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

Percentage Accurate: 66.5% → 99.5%

Time: 8.9s

Alternatives: 11

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: 66.5% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+21}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{elif}\;y \leq 200000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{x + -1}{y} \cdot \left(-1 + \frac{1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.45e+21)
   (+ x (/ 1.0 y))
   (if (<= y 200000.0)
     (+ 1.0 (/ (* y (- 1.0 x)) (- -1.0 y)))
     (+ x (* (/ (+ x -1.0) y) (+ -1.0 (/ 1.0 y)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.45e+21) {
		tmp = x + (1.0 / y);
	} else if (y <= 200000.0) {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	} else {
		tmp = x + (((x + -1.0) / y) * (-1.0 + (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 <= (-1.45d+21)) then
        tmp = x + (1.0d0 / y)
    else if (y <= 200000.0d0) then
        tmp = 1.0d0 + ((y * (1.0d0 - x)) / ((-1.0d0) - y))
    else
        tmp = x + (((x + (-1.0d0)) / y) * ((-1.0d0) + (1.0d0 / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.45e+21) {
		tmp = x + (1.0 / y);
	} else if (y <= 200000.0) {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	} else {
		tmp = x + (((x + -1.0) / y) * (-1.0 + (1.0 / y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.45e+21:
		tmp = x + (1.0 / y)
	elif y <= 200000.0:
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y))
	else:
		tmp = x + (((x + -1.0) / y) * (-1.0 + (1.0 / y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.45e+21)
		tmp = Float64(x + Float64(1.0 / y));
	elseif (y <= 200000.0)
		tmp = Float64(1.0 + Float64(Float64(y * Float64(1.0 - x)) / Float64(-1.0 - y)));
	else
		tmp = Float64(x + Float64(Float64(Float64(x + -1.0) / y) * Float64(-1.0 + Float64(1.0 / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.45e+21)
		tmp = x + (1.0 / y);
	elseif (y <= 200000.0)
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	else
		tmp = x + (((x + -1.0) / y) * (-1.0 + (1.0 / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.45e+21], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 200000.0], N[(1.0 + N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision] * N[(-1.0 + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.45e21

   1. Initial program 28.3%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate--r- N/A

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

      6. div-sub N/A

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

      7. neg-sub0 N/A

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

      8. mul-1-neg N/A

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

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(-1 \cdot \frac{x - 1}{y}\right)}\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{-1 \cdot \left(x - 1\right)}{\color{blue}{y}}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(-1 \cdot \left(x - 1\right)\right), \color{blue}{y}\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(x - 1\right)\right)\right), y\right)\right) \]

      13. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(0 - \left(x - 1\right)\right), y\right)\right) \]

      14. associate-+l- N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(\left(0 - x\right) + 1\right), y\right)\right) \]

      15. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(x\right)\right) + 1\right), y\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(x\right)\right)\right), y\right)\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(1 - x\right), y\right)\right) \]

      18. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, x\right), y\right)\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{1}{y}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{y}\right)\right) \]

   8. Simplified 100.0%

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

   if -1.45e21 < y < 2e5

   1. Initial program 100.0%

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

   if 2e5 < y

   1. Initial program 36.2%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+21}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{elif}\;y \leq 200000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{x + -1}{y} \cdot \left(-1 + \frac{1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+21}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{elif}\;y \leq 130000000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.45e+21)
   (+ x (/ 1.0 y))
   (if (<= y 130000000.0)
     (+ 1.0 (/ (* y (- 1.0 x)) (- -1.0 y)))
     (+ x (/ (- 1.0 x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.45e+21) {
		tmp = x + (1.0 / y);
	} else if (y <= 130000000.0) {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	} else {
		tmp = x + ((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 <= (-1.45d+21)) then
        tmp = x + (1.0d0 / y)
    else if (y <= 130000000.0d0) then
        tmp = 1.0d0 + ((y * (1.0d0 - x)) / ((-1.0d0) - y))
    else
        tmp = x + ((1.0d0 - x) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.45e+21) {
		tmp = x + (1.0 / y);
	} else if (y <= 130000000.0) {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	} else {
		tmp = x + ((1.0 - x) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.45e+21:
		tmp = x + (1.0 / y)
	elif y <= 130000000.0:
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y))
	else:
		tmp = x + ((1.0 - x) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.45e+21)
		tmp = Float64(x + Float64(1.0 / y));
	elseif (y <= 130000000.0)
		tmp = Float64(1.0 + Float64(Float64(y * Float64(1.0 - x)) / Float64(-1.0 - y)));
	else
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.45e+21)
		tmp = x + (1.0 / y);
	elseif (y <= 130000000.0)
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	else
		tmp = x + ((1.0 - x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.45e+21], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 130000000.0], N[(1.0 + N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.45e21

