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

Percentage Accurate: 66.1% → 99.5%

Time: 7.3s

Alternatives: 12

Speedup: 1.2×

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: 66.1% 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: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{+15}:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{elif}\;y \leq 8200000:\\ \;\;\;\;1 - \frac{\left(x - 1\right) \cdot y}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\frac{y - 1}{y \cdot y} - 1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.7e+15)
   (- x (/ (- x 1.0) y))
   (if (<= y 8200000.0)
     (- 1.0 (/ (* (- x 1.0) y) (- -1.0 y)))
     (- x (/ (- (/ (- y 1.0) (* y y)) 1.0) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.7e+15) {
		tmp = x - ((x - 1.0) / y);
	} else if (y <= 8200000.0) {
		tmp = 1.0 - (((x - 1.0) * y) / (-1.0 - y));
	} else {
		tmp = x - ((((y - 1.0) / (y * y)) - 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 <= (-4.7d+15)) then
        tmp = x - ((x - 1.0d0) / y)
    else if (y <= 8200000.0d0) then
        tmp = 1.0d0 - (((x - 1.0d0) * y) / ((-1.0d0) - y))
    else
        tmp = x - ((((y - 1.0d0) / (y * y)) - 1.0d0) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.7e+15) {
		tmp = x - ((x - 1.0) / y);
	} else if (y <= 8200000.0) {
		tmp = 1.0 - (((x - 1.0) * y) / (-1.0 - y));
	} else {
		tmp = x - ((((y - 1.0) / (y * y)) - 1.0) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.7e+15:
		tmp = x - ((x - 1.0) / y)
	elif y <= 8200000.0:
		tmp = 1.0 - (((x - 1.0) * y) / (-1.0 - y))
	else:
		tmp = x - ((((y - 1.0) / (y * y)) - 1.0) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.7e+15)
		tmp = Float64(x - Float64(Float64(x - 1.0) / y));
	elseif (y <= 8200000.0)
		tmp = Float64(1.0 - Float64(Float64(Float64(x - 1.0) * y) / Float64(-1.0 - y)));
	else
		tmp = Float64(x - Float64(Float64(Float64(Float64(y - 1.0) / Float64(y * y)) - 1.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.7e+15)
		tmp = x - ((x - 1.0) / y);
	elseif (y <= 8200000.0)
		tmp = 1.0 - (((x - 1.0) * y) / (-1.0 - y));
	else
		tmp = x - ((((y - 1.0) / (y * y)) - 1.0) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.7e+15], N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8200000.0], N[(1.0 - N[(N[(N[(x - 1.0), $MachinePrecision] * y), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(N[(y - 1.0), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.7e15

   1. Initial program 19.5%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -4.7e15 < y < 8.2e6

   1. Initial program 99.9%

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

   if 8.2e6 < y

   1. Initial program 33.1%

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

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.9%

      \[\leadsto x - \frac{\frac{\frac{y - 1}{y}}{y} - 1}{y} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{+15}:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{elif}\;y \leq 8200000:\\ \;\;\;\;1 - \frac{\left(x - 1\right) \cdot y}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\frac{y - 1}{y \cdot y} - 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{+15}:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{elif}\;y \leq 8500000:\\ \;\;\;\;1 - \frac{\left(x - 1\right) \cdot y}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\frac{1}{y} - 1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.7e+15)
   (- x (/ (- x 1.0) y))
   (if (<= y 8500000.0)
     (- 1.0 (/ (* (- x 1.0) y) (- -1.0 y)))
     (- x (/ (- (/ 1.0 y) 1.0) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.7e+15) {
		tmp = x - ((x - 1.0) / y);
	} else if (y <= 8500000.0) {
		tmp = 1.0 - (((x - 1.0) * y) / (-1.0 - y));
	} else {
		tmp = x - (((1.0 / y) - 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 <= (-4.7d+15)) then
        tmp = x - ((x - 1.0d0) / y)
    else if (y <= 8500000.0d0) then
        tmp = 1.0d0 - (((x - 1.0d0) * y) / ((-1.0d0) - y))
    else
        tmp = x - (((1.0d0 / y) - 1.0d0) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.7e+15) {
		tmp = x - ((x - 1.0) / y);
	} else if (y <= 8500000.0) {
		tmp = 1.0 - (((x - 1.0) * y) / (-1.0 - y));
	} else {
		tmp = x - (((1.0 / y) - 1.0) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.7e+15:
		tmp = x - ((x - 1.0) / y)
	elif y <= 8500000.0:
		tmp = 1.0 - (((x - 1.0) * y) / (-1.0 - y))
	else:
		tmp = x - (((1.0 / y) - 1.0) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.7e+15)
		tmp = Float64(x - Float64(Float64(x - 1.0) / y));
	elseif (y <= 8500000.0)
		tmp = Float64(1.0 - Float64(Float64(Float64(x - 1.0) * y) / Float64(-1.0 - y)));
	else
		tmp = Float64(x - Float64(Float64(Float64(1.0 / y) - 1.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.7e+15)
		tmp = x - ((x - 1.0) / y);
	elseif (y <= 8500000.0)
		tmp = 1.0 - (((x - 1.0) * y) / (-1.0 - y));
	else
		tmp = x - (((1.0 / y) - 1.0) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.7e+15], N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8500000.0], N[(1.0 - N[(N[(N[(x - 1.0), $MachinePrecision] * y), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(1.0 / y), $MachinePrecision] - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.7e15

