) / y)) / y)
    else
        tmp = 1.0d0 + ((y * (1.0d0 - x)) / ((-1.0d0) - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -300000.0) || !(y <= 340000.0)) {
		tmp = x + (((1.0 - x) + ((x + -1.0) / y)) / y);
	} else {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -300000.0) or not (y <= 340000.0):
		tmp = x + (((1.0 - x) + ((x + -1.0) / y)) / y)
	else:
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -300000.0) || !(y <= 340000.0))
		tmp = Float64(x + Float64(Float64(Float64(1.0 - x) + Float64(Float64(x + -1.0) / y)) / y));
	else
		tmp = Float64(1.0 + Float64(Float64(y * Float64(1.0 - x)) / Float64(-1.0 - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -300000.0) || ~((y <= 340000.0)))
		tmp = x + (((1.0 - x) + ((x + -1.0) / y)) / y);
	else
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -300000.0], N[Not[LessEqual[y, 340000.0]], $MachinePrecision]], N[(x + N[(N[(N[(1.0 - x), $MachinePrecision] + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3e5 or 3.4e5 < y

   1. Initial program 29.9%

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

      1. associate-/l* 58.8%

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

      2. +-commutative 58.8%

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

   3. Simplified 58.8%

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

   5. Taylor expanded in y around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   if -3e5 < y < 3.4e5

   1. Initial program 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -300000 \lor \neg \left(y \leq 340000\right):\\ \;\;\;\;x + \frac{\left(1 - x\right) + \frac{x + -1}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -85000000 \lor \neg \left(y \leq 160000000\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -85000000.0) (not (<= y 160000000.0)))
   (+ x (/ (- 1.0 x) y))
   (+ 1.0 (/ (* y (- 1.0 x)) (- -1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -85000000.0) || !(y <= 160000000.0)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = 1.0 + ((y * (1.0 - 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 <= (-85000000.0d0)) .or. (.not. (y <= 160000000.0d0))) then
        tmp = x + ((1.0d0 - x) / y)
    else
        tmp = 1.0d0 + ((y * (1.0d0 - x)) / ((-1.0d0) - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -85000000.0) || !(y <= 160000000.0)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -85000000.0) or not (y <= 160000000.0):
		tmp = x + ((1.0 - x) / y)
	else:
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -85000000.0) || !(y <= 160000000.0))
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	else
		tmp = Float64(1.0 + Float64(Float64(y * Float64(1.0 - x)) / Float64(-1.0 - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -85000000.0) || ~((y <= 160000000.0)))
		tmp = x + ((1.0 - x) / y);
	else
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -85000000.0], N[Not[LessEqual[y, 160000000.0]], $MachinePrecision]], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.5e7 or 1.6e8 < y

   1. Initial program 27.9%

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

      1. associate-/l* 57.6%

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

      2. +-commutative 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in y around inf 99.8%

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

      1. associate--l+ 99.8%

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

      2. div-sub 99.8%

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

   7. Simplified 99.8%

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

   if -8.5e7 < y < 1.6e8

   1. Initial program 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -85000000 \lor \neg \left(y \leq 160000000\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -160000000 \lor \neg \left(y \leq 310000000\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y}{y + 1} \cdot \left(x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -160000000.0) (not (<= y 310000000.0)))
   (+ x (/ (- 1.0 x) y))
   (+ 1.0 (* (/ y (+ y 1.0)) (+ x -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -160000000.0) || !(y <= 310000000.0)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = 1.0 + ((y / (y + 1.0)) * (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 <= (-160000000.0d0)) .or. (.not. (y <= 310000000.0d0))) then
        tmp = x + ((1.0d0 - x) / y)
    else
        tmp = 1.0d0 + ((y / (y + 1.0d0)) * (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -160000000.0) || !(y <= 310000000.0)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = 1.0 + ((y / (y + 1.0)) * (x + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -160000000.0) or not (y <= 310000000.0):
		tmp = x + ((1.0 - x) / y)
	else:
		tmp = 1.0 + ((y / (y + 1.0)) * (x + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -160000000.0) || !(y <= 310000000.0))
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	else
		tmp = Float64(1.0 + Float64(Float64(y / Float64(y + 1.0)) * Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -160000000.0) || ~((y <= 310000000.0)))
		tmp = x + ((1.0 - x) / y);
	else
		tmp = 1.0 + ((y / (y + 1.0)) * (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -160000000.0], N[Not[LessEqual[y, 310000000.0]], $MachinePrecision]], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6e8 or 3.1e8 < y

   1. Initial program 27.9%

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

      1. associate-/l* 57.6%

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

      2. +-commutative 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in y around inf 99.8%