   1. Initial program 28.3%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate--r- N/A

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

      6. div-sub N/A

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

      7. neg-sub0 N/A

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

      8. mul-1-neg N/A

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

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(-1 \cdot \frac{x - 1}{y}\right)}\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{-1 \cdot \left(x - 1\right)}{\color{blue}{y}}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(-1 \cdot \left(x - 1\right)\right), \color{blue}{y}\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(x - 1\right)\right)\right), y\right)\right) \]

      13. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(0 - \left(x - 1\right)\right), y\right)\right) \]

      14. associate-+l- N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(\left(0 - x\right) + 1\right), y\right)\right) \]

      15. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(x\right)\right) + 1\right), y\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(x\right)\right)\right), y\right)\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(1 - x\right), y\right)\right) \]

      18. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, x\right), y\right)\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{1}{y}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{y}\right)\right) \]

   8. Simplified 100.0%

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

   if -1.45e21 < y < 1.3e8

   1. Initial program 100.0%

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

   if 1.3e8 < y

   1. Initial program 36.2%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate--r- N/A

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

      6. div-sub N/A

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

      7. neg-sub0 N/A

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

      8. mul-1-neg N/A

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

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(-1 \cdot \frac{x - 1}{y}\right)}\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{-1 \cdot \left(x - 1\right)}{\color{blue}{y}}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(-1 \cdot \left(x - 1\right)\right), \color{blue}{y}\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(x - 1\right)\right)\right), y\right)\right) \]

      13. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(0 - \left(x - 1\right)\right), y\right)\right) \]

      14. associate-+l- N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(\left(0 - x\right) + 1\right), y\right)\right) \]

      15. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(x\right)\right) + 1\right), y\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(x\right)\right)\right), y\right)\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(1 - x\right), y\right)\right) \]

      18. --lowering--.f64 99.5%

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, x\right), y\right)\right) \]

   5. Simplified 99.5%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+21}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{elif}\;y \leq 130000000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 33.9%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate--r- N/A

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

      6. div-sub N/A

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

      7. neg-sub0 N/A

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

      8. mul-1-neg N/A

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

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(-1 \cdot \frac{x - 1}{y}\right)}\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{-1 \cdot \left(x - 1\right)}{\color{blue}{y}}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(-1 \cdot \left(x - 1\right)\right), \color{blue}{y}\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(x - 1\right)\right)\right), y\right)\right) \]

      13. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(0 - \left(x - 1\right)\right), y\right)\right) \]

      14. associate-+l- N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(\left(0 - x\right) + 1\right), y\right)\right) \]

      15. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(x\right)\right) + 1\right), y\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(x\right)\right)\right), y\right)\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(1 - x\right), y\right)\right) \]

      18. --lowering--.f64 99.3%

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, x\right), y\right)\right) \]

   5. Simplified 99.3%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(y \cdot \left(\left(1 + y \cdot \left(x - 1\right)\right) - x\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(\left(1 + y \cdot \left(x - 1\right)\right) - x\right)}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \left(1 + \color{blue}{\left(y \cdot \left(x - 1\right) - x\right)}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \left(\left(y \cdot \left(x - 1\right) - x\right) + \color{blue}{1}\right)\right)\right) \]

      4. associate-+l- N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \left(x - 1\right) - \color{blue}{\left(x - 1\right)}\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \left(x - 1\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - 1\right)\right)\right)}\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \left(x - 1\right) + -1 \cdot \color{blue}{\left(x - 1\right)}\right)\right)\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \left(\left(x - 1\right) \cdot \color{blue}{\left(y + -1\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(x - 1\right), \color{blue}{\left(y + -1\right)}\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\color{blue}{y} + -1\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(x + -1\right), \left(y + -1\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(\color{blue}{y} + -1\right)\right)\right)\right) \]