   1. Initial program 19.5%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -4.7e15 < y < 8.5e6

   1. Initial program 99.9%

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

   if 8.5e6 < y

   1. Initial program 33.1%

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

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 99.7%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{+15}:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{elif}\;y \leq 8500000:\\ \;\;\;\;1 - \frac{\left(x - 1\right) \cdot y}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\frac{1}{y} - 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{x - 1}{y}\\ \mathbf{if}\;y \leq -4.7 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 190000000:\\ \;\;\;\;1 - \frac{\left(x - 1\right) \cdot y}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ (- x 1.0) y))))
   (if (<= y -4.7e+15)
     t_0
     (if (<= y 190000000.0) (- 1.0 (/ (* (- x 1.0) y) (- -1.0 y))) t_0))))

C

double code(double x, double y) {
	double t_0 = x - ((x - 1.0) / y);
	double tmp;
	if (y <= -4.7e+15) {
		tmp = t_0;
	} else if (y <= 190000000.0) {
		tmp = 1.0 - (((x - 1.0) * y) / (-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 - ((x - 1.0d0) / y)
    if (y <= (-4.7d+15)) then
        tmp = t_0
    else if (y <= 190000000.0d0) then
        tmp = 1.0d0 - (((x - 1.0d0) * y) / ((-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 - ((x - 1.0) / y);
	double tmp;
	if (y <= -4.7e+15) {
		tmp = t_0;
	} else if (y <= 190000000.0) {
		tmp = 1.0 - (((x - 1.0) * y) / (-1.0 - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x - ((x - 1.0) / y)
	tmp = 0
	if y <= -4.7e+15:
		tmp = t_0
	elif y <= 190000000.0:
		tmp = 1.0 - (((x - 1.0) * y) / (-1.0 - y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x - Float64(Float64(x - 1.0) / y))
	tmp = 0.0
	if (y <= -4.7e+15)
		tmp = t_0;
	elseif (y <= 190000000.0)
		tmp = Float64(1.0 - Float64(Float64(Float64(x - 1.0) * y) / Float64(-1.0 - y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x - ((x - 1.0) / y);
	tmp = 0.0;
	if (y <= -4.7e+15)
		tmp = t_0;
	elseif (y <= 190000000.0)
		tmp = 1.0 - (((x - 1.0) * y) / (-1.0 - y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.7e+15], t$95$0, If[LessEqual[y, 190000000.0], N[(1.0 - N[(N[(N[(x - 1.0), $MachinePrecision] * y), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.7e15 or 1.9e8 < y

   1. Initial program 25.5%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if -4.7e15 < y < 1.9e8

   1. Initial program 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{+15}:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{elif}\;y \leq 190000000:\\ \;\;\;\;1 - \frac{\left(x - 1\right) \cdot y}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{x - 1}{y}\\ \mathbf{if}\;y \leq -4.7 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 180000000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1 - x}{-1 - y}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ (- x 1.0) y))))
   (if (<= y -4.7e+15)
     t_0
     (if (<= y 180000000.0) (fma y (/ (- 1.0 x) (- -1.0 y)) 1.0) t_0))))

C

double code(double x, double y) {
	double t_0 = x - ((x - 1.0) / y);
	double tmp;
	if (y <= -4.7e+15) {
		tmp = t_0;
	} else if (y <= 180000000.0) {
		tmp = fma(y, ((1.0 - x) / (-1.0 - y)), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x - Float64(Float64(x - 1.0) / y))
	tmp = 0.0
	if (y <= -4.7e+15)
		tmp = t_0;
	elseif (y <= 180000000.0)
		tmp = fma(y, Float64(Float64(1.0 - x) / Float64(-1.0 - y)), 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.7e+15], t$95$0, If[LessEqual[y, 180000000.0], N[(y * N[(N[(1.0 - x), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.7e15 or 1.8e8 < y

   1. Initial program 25.5%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if -4.7e15 < y < 1.8e8

   1. Initial program 99.4%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 99.4

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

   4. Applied rewrites 99.4%

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

Alternative 5: 98.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{x - 1}{y}\\ \mathbf{if}\;y \leq -3250:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1100000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{y - -1}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ (- x 1.0) y))))
   (if (<= y -3250.0)
     t_0
     (if (<= y 1100000.0) (fma y (/ x (- y -1.0)) 1.0) t_0))))

