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

Percentage Accurate: 66.2% → 99.7%

Time: 11.3s

Alternatives: 10

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: 66.2% 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: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{1 - x}{y}\\ \mathbf{if}\;y \leq -170000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 175000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{y \cdot \left(1 - x\right)}{y \cdot y + -1}, 1 - y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (/ (- 1.0 x) y))))
   (if (<= y -170000000.0)
     t_0
     (if (<= y 175000000.0)
       (fma (/ (* y (- 1.0 x)) (+ (* y y) -1.0)) (- 1.0 y) 1.0)
       t_0))))

C

double code(double x, double y) {
	double t_0 = x + ((1.0 - x) / y);
	double tmp;
	if (y <= -170000000.0) {
		tmp = t_0;
	} else if (y <= 175000000.0) {
		tmp = fma(((y * (1.0 - x)) / ((y * y) + -1.0)), (1.0 - 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 <= -170000000.0)
		tmp = t_0;
	elseif (y <= 175000000.0)
		tmp = fma(Float64(Float64(y * Float64(1.0 - x)) / Float64(Float64(y * y) + -1.0)), 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[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -170000000.0], t$95$0, If[LessEqual[y, 175000000.0], N[(N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(N[(y * y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 - y), $MachinePrecision] + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.7e8 or 1.75e8 < y

   1. Initial program 31.0%

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

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

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

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

      9. associate-*r/ N/A

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

      10. /-lowering-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. sub-neg N/A

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

      17. --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%

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

   if -1.7e8 < y < 1.75e8

   1. Initial program 99.7%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. associate-/r/ N/A

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. *-rgt-identity N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -170000000:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 175000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{y \cdot \left(1 - x\right)}{y \cdot y + -1}, 1 - y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - x}{y}\\ \mathbf{if}\;y \leq -170000000:\\ \;\;\;\;x + t\_0\\ \mathbf{elif}\;y \leq 14000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(1 - x\right) - t\_0 \cdot \left(1 + \frac{-1}{y}\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 x) y)))
   (if (<= y -170000000.0)
     (+ x t_0)
     (if (<= y 14000.0)
       (+ 1.0 (/ (* y (- 1.0 x)) (- -1.0 y)))
       (+ x (/ (- (- 1.0 x) (* t_0 (+ 1.0 (/ -1.0 y)))) y))))))

C

double code(double x, double y) {
	double t_0 = (1.0 - x) / y;
	double tmp;
	if (y <= -170000000.0) {
		tmp = x + t_0;
	} else if (y <= 14000.0) {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	} else {
		tmp = x + (((1.0 - x) - (t_0 * (1.0 + (-1.0 / y)))) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 - x) / y
    if (y <= (-170000000.0d0)) then
        tmp = x + t_0
    else if (y <= 14000.0d0) then
        tmp = 1.0d0 + ((y * (1.0d0 - x)) / ((-1.0d0) - y))
    else
        tmp = x + (((1.0d0 - x) - (t_0 * (1.0d0 + ((-1.0d0) / y)))) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (1.0 - x) / y;
	double tmp;
	if (y <= -170000000.0) {
		tmp = x + t_0;
	} else if (y <= 14000.0) {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	} else {
		tmp = x + (((1.0 - x) - (t_0 * (1.0 + (-1.0 / y)))) / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.0 - x) / y
	tmp = 0
	if y <= -170000000.0:
		tmp = x + t_0
	elif y <= 14000.0:
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y))
	else:
		tmp = x + (((1.0 - x) - (t_0 * (1.0 + (-1.0 / y)))) / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 - x) / y)
	tmp = 0.0
	if (y <= -170000000.0)
		tmp = Float64(x + t_0);
	elseif (y <= 14000.0)
		tmp = Float64(1.0 + Float64(Float64(y * Float64(1.0 - x)) / Float64(-1.0 - y)));
	else
		tmp = Float64(x + Float64(Float64(Float64(1.0 - x) - Float64(t_0 * Float64(1.0 + Float64(-1.0 / y)))) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (1.0 - x) / y;
	tmp = 0.0;
	if (y <= -170000000.0)
		tmp = x + t_0;
	elseif (y <= 14000.0)
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	else
		tmp = x + (((1.0 - x) - (t_0 * (1.0 + (-1.0 / y)))) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -170000000.0], N[(x + t$95$0), $MachinePrecision], If[LessEqual[y, 14000.0], N[(1.0 + N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(1.0 - x), $MachinePrecision] - N[(t$95$0 * N[(1.0 + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.7e8

   1. Initial program 35.1%

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

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

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

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

      9. associate-*r/ N/A

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

      10. /-lowering-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. sub-neg N/A

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

      17. --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%

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

   if -1.7e8 < y < 14000

   1. Initial program 100.0%

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

   if 14000 < y

   1. Initial program 29.3%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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. Simplified 99.6%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -170000000:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 14000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(1 - x\right) - \frac{1 - x}{y} \cdot \left(1 + \frac{-1}{y}\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{1 - x}{y}\\ \mathbf{if}\;y \leq -155000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 210000000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (/ (- 1.0 x) y))))
   (if (<= y -155000000.0)
     t_0
     (if (<= y 210000000.0) (+ 1.0 (/ (* y (- 1.0 x)) (- -1.0 y))) t_0))))