      12. +-lowering-+.f64 99.8%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(y, \color{blue}{-1}\right)\right)\right)\right) \]

   5. Simplified 99.8%

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

4. Final simplification 99.5%

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

Alternative 4: 98.2% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 33.9%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate--r- N/A

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

      6. div-sub N/A

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

      7. neg-sub0 N/A

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

      8. mul-1-neg N/A

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

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(-1 \cdot \frac{x - 1}{y}\right)}\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{-1 \cdot \left(x - 1\right)}{\color{blue}{y}}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(-1 \cdot \left(x - 1\right)\right), \color{blue}{y}\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(x - 1\right)\right)\right), y\right)\right) \]

      13. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(0 - \left(x - 1\right)\right), y\right)\right) \]

      14. associate-+l- N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(\left(0 - x\right) + 1\right), y\right)\right) \]

      15. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(x\right)\right) + 1\right), y\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(x\right)\right)\right), y\right)\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(1 - x\right), y\right)\right) \]

      18. --lowering--.f64 99.3%

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, x\right), y\right)\right) \]

   5. Simplified 99.3%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(y \cdot \left(1 - x\right)\right)}\right) \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(y \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(y \cdot \left(\left(\mathsf{neg}\left(x\right)\right) + \color{blue}{1}\right)\right)\right) \]

      3. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(y \cdot \left(\left(0 - x\right) + 1\right)\right)\right) \]

      4. associate-+l- N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(y \cdot \left(0 - \color{blue}{\left(x - 1\right)}\right)\right)\right) \]

      5. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(y \cdot \left(\mathsf{neg}\left(\left(x - 1\right)\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(y \cdot \left(-1 \cdot \color{blue}{\left(x - 1\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(-1 \cdot \left(x - 1\right)\right)}\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \left(\mathsf{neg}\left(\left(x - 1\right)\right)\right)\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \left(0 - \color{blue}{\left(x - 1\right)}\right)\right)\right) \]

      10. associate-+l- N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \left(\left(0 - x\right) + \color{blue}{1}\right)\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \left(\left(\mathsf{neg}\left(x\right)\right) + 1\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \left(1 + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \left(1 - \color{blue}{x}\right)\right)\right) \]

      14. --lowering--.f64 99.3%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \color{blue}{x}\right)\right)\right) \]

   5. Simplified 99.3%

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

4. Final simplification 99.3%

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

Alternative 5: 97.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{1 - x}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.15:\\ \;\;\;\;1 + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (/ (- 1.0 x) y))))
   (if (<= y -1.0) t_0 (if (<= y 1.15) (+ 1.0 (* y x)) t_0))))

C

double code(double x, double y) {
	double t_0 = x + ((1.0 - x) / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.15) {
		tmp = 1.0 + (y * x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + ((1.0d0 - x) / y)
    if (y <= (-1.0d0)) then
        tmp = t_0
    else if (y <= 1.15d0) then
        tmp = 1.0d0 + (y * x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x + ((1.0 - x) / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.15) {
		tmp = 1.0 + (y * x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + ((1.0 - x) / y)
	tmp = 0
	if y <= -1.0:
		tmp = t_0
	elif y <= 1.15:
		tmp = 1.0 + (y * x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(Float64(1.0 - x) / y))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 1.15)
		tmp = Float64(1.0 + Float64(y * x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + ((1.0 - x) / y);
	tmp = 0.0;
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 1.15)
		tmp = 1.0 + (y * x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 1.15], N[(1.0 + N[(y * x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.1499999999999999 < y

   1. Initial program 33.9%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate--r- N/A

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

      6. div-sub N/A

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

      7. neg-sub0 N/A

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

      8. mul-1-neg N/A

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

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(-1 \cdot \frac{x - 1}{y}\right)}\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{-1 \cdot \left(x - 1\right)}{\color{blue}{y}}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(-1 \cdot \left(x - 1\right)\right), \color{blue}{y}\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(x - 1\right)\right)\right), y\right)\right) \]

      13. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(0 - \left(x - 1\right)\right), y\right)\right) \]

      14. associate-+l- N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(\left(0 - x\right) + 1\right), y\right)\right) \]

