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

Percentage Accurate: 64.9% → 99.8%

Time: 12.4s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 64.9% 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: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - x}{y}\\ \mathbf{if}\;y \leq -1800000:\\ \;\;\;\;x + \left(t\_0 - \frac{t\_0}{y}\right)\\ \mathbf{elif}\;y \leq 140000000:\\ \;\;\;\;1 + \left(1 - x\right) \cdot \frac{y}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 x) y)))
   (if (<= y -1800000.0)
     (+ x (- t_0 (/ t_0 y)))
     (if (<= y 140000000.0)
       (+ 1.0 (* (- 1.0 x) (/ y (- -1.0 y))))
       (+ x t_0)))))

C

double code(double x, double y) {
	double t_0 = (1.0 - x) / y;
	double tmp;
	if (y <= -1800000.0) {
		tmp = x + (t_0 - (t_0 / y));
	} else if (y <= 140000000.0) {
		tmp = 1.0 + ((1.0 - x) * (y / (-1.0 - y)));
	} else {
		tmp = x + 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 - x) / y
    if (y <= (-1800000.0d0)) then
        tmp = x + (t_0 - (t_0 / y))
    else if (y <= 140000000.0d0) then
        tmp = 1.0d0 + ((1.0d0 - x) * (y / ((-1.0d0) - y)))
    else
        tmp = x + t_0
    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 <= -1800000.0) {
		tmp = x + (t_0 - (t_0 / y));
	} else if (y <= 140000000.0) {
		tmp = 1.0 + ((1.0 - x) * (y / (-1.0 - y)));
	} else {
		tmp = x + t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.0 - x) / y
	tmp = 0
	if y <= -1800000.0:
		tmp = x + (t_0 - (t_0 / y))
	elif y <= 140000000.0:
		tmp = 1.0 + ((1.0 - x) * (y / (-1.0 - y)))
	else:
		tmp = x + t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 - x) / y)
	tmp = 0.0
	if (y <= -1800000.0)
		tmp = Float64(x + Float64(t_0 - Float64(t_0 / y)));
	elseif (y <= 140000000.0)
		tmp = Float64(1.0 + Float64(Float64(1.0 - x) * Float64(y / Float64(-1.0 - y))));
	else
		tmp = Float64(x + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (1.0 - x) / y;
	tmp = 0.0;
	if (y <= -1800000.0)
		tmp = x + (t_0 - (t_0 / y));
	elseif (y <= 140000000.0)
		tmp = 1.0 + ((1.0 - x) * (y / (-1.0 - y)));
	else
		tmp = x + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -1800000.0], N[(x + N[(t$95$0 - N[(t$95$0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 140000000.0], N[(1.0 + N[(N[(1.0 - x), $MachinePrecision] * N[(y / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.8e6

   1. Initial program 21.0%

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

      1. associate-/l* 52.6%

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

      2. remove-double-neg 52.6%

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

      3. remove-double-neg 52.6%

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

      4. +-commutative 52.6%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in y around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. neg-mul-1 100.0%

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

      3. sub-neg 100.0%

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

      4. associate--l+ 100.0%

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

      5. div-sub 100.0%

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

      6. sub-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. neg-sub0 100.0%

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

      9. associate-+l- 100.0%

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

      10. neg-sub0 100.0%

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

      11. mul-1-neg 100.0%

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

      12. associate-*r/ 100.0%

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

      13. +-commutative 100.0%

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

   7. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. unpow2 100.0%

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

      3. times-frac 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. associate-*l/ 100.0%