C

double code(double x, double y) {
	double t_0 = x + ((1.0 - x) / y);
	double tmp;
	if (y <= -155000000.0) {
		tmp = t_0;
	} else if (y <= 210000000.0) {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + ((1.0d0 - x) / y)
    if (y <= (-155000000.0d0)) then
        tmp = t_0
    else if (y <= 210000000.0d0) then
        tmp = 1.0d0 + ((y * (1.0d0 - x)) / ((-1.0d0) - y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x + ((1.0 - x) / y);
	double tmp;
	if (y <= -155000000.0) {
		tmp = t_0;
	} else if (y <= 210000000.0) {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + ((1.0 - x) / y)
	tmp = 0
	if y <= -155000000.0:
		tmp = t_0
	elif y <= 210000000.0:
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y))
	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 <= -155000000.0)
		tmp = t_0;
	elseif (y <= 210000000.0)
		tmp = Float64(1.0 + Float64(Float64(y * Float64(1.0 - x)) / Float64(-1.0 - y)));
	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 <= -155000000.0)
		tmp = t_0;
	elseif (y <= 210000000.0)
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	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, -155000000.0], t$95$0, If[LessEqual[y, 210000000.0], N[(1.0 + N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.55e8 or 2.1e8 < y

   1. Initial program 31.0%

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

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

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

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

      9. associate-*r/ N/A

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

      10. /-lowering-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. sub-neg N/A

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

      17. --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%

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

   if -1.55e8 < y < 2.1e8

   1. Initial program 99.7%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -155000000:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 210000000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% 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 32.9%

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

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

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

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

      9. associate-*r/ N/A

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

      10. /-lowering-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. sub-neg N/A

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

      17. --lowering--.f64 97.9%

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

   5. Simplified 97.9%

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

   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 97.1%

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

   5. Simplified 97.1%

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

4. Final simplification 97.5%

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

Alternative 5: 86.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{1 - x}{y}\\ \mathbf{if}\;y \leq -125:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 420000:\\ \;\;\;\;\frac{1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (/ (- 1.0 x) y))))
   (if (<= y -125.0) t_0 (if (<= y 420000.0) (/ 1.0 (+ y 1.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = x + ((1.0 - x) / y);
	double tmp;
	if (y <= -125.0) {
		tmp = t_0;
	} else if (y <= 420000.0) {
		tmp = 1.0 / (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 <= (-125.0d0)) then
        tmp = t_0
    else if (y <= 420000.0d0) then
        tmp = 1.0d0 / (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 <= -125.0) {
		tmp = t_0;
	} else if (y <= 420000.0) {
		tmp = 1.0 / (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 <= -125.0:
		tmp = t_0
	elif y <= 420000.0:
		tmp = 1.0 / (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 <= -125.0)
		tmp = t_0;
	elseif (y <= 420000.0)
		tmp = Float64(1.0 / 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 <= -125.0)
		tmp = t_0;
	elseif (y <= 420000.0)
		tmp = 1.0 / (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, -125.0], t$95$0, If[LessEqual[y, 420000.0], N[(1.0 / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -125 or 4.2e5 < y

   1. Initial program 31.6%

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

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

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

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

      9. associate-*r/ N/A

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

      10. /-lowering-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. sub-neg N/A

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

      17. --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%

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

   if -125 < y < 4.2e5

   1. Initial program 99.7%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

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

      4. clear-num N/A

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

      5. flip-- N/A

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

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

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

      7. sub-neg N/A

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

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

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

      9. distribute-neg-frac2 N/A

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

      10. /-lowering-/.f64 N/A

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

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

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

      12. --lowering--.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. metadata-eval N/A

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in y around 0

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

      1. remove-double-neg N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. sub-neg N/A