      15. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(x\right)\right) + 1\right), y\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(x\right)\right)\right), y\right)\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(1 - x\right), y\right)\right) \]

      18. --lowering--.f64 99.3%

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, x\right), y\right)\right) \]

   5. Simplified 99.3%

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

   if -1 < y < 1.1499999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(-1 \cdot \frac{x \cdot y}{1 + y}\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(\frac{x \cdot y}{1 + y}\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(x \cdot \frac{y}{1 + y}\right)\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{y}{1 + y}\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{y}{1 + y}\right)\right)}\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{y}{\color{blue}{\mathsf{neg}\left(\left(1 + y\right)\right)}}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(\left(1 + y\right)\right)\right)}\right)\right)\right) \]

      7. distribute-neg-in N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \left(-1 + \left(\mathsf{neg}\left(\color{blue}{y}\right)\right)\right)\right)\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \left(-1 - \color{blue}{y}\right)\right)\right)\right) \]

      10. --lowering--.f64 98.3%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(-1, \color{blue}{y}\right)\right)\right)\right) \]

   5. Simplified 98.3%

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

   6. Taylor expanded in y around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot y\right)}\right) \]

      2. *-lowering-*.f64 98.2%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right) \]

   8. Simplified 98.2%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 1.15:\\ \;\;\;\;1 + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (1.0d0 / y)
    if (y <= (-1.0d0)) then
        tmp = t_0
    else if (y <= 1.0d0) then
        tmp = 1.0d0 + (y * x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 33.9%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate--r- N/A

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

      6. div-sub N/A

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

      7. neg-sub0 N/A

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

      8. mul-1-neg N/A

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

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(-1 \cdot \frac{x - 1}{y}\right)}\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{-1 \cdot \left(x - 1\right)}{\color{blue}{y}}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(-1 \cdot \left(x - 1\right)\right), \color{blue}{y}\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(x - 1\right)\right)\right), y\right)\right) \]

      13. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(0 - \left(x - 1\right)\right), y\right)\right) \]

      14. associate-+l- N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(\left(0 - x\right) + 1\right), y\right)\right) \]

      15. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(x\right)\right) + 1\right), y\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(x\right)\right)\right), y\right)\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(1 - x\right), y\right)\right) \]

      18. --lowering--.f64 99.3%

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, x\right), y\right)\right) \]

   5. Simplified 99.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{1}{y}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 98.9%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{y}\right)\right) \]

   8. Simplified 98.9%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(-1 \cdot \frac{x \cdot y}{1 + y}\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(\frac{x \cdot y}{1 + y}\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(x \cdot \frac{y}{1 + y}\right)\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{y}{1 + y}\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{y}{1 + y}\right)\right)}\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{y}{\color{blue}{\mathsf{neg}\left(\left(1 + y\right)\right)}}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(\left(1 + y\right)\right)\right)}\right)\right)\right) \]

      7. distribute-neg-in N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \left(-1 + \left(\mathsf{neg}\left(\color{blue}{y}\right)\right)\right)\right)\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \left(-1 - \color{blue}{y}\right)\right)\right)\right) \]

      10. --lowering--.f64 98.3%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(-1, \color{blue}{y}\right)\right)\right)\right) \]

   5. Simplified 98.3%

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

   6. Taylor expanded in y around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot y\right)}\right) \]

      2. *-lowering-*.f64 98.2%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right) \]

   8. Simplified 98.2%

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

4. Final simplification 98.5%

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

Alternative 7: 86.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 85:\\ \;\;\;\;1 + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 85.0) (+ 1.0 (* y x)) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 85.0) {
		tmp = 1.0 + (y * x);
	} 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 <= 85.0d0) then
        tmp = 1.0d0 + (y * x)
    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 <= 85.0) {
		tmp = 1.0 + (y * x);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 85.0:
		tmp = 1.0 + (y * x)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 85.0)
		tmp = Float64(1.0 + Float64(y * x));
	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 <= 85.0)
		tmp = 1.0 + (y * x);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 85.0], N[(1.0 + N[(y * x), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 85 < y