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

       2. +-commutative 100.0%

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

       3. metadata-eval 100.0%

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

       4. sub-neg 100.0%

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

       5. *-lft-identity 100.0%

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

       6. sub-neg 100.0%

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

       7. metadata-eval 100.0%

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

       8. +-commutative 100.0%

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

   11. Simplified 100.0%

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

   if -1.8e6 < y < 1.4e8

   1. Initial program 99.7%

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

      1. associate-/l* 99.8%

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

      2. remove-double-neg 99.8%

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

      3. remove-double-neg 99.8%

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

      4. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   if 1.4e8 < y

   1. Initial program 24.7%

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

      1. associate-/l* 49.6%

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

      2. remove-double-neg 49.6%

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

      3. remove-double-neg 49.6%

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

      4. +-commutative 49.6%

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

   3. Simplified 49.6%

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

   5. Taylor expanded in y around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. div-sub 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1800000:\\ \;\;\;\;x + \left(\frac{1 - x}{y} - \frac{\frac{1 - x}{y}}{y}\right)\\ \mathbf{elif}\;y \leq 140000000:\\ \;\;\;\;1 + \left(1 - x\right) \cdot \frac{y}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -220000000 \lor \neg \left(y \leq 140000000\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(1 - x\right) \cdot \frac{y}{-1 - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -220000000.0) (not (<= y 140000000.0)))
   (+ x (/ (- 1.0 x) y))
   (+ 1.0 (* (- 1.0 x) (/ y (- -1.0 y))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -220000000.0) || !(y <= 140000000.0)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = 1.0 + ((1.0 - x) * (y / (-1.0 - y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-220000000.0d0)) .or. (.not. (y <= 140000000.0d0))) then
        tmp = x + ((1.0d0 - x) / y)
    else
        tmp = 1.0d0 + ((1.0d0 - x) * (y / ((-1.0d0) - y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -220000000.0) || !(y <= 140000000.0)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = 1.0 + ((1.0 - x) * (y / (-1.0 - y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -220000000.0) or not (y <= 140000000.0):
		tmp = x + ((1.0 - x) / y)
	else:
		tmp = 1.0 + ((1.0 - x) * (y / (-1.0 - y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -220000000.0) || !(y <= 140000000.0))
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	else
		tmp = Float64(1.0 + Float64(Float64(1.0 - x) * Float64(y / Float64(-1.0 - y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -220000000.0) || ~((y <= 140000000.0)))
		tmp = x + ((1.0 - x) / y);
	else
		tmp = 1.0 + ((1.0 - x) * (y / (-1.0 - y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -220000000.0], N[Not[LessEqual[y, 140000000.0]], $MachinePrecision]], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(1.0 - x), $MachinePrecision] * N[(y / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.2e8 or 1.4e8 < y

   1. Initial program 22.7%

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

      1. associate-/l* 51.2%

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

      2. remove-double-neg 51.2%

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

      3. remove-double-neg 51.2%

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

      4. +-commutative 51.2%

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

   3. Simplified 51.2%

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

   5. Taylor expanded in y around inf 99.9%

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

      1. associate--l+ 99.9%

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

      2. div-sub 99.9%

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

   7. Simplified 99.9%

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

   if -2.2e8 < y < 1.4e8

   1. Initial program 99.7%

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

      1. associate-/l* 99.8%

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

      2. remove-double-neg 99.8%

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

      3. remove-double-neg 99.8%

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

      4. +-commutative 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -220000000 \lor \neg \left(y \leq 140000000\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(1 - x\right) \cdot \frac{y}{-1 - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1800000:\\ \;\;\;\;x + \frac{\left(1 - x\right) + \frac{x + -1}{y}}{y}\\ \mathbf{elif}\;y \leq 140000000:\\ \;\;\;\;1 + \left(1 - x\right) \cdot \frac{y}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1800000.0)
   (+ x (/ (+ (- 1.0 x) (/ (+ x -1.0) y)) y))
   (if (<= y 140000000.0)
     (+ 1.0 (* (- 1.0 x) (/ y (- -1.0 y))))
     (+ x (/ (- 1.0 x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1800000.0) {
		tmp = x + (((1.0 - x) + ((x + -1.0) / y)) / y);
	} else if (y <= 140000000.0) {
		tmp = 1.0 + ((1.0 - x) * (y / (-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 <= (-1800000.0d0)) then
        tmp = x + (((1.0d0 - x) + ((x + (-1.0d0)) / y)) / y)
    else if (y <= 140000000.0d0) then
        tmp = 1.0d0 + ((1.0d0 - x) * (y / ((-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 <= -1800000.0) {
		tmp = x + (((1.0 - x) + ((x + -1.0) / y)) / y);
	} else if (y <= 140000000.0) {
		tmp = 1.0 + ((1.0 - x) * (y / (-1.0 - y)));
	} else {
		tmp = x + ((1.0 - x) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1800000.0:
		tmp = x + (((1.0 - x) + ((x + -1.0) / y)) / y)
	elif y <= 140000000.0:
		tmp = 1.0 + ((1.0 - x) * (y / (-1.0 - y)))
	else:
		tmp = x + ((1.0 - x) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1800000.0)
		tmp = Float64(x + Float64(Float64(Float64(1.0 - x) + Float64(Float64(x + -1.0) / y)) / y));
	elseif (y <= 140000000.0)
		tmp = Float64(1.0 + Float64(Float64(1.0 - x) * Float64(y / 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 <= -1800000.0)
		tmp = x + (((1.0 - x) + ((x + -1.0) / y)) / y);
	elseif (y <= 140000000.0)
		tmp = 1.0 + ((1.0 - x) * (y / (-1.0 - y)));
	else
		tmp = x + ((1.0 - x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1800000.0], N[(x + N[(N[(N[(1.0 - x), $MachinePrecision] + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 140000000.0], N[(1.0 + N[(N[(1.0 - x), $MachinePrecision] * N[(y / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.8e6