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

      1. associate--l+ 99.8%

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

      2. div-sub 99.8%

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

   7. Simplified 99.8%

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

   if -1.6e8 < y < 3.1e8

   1. Initial program 99.6%

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

      1. associate-/l* 99.6%

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

      2. +-commutative 99.6%

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

   3. Simplified 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -160000000 \lor \neg \left(y \leq 310000000\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y}{y + 1} \cdot \left(x + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 1.0))
		tmp = Float64(x + Float64(Float64(1.0 - x) / 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 <= 1.0)))
		tmp = x + ((1.0 - x) / 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, 1.0]], $MachinePrecision]], N[(x + N[(N[(1.0 - x), $MachinePrecision] / 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 1 < y

   1. Initial program 32.4%

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

      1. associate-/l* 60.1%

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

      2. +-commutative 60.1%

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

   3. Simplified 60.1%

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

   5. Taylor expanded in y around inf 96.7%

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

      1. associate--l+ 96.7%

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

      2. div-sub 96.7%

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

   7. Simplified 96.7%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 99.3%

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

4. Final simplification 98.0%

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

Alternative 5: 97.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.2\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.2))) (+ x (/ (- 1.0 x) y)) (+ 1.0 (* x y))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.2)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = 1.0 + (x * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.0d0)) .or. (.not. (y <= 1.2d0))) then
        tmp = x + ((1.0d0 - x) / y)
    else
        tmp = 1.0d0 + (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.2)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = 1.0 + (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 1.2):
		tmp = x + ((1.0 - x) / y)
	else:
		tmp = 1.0 + (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 1.2))
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	else
		tmp = Float64(1.0 + Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 1.2)))
		tmp = x + ((1.0 - x) / y);
	else
		tmp = 1.0 + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.2]], $MachinePrecision]], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.19999999999999996 < y

   1. Initial program 32.4%

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

      1. associate-/l* 60.1%

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

      2. +-commutative 60.1%

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

   3. Simplified 60.1%

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

   5. Taylor expanded in y around inf 96.7%

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

      1. associate--l+ 96.7%

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

      2. div-sub 96.7%

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

   7. Simplified 96.7%

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

   if -1 < y < 1.19999999999999996

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 99.3%

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

   6. Taylor expanded in x around inf 99.0%

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

      1. neg-mul-1 99.0%

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

   8. Simplified 99.0%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.2\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.7% accurate, 0.7× speedup

Math

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

FPCore

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

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 + (x * y);
	}
	return tmp;
}

Fortran

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

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = x + (1.0 / y);
	} else {
		tmp = 1.0 + (x * y);
	}
	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 + (x * y)
	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(x * y));
	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 + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 32.4%

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

      1. associate-/l* 60.1%

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

      2. +-commutative 60.1%

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

   3. Simplified 60.1%

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

   5. Taylor expanded in y around inf 96.7%

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

      1. associate--l+ 96.7%

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

      2. div-sub 96.7%

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

   7. Simplified 96.7%

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

   8. Taylor expanded in x around 0 95.2%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 99.3%

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

   6. Taylor expanded in x around inf 99.0%

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

      1. neg-mul-1 99.0%

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

   8. Simplified 99.0%

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

4. Final simplification 97.0%

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

Alternative 7: 86.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -750000000000 \lor \neg \left(y \leq 170\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -750000000000.0) (not (<= y 170.0)))
   (+ x (/ 1.0 y))
   (/ 1.0 (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -750000000000.0) || !(y <= 170.0)) {
		tmp = x + (1.0 / y);
	} else {
		tmp = 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 <= (-750000000000.0d0)) .or. (.not. (y <= 170.0d0))) then
        tmp = x + (1.0d0 / y)
    else
        tmp = 1.0d0 / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -750000000000.0) || !(y <= 170.0)) {
		tmp = x + (1.0 / y);
	} else {
		tmp = 1.0 / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -750000000000.0) or not (y <= 170.0):
		tmp = x + (1.0 / y)
	else:
		tmp = 1.0 / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -750000000000.0) || !(y <= 170.0))
		tmp = Float64(x + Float64(1.0 / y));
	else
		tmp = Float64(1.0 / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -750000000000.0) || ~((y <= 170.0)))
		tmp = x + (1.0 / y);
	else
		tmp = 1.0 / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -750000000000.0], N[Not[LessEqual[y, 170.0]], $MachinePrecision]], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.5e11 or 170 < y

   1. Initial program 30.4%

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

      1. associate-/l* 59.2%

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

      2. +-commutative 59.2%

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

   3. Simplified 59.2%

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

   5. Taylor expanded in y around inf 98.7%

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

      1. associate--l+ 98.7%

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

      2. div-sub 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in x around 0 97.3%