   1. Initial program 33.9%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 77.4%

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

   if -1 < y < 85

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(-1 \cdot \frac{x \cdot y}{1 + y}\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(\frac{x \cdot y}{1 + y}\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(x \cdot \frac{y}{1 + y}\right)\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{y}{1 + y}\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{y}{1 + y}\right)\right)}\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{y}{\color{blue}{\mathsf{neg}\left(\left(1 + y\right)\right)}}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(\left(1 + y\right)\right)\right)}\right)\right)\right) \]

      7. distribute-neg-in N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \left(-1 + \left(\mathsf{neg}\left(\color{blue}{y}\right)\right)\right)\right)\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \left(-1 - \color{blue}{y}\right)\right)\right)\right) \]

      10. --lowering--.f64 98.3%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(-1, \color{blue}{y}\right)\right)\right)\right) \]

   5. Simplified 98.3%

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

   6. Taylor expanded in y around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot y\right)}\right) \]

      2. *-lowering-*.f64 98.2%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right) \]

   8. Simplified 98.2%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 85:\\ \;\;\;\;1 + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-12}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 8e-12) (- 1.0 y) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 8e-12) {
		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 <= 8d-12) 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 <= 8e-12) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 8e-12:
		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 <= 8e-12)
		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 <= 8e-12)
		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, 8e-12], N[(1.0 - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 7.99999999999999984e-12 < y

   1. Initial program 34.4%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 76.9%

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

   if -1 < y < 7.99999999999999984e-12

   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

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(y \cdot \left(1 - x\right)\right)}\right) \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(y \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(y \cdot \left(\left(\mathsf{neg}\left(x\right)\right) + \color{blue}{1}\right)\right)\right) \]

      3. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(y \cdot \left(\left(0 - x\right) + 1\right)\right)\right) \]

      4. associate-+l- N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(y \cdot \left(0 - \color{blue}{\left(x - 1\right)}\right)\right)\right) \]

      5. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(y \cdot \left(\mathsf{neg}\left(\left(x - 1\right)\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(y \cdot \left(-1 \cdot \color{blue}{\left(x - 1\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(-1 \cdot \left(x - 1\right)\right)}\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \left(\mathsf{neg}\left(\left(x - 1\right)\right)\right)\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \left(0 - \color{blue}{\left(x - 1\right)}\right)\right)\right) \]

      10. associate-+l- N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \left(\left(0 - x\right) + \color{blue}{1}\right)\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \left(\left(\mathsf{neg}\left(x\right)\right) + 1\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \left(1 + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \left(1 - \color{blue}{x}\right)\right)\right) \]

      14. --lowering--.f64 99.5%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \color{blue}{x}\right)\right)\right) \]

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0

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

      1. --lowering--.f64 83.3%

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{y}\right) \]

   8. Simplified 83.3%

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

Alternative 9: 73.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-12}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1.0) x (if (<= y 8e-12) 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 8e-12) {
		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 <= 8d-12) 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 <= 8e-12) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 8e-12:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 8e-12)
		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 <= 8e-12)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 8e-12], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 7.99999999999999984e-12 < y

   1. Initial program 34.4%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 76.9%

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

   if -1 < y < 7.99999999999999984e-12

   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

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

   5. Simplified 82.2%

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

Alternative 10: 38.9% 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 67.4%

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

3. Taylor expanded in y around 0

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

5. Simplified 43.2%

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

Alternative 11: 3.1% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.0)

C

double code(double x, double y) {
	return 0.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.0d0
end function

Java

public static double code(double x, double y) {
	return 0.0;
}

Python

def code(x, y):
	return 0.0

Julia

function code(x, y)
	return 0.0
end

MATLAB

function tmp = code(x, y)
	tmp = 0.0;
end

Wolfram

code[x_, y_] := 0.0

Derivation

1. Initial program 67.4%

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

3. Taylor expanded in y around inf

   \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(1 - x\right)}\right) \]
4. Step-by-step derivation

   1. --lowering--.f64 26.9%

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{\_.f64}\left(1, \color{blue}{x}\right)\right) \]

5. Simplified 26.9%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{1}\right) \]
7. Step-by-step derivation

8. Simplified 3.2%

   \[\leadsto 1 - \color{blue}{1} \]
9. Step-by-step derivation

   1. metadata-eval 3.2%

      \[\leadsto 0 \]

10. Applied egg-rr 3.2%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Developer Target 1: 99.6% 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 2024163 
(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))))