C

double code(double x, double y) {
	double t_0 = x - ((x - 1.0) / y);
	double tmp;
	if (y <= -3250.0) {
		tmp = t_0;
	} else if (y <= 1100000.0) {
		tmp = fma(y, (x / (y - -1.0)), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x - Float64(Float64(x - 1.0) / y))
	tmp = 0.0
	if (y <= -3250.0)
		tmp = t_0;
	elseif (y <= 1100000.0)
		tmp = fma(y, Float64(x / Float64(y - -1.0)), 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3250.0], t$95$0, If[LessEqual[y, 1100000.0], N[(y * N[(x / N[(y - -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3250 or 1.1e6 < y

   1. Initial program 27.3%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 98.6

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

   5. Applied rewrites 98.6%

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

   if -3250 < y < 1.1e6

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 98.7

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

   7. Applied rewrites 98.7%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{x}{1 + y}}, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3250:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{elif}\;y \leq 1100000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{y - -1}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{x - 1}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\left(y - 1\right) \cdot \left(1 - x\right), y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ (- x 1.0) y))))
   (if (<= y -1.0)
     t_0
     (if (<= y 1.0) (fma (* (- y 1.0) (- 1.0 x)) y 1.0) t_0))))

C

double code(double x, double y) {
	double t_0 = x - ((x - 1.0) / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = fma(((y - 1.0) * (1.0 - x)), y, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 29.5%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 97.0

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

   5. Applied rewrites 97.0%

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

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 99.4%

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

4. Final simplification 98.2%

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

Alternative 7: 98.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{x - 1}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(x - 1, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ (- x 1.0) y))))
   (if (<= y -1.0) t_0 (if (<= y 1.0) (fma (- x 1.0) y 1.0) t_0))))

C

double code(double x, double y) {
	double t_0 = x - ((x - 1.0) / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = fma((x - 1.0), y, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 29.5%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 97.0

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

   5. Applied rewrites 97.0%

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

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 99.0

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

   5. Applied rewrites 99.0%

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

Alternative 8: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{x}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.1:\\ \;\;\;\;\mathsf{fma}\left(x - 1, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ x y))))
   (if (<= y -1.0) t_0 (if (<= y 1.1) (fma (- x 1.0) y 1.0) t_0))))

C

double code(double x, double y) {
	double t_0 = x - (x / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.1) {
		tmp = fma((x - 1.0), y, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x - Float64(x / y))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 1.1)
		tmp = fma(Float64(x - 1.0), y, 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x - N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 1.1], N[(N[(x - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.1000000000000001 < y

   1. Initial program 29.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 75.2

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

   5. Applied rewrites 75.2%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 73.6%

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

   if -1 < y < 1.1000000000000001

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 99.0

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

   5. Applied rewrites 99.0%

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

Alternative 9: 74.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \left(1 - x\right)\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(x - 1, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (- 1.0 x))))
   (if (<= y -1.0) t_0 (if (<= y 1.0) (fma (- x 1.0) y 1.0) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (1.0 - x);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = fma((x - 1.0), y, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(1.0 - x))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = fma(Float64(x - 1.0), y, 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 29.5%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 56.3

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

   5. Applied rewrites 56.3%

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

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 99.0

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

   5. Applied rewrites 99.0%

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

Alternative 10: 63.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \left(1 - x\right)\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.74:\\ \;\;\;\;\mathsf{fma}\left(y - 1, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (- 1.0 x))))
   (if (<= y -1.0) t_0 (if (<= y 0.74) (fma (- y 1.0) y 1.0) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (1.0 - x);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 0.74) {
		tmp = fma((y - 1.0), y, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(1.0 - x))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 0.74)
		tmp = fma(Float64(y - 1.0), y, 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(1.0 - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 0.74], N[(N[(y - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.73999999999999999 < y

   1. Initial program 29.5%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 56.3

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

   5. Applied rewrites 56.3%

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

   if -1 < y < 0.73999999999999999

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(y - 1, y, 1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 80.8%

      \[\leadsto \mathsf{fma}\left(y - 1, y, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 49.1% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.2%

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

3. Taylor expanded in y around inf

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

   1. lower--.f64 30.2

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

5. Applied rewrites 30.2%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 56.2%

   \[\leadsto 1 - \left(-x\right) \]
9. Add Preprocessing

Alternative 12: 3.1% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.2%

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

3. Taylor expanded in y around inf

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

   1. lower--.f64 30.2

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

5. Applied rewrites 30.2%

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

6. Taylor expanded in x around 0

   \[\leadsto 1 - 1 \]
7. Step-by-step derivation

8. Applied rewrites 3.0%

   \[\leadsto 1 - 1 \]
9. Add Preprocessing

Developer Target 1: 99.6% accurate, 0.6× 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 2024327 
(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))))