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

   if -7.5e11 < y < 170

   1. Initial program 99.3%

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

      1. sub-neg 99.3%

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

      2. +-commutative 99.3%

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

      3. *-commutative 99.3%

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

      4. associate-/l* 99.3%

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

      5. distribute-rgt-neg-in 99.3%

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

      6. fma-define 99.3%

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

      7. distribute-frac-neg2 99.3%

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

      8. +-commutative 99.3%

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

      9. distribute-neg-in 99.3%

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

      10. metadata-eval 99.3%

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

      11. unsub-neg 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in x around inf 99.2%

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

      1. +-commutative 99.2%

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

      2. +-commutative 99.2%

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

      3. associate-+l+ 99.2%

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

      4. +-commutative 99.2%

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

      5. mul-1-neg 99.2%

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

      6. distribute-neg-frac2 99.2%

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

      7. distribute-rgt-neg-out 99.2%

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

      8. distribute-neg-in 99.2%

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

      9. metadata-eval 99.2%

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

      10. sub-neg 99.2%

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

   7. Simplified 99.2%

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

      1. associate-/r* 99.1%

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

      2. frac-add 99.8%

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

      3. *-un-lft-identity 99.8%

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

   9. Applied egg-rr 99.8%

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

       1. associate-*r/ 99.8%

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

       2. *-rgt-identity 99.8%

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

       3. times-frac 99.8%

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

       4. *-rgt-identity 99.8%

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

       5. associate-*r/ 99.6%

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

       6. rgt-mult-inverse 99.8%

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

       7. rem-square-sqrt 53.4%

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

       8. associate-*r/ 53.4%

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

       9. /-rgt-identity 53.4%

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

       10. rem-square-sqrt 99.8%

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

       11. *-lft-identity 99.8%

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

       12. associate-+l- 99.8%

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

       13. +-inverses 99.8%

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

       14. metadata-eval 99.8%

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

   11. Simplified 99.8%

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

   12. Taylor expanded in x around 0 81.7%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -750000000000 \lor \neg \left(y \leq 170\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.125\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 0.125))) (+ x (/ 1.0 y)) (- 1.0 y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.125)) {
		tmp = x + (1.0 / y);
	} else {
		tmp = 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)) .or. (.not. (y <= 0.125d0))) then
        tmp = x + (1.0d0 / y)
    else
        tmp = 1.0d0 - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.125)) {
		tmp = x + (1.0 / y);
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 0.125):
		tmp = x + (1.0 / y)
	else:
		tmp = 1.0 - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 0.125))
		tmp = Float64(x + Float64(1.0 / y));
	else
		tmp = Float64(1.0 - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 0.125)))
		tmp = x + (1.0 / y);
	else
		tmp = 1.0 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 0.125]], $MachinePrecision]], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(1.0 - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.125 < y

   1. Initial program 32.4%

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

      1. associate-/l* 60.1%

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

      2. +-commutative 60.1%

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

   3. Simplified 60.1%

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

   5. Taylor expanded in y around inf 96.7%

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

      1. associate--l+ 96.7%

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

      2. div-sub 96.7%

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

   7. Simplified 96.7%

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

   8. Taylor expanded in x around 0 95.2%

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

   if -1 < y < 0.125

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 82.6%

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

      1. +-commutative 82.6%

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

   7. Simplified 82.6%

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

   8. Taylor expanded in y around 0 82.1%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.125\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.1%