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

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

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

      10. mul-1-neg N/A

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

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

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

      12. mul-1-neg N/A

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

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

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

      14. sub-neg N/A

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

      15. remove-double-neg N/A

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

      16. mul-1-neg N/A

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

      17. distribute-neg-in N/A

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

      18. +-commutative N/A

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

      19. mul-1-neg N/A

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

      20. sub-neg N/A

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

      21. remove-double-neg N/A

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

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

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

   7. Simplified 77.0%

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

   8. Taylor expanded in x around 0

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

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

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

      2. +-lowering-+.f64 77.0%

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

   10. Simplified 77.0%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -125:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 420000:\\ \;\;\;\;\frac{1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{1}{y}\\ \mathbf{if}\;y \leq -125:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 8500000:\\ \;\;\;\;\frac{1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (/ 1.0 y))))
   (if (<= y -125.0) t_0 (if (<= y 8500000.0) (/ 1.0 (+ y 1.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = x + (1.0 / y);
	double tmp;
	if (y <= -125.0) {
		tmp = t_0;
	} else if (y <= 8500000.0) {
		tmp = 1.0 / (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 / y)
    if (y <= (-125.0d0)) then
        tmp = t_0
    else if (y <= 8500000.0d0) then
        tmp = 1.0d0 / (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 / y);
	double tmp;
	if (y <= -125.0) {
		tmp = t_0;
	} else if (y <= 8500000.0) {
		tmp = 1.0 / (y + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + (1.0 / y)
	tmp = 0
	if y <= -125.0:
		tmp = t_0
	elif y <= 8500000.0:
		tmp = 1.0 / (y + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(1.0 / y))
	tmp = 0.0
	if (y <= -125.0)
		tmp = t_0;
	elseif (y <= 8500000.0)
		tmp = Float64(1.0 / Float64(y + 1.0));
	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 <= -125.0)
		tmp = t_0;
	elseif (y <= 8500000.0)
		tmp = 1.0 / (y + 1.0);
	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, -125.0], t$95$0, If[LessEqual[y, 8500000.0], N[(1.0 / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -125 or 8.5e6 < y

   1. Initial program 31.6%

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

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

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

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

      9. associate-*r/ N/A

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

      10. /-lowering-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. sub-neg N/A

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

      17. --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%

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

   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 98.4%

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

   8. Simplified 98.4%

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

   if -125 < y < 8.5e6

   1. Initial program 99.7%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

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

      4. clear-num N/A

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

      5. flip-- N/A

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

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

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

      7. sub-neg N/A

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

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

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

      9. distribute-neg-frac2 N/A

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

      10. /-lowering-/.f64 N/A

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

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

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

      12. --lowering--.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. metadata-eval N/A

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in y around 0

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

      1. remove-double-neg N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. sub-neg N/A

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

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

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

      10. mul-1-neg N/A

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

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

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

      12. mul-1-neg N/A

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

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

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

      14. sub-neg N/A

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

      15. remove-double-neg N/A

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

      16. mul-1-neg N/A

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

      17. distribute-neg-in N/A

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

      18. +-commutative N/A

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

      19. mul-1-neg N/A

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

      20. sub-neg N/A

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

      21. remove-double-neg N/A

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

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

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

   7. Simplified 77.0%

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

   8. Taylor expanded in x around 0

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

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

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

      2. +-lowering-+.f64 77.0%

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

   10. Simplified 77.0%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -125:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{elif}\;y \leq 8500000:\\ \;\;\;\;\frac{1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{1}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.112:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (/ 1.0 y))))
   (if (<= y -1.0) t_0 (if (<= y 0.112) (- 1.0 y) t_0))))

C

double code(double x, double y) {
	double t_0 = x + (1.0 / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 0.112) {
		tmp = 1.0 - y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (1.0d0 / y)
    if (y <= (-1.0d0)) then
        tmp = t_0
    else if (y <= 0.112d0) then
        tmp = 1.0d0 - y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x + (1.0 / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 0.112) {
		tmp = 1.0 - y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + (1.0 / y)
	tmp = 0
	if y <= -1.0:
		tmp = t_0
	elif y <= 0.112:
		tmp = 1.0 - y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(1.0 / y))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 0.112)
		tmp = Float64(1.0 - y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + (1.0 / y);
	tmp = 0.0;
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 0.112)
		tmp = 1.0 - y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 0.112], N[(1.0 - y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.112000000000000002 < y

   1. Initial program 32.9%

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

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

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

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

      9. associate-*r/ N/A

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

      10. /-lowering-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. sub-neg N/A

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

      17. --lowering--.f64 97.9%

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

   5. Simplified 97.9%

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

   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 96.9%

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

   8. Simplified 96.9%

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

   if -1 < y < 0.112000000000000002

   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 97.1%

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

   5. Simplified 97.1%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 74.3%

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

Alternative 8: 74.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.76:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 0.76) (- 1.0 y) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.76) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x
    else if (y <= 0.76d0) then
        tmp = 1.0d0 - y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.76) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 0.76:
		tmp = 1.0 - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 0.76)
		tmp = Float64(1.0 - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 0.76)
		tmp = 1.0 - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 0.76], N[(1.0 - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.76000000000000001 < y

   1. Initial program 32.9%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 71.8%

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

   if -1 < y < 0.76000000000000001

   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 97.1%

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

   5. Simplified 97.1%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 74.3%

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

Alternative 9: 74.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 28000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1.0) x (if (<= y 28000.0) 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 28000.0) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x
    else if (y <= 28000.0d0) then
        tmp = 1.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 28000.0) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 28000.0:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 28000.0)
		tmp = 1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 28000.0)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 28000.0], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 28000 < y

   1. Initial program 31.6%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 73.5%

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

   if -1 < y < 28000

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 72.4%

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

Alternative 10: 39.1% accurate, 11.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 67.0%

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

3. Taylor expanded in y around 0

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

5. Simplified 39.5%

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

Developer Target 1: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / y) - ((x / y) - x)
    if (y < (-3693.8482788297247d0)) then
        tmp = t_0
    else if (y < 6799310503.41891d0) then
        tmp = 1.0d0 - (((1.0d0 - x) * y) / (y + 1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024192 
(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))))