   1. Initial program 21.0%

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

      1. associate-/l* 52.6%

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

      2. remove-double-neg 52.6%

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

      3. remove-double-neg 52.6%

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

      4. +-commutative 52.6%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in y around -inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. unsub-neg 99.9%

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

   7. Simplified 99.9%

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

   if -1.8e6 < y < 1.4e8

   1. Initial program 99.7%

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

      1. associate-/l* 99.8%

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

      2. remove-double-neg 99.8%

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

      3. remove-double-neg 99.8%

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

      4. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   if 1.4e8 < y

   1. Initial program 24.7%

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

      1. associate-/l* 49.6%

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

      2. remove-double-neg 49.6%

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

      3. remove-double-neg 49.6%

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

      4. +-commutative 49.6%

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

   3. Simplified 49.6%

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

   5. Taylor expanded in y around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. div-sub 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1800000:\\ \;\;\;\;x + \frac{\left(1 - x\right) + \frac{x + -1}{y}}{y}\\ \mathbf{elif}\;y \leq 140000000:\\ \;\;\;\;1 + \left(1 - x\right) \cdot \frac{y}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{x}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+52}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ x y))))
   (if (<= y -1.0)
     t_0
     (if (<= y 0.75) (- 1.0 y) (if (<= y 1.65e+52) (/ 1.0 y) t_0)))))

C

double code(double x, double y) {
	double t_0 = x - (x / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 0.75) {
		tmp = 1.0 - y;
	} else if (y <= 1.65e+52) {
		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 - (x / y)
    if (y <= (-1.0d0)) then
        tmp = t_0
    else if (y <= 0.75d0) then
        tmp = 1.0d0 - y
    else if (y <= 1.65d+52) 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 - (x / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 0.75) {
		tmp = 1.0 - y;
	} else if (y <= 1.65e+52) {
		tmp = 1.0 / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x - (x / y)
	tmp = 0
	if y <= -1.0:
		tmp = t_0
	elif y <= 0.75:
		tmp = 1.0 - y
	elif y <= 1.65e+52:
		tmp = 1.0 / y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x - Float64(x / y))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 0.75)
		tmp = Float64(1.0 - y);
	elseif (y <= 1.65e+52)
		tmp = Float64(1.0 / y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x - (x / y);
	tmp = 0.0;
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 0.75)
		tmp = 1.0 - y;
	elseif (y <= 1.65e+52)
		tmp = 1.0 / y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x - N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 0.75], N[(1.0 - y), $MachinePrecision], If[LessEqual[y, 1.65e+52], N[(1.0 / y), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 1.65e52 < y

   1. Initial program 24.5%

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

      1. sub-neg 24.5%

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

      2. +-commutative 24.5%

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

      3. *-commutative 24.5%

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

      4. associate-/l* 53.4%

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

      5. distribute-rgt-neg-in 53.4%

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

      6. fma-define 53.3%

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

      7. distribute-frac-neg2 53.3%

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

      8. +-commutative 53.3%

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

      9. distribute-neg-in 53.3%

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

      10. metadata-eval 53.3%

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

      11. unsub-neg 53.3%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in x around inf 49.5%

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

      1. *-commutative 49.5%

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

      2. *-lft-identity 49.5%

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

      3. times-frac 62.9%

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

      4. /-rgt-identity 62.9%

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

      5. +-commutative 62.9%

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

   7. Simplified 62.9%

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

   8. Taylor expanded in y around inf 78.0%

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

      1. neg-mul-1 78.0%

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

      2. unsub-neg 78.0%

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

   10. Simplified 78.0%

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

   if -1 < y < 0.75

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 77.6%

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

      1. +-commutative 77.6%

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

   7. Simplified 77.6%

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

   8. Taylor expanded in y around 0 77.6%

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

      1. neg-mul-1 77.6%

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

      2. unsub-neg 77.6%

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

   10. Simplified 77.6%

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

   if 0.75 < y < 1.65e52

   1. Initial program 41.1%

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

      1. associate-/l* 41.1%

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

      2. remove-double-neg 41.1%

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

      3. remove-double-neg 41.1%

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

      4. +-commutative 41.1%

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

   3. Simplified 41.1%

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

   5. Taylor expanded in x around 0 27.3%

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

      1. +-commutative 27.3%

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

   7. Simplified 27.3%

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

   8. Taylor expanded in y around inf 68.1%

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

Alternative 5: 73.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+52}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   x
   (if (<= y 0.75) (- 1.0 y) (if (<= y 1.65e+52) (/ 1.0 y) x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.75) {
		tmp = 1.0 - y;
	} else if (y <= 1.65e+52) {
		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 <= 0.75d0) then
        tmp = 1.0d0 - y
    else if (y <= 1.65d+52) 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 <= 0.75) {
		tmp = 1.0 - y;
	} else if (y <= 1.65