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

   1. sub-neg 65.1%

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

   2. +-commutative 65.1%

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

   3. *-commutative 65.1%

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

   4. associate-/l* 79.4%

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

   5. distribute-rgt-neg-in 79.4%

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

   6. fma-define 79.4%

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

   7. distribute-frac-neg2 79.4%

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

   8. +-commutative 79.4%

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

   9. distribute-neg-in 79.4%

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

   10. metadata-eval 79.4%

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

   11. unsub-neg 79.4%

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

3. Simplified 79.4%

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

5. Taylor expanded in x around inf 79.5%

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

   1. +-commutative 79.5%

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

   2. +-commutative 79.5%

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

   3. associate-+l+ 85.9%

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

   4. +-commutative 85.9%

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

   5. mul-1-neg 85.9%

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

   6. distribute-neg-frac2 85.9%

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

   7. distribute-rgt-neg-out 85.9%

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

   8. distribute-neg-in 85.9%

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

   9. metadata-eval 85.9%

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

   10. sub-neg 85.9%

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

7. Simplified 85.9%

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

   1. associate-/r* 82.2%

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

   2. frac-add 83.8%

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

   3. *-un-lft-identity 83.8%

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

9. Applied egg-rr 83.8%

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

    1. associate-*r/ 73.7%

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

    2. *-rgt-identity 73.7%

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

    3. times-frac 89.4%

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

    4. *-rgt-identity 89.4%

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

    5. associate-*r/ 87.9%

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

    6. rgt-mult-inverse 89.4%

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

    7. rem-square-sqrt 42.0%

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

    8. associate-*r/ 42.0%

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

    9. /-rgt-identity 42.0%

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

    10. rem-square-sqrt 89.4%

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

    11. *-lft-identity 89.4%

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

    12. associate-+l- 99.8%

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

    13. +-inverses 99.8%

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

    14. metadata-eval 99.8%

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

11. Simplified 99.8%

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

12. Final simplification 99.8%

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

Alternative 10: 74.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.235:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 0.235) (- 1.0 y) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.235) {
		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 <= 0.235d0) 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 <= 0.235) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 0.235:
		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 <= 0.235)
		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 <= 0.235)
		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, 0.235], N[(1.0 - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.23499999999999999 < y

   1. Initial program 32.4%

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

      1. associate-/l* 60.1%

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

      2. +-commutative 60.1%

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

   3. Simplified 60.1%

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

   5. Taylor expanded in y around inf 72.9%

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

   if -1 < y < 0.23499999999999999

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 82.6%

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

      1. +-commutative 82.6%

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

   7. Simplified 82.6%

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

   8. Taylor expanded in y around 0 82.1%

      \[\leadsto 1 - \color{blue}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 73.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 150:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1.0) x (if (<= y 150.0) 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 150.0) {
		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 <= 150.0d0) 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 <= 150.0) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 150.0:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 150.0)
		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 <= 150.0)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 150.0], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 150 < y

   1. Initial program 31.9%

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

      1. associate-/l* 59.9%

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

      2. +-commutative 59.9%

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

   3. Simplified 59.9%

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

   5. Taylor expanded in y around inf 73.4%

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

   if -1 < y < 150

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. +-commutative 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-/l* 99.9%

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

      5. distribute-rgt-neg-in 99.9%

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

      6. fma-define 99.9%

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

      7. distribute-frac-neg2 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. metadata-eval 99.9%

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

      11. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around -inf 99.8%

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

      1. associate-*r* 99.8%

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

      2. neg-mul-1 99.8%

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

      3. distribute-lft-out 99.8%

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

      4. associate-*r* 99.8%

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

      5. distribute-lft-neg-in 99.8%

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

      6. *-commutative 99.8%

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

      7. neg-mul-1 99.8%

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

      8. remove-double-neg 99.8%

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

      9. +-commutative 99.8%

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

      10. mul-1-neg 99.8%

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

      11. sub-neg 99.8%

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

      12. sub-neg 99.8%

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

      13. distribute-frac-neg2 99.8%

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

      14. distribute-neg-in 99.8%

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

      15. metadata-eval 99.8%

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

      16. sub-neg 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in y around 0 81.3%

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

Alternative 12: 38.5% 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 65.1%

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

   1. sub-neg 65.1%

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

   2. +-commutative 65.1%

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

   3. *-commutative 65.1%

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

   4. associate-/l* 79.4%

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

   5. distribute-rgt-neg-in 79.4%

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

   6. fma-define 79.4%

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

   7. distribute-frac-neg2 79.4%

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

   8. +-commutative 79.4%

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

   9. distribute-neg-in 79.4%

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

   10. metadata-eval 79.4%

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

   11. unsub-neg 79.4%

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

3. Simplified 79.4%

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

5. Taylor expanded in x around -inf 88.7%

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

   1. associate-*r* 88.7%

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

   2. neg-mul-1 88.7%

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

   3. distribute-lft-out 88.7%

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

   4. associate-*r* 88.7%

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

   5. distribute-lft-neg-in 88.7%

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

   6. *-commutative 88.7%

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

   7. neg-mul-1 88.7%

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

   8. remove-double-neg 88.7%

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

   9. +-commutative 88.7%

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

   10. mul-1-neg 88.7%

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

   11. sub-neg 88.7%

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

   12. sub-neg 88.7%

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

   13. distribute-frac-neg2 88.7%

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

   14. distribute-neg-in 88.7%

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

   15. metadata-eval 88.7%

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

   16. sub-neg 88.7%

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

7. Simplified 88.7%

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

8. Taylor expanded in y around 0 41.4%

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

Developer Target 1: 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 2024148 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -36938482788297247/10000000000000) (- (/ 1 y) (- (/ x y) x)) (if (< y 679931050341891/100000) (- 1 (/ (* (- 1 x) y) (+ y 1))) (- (/ 1 y) (- (/ x y) x)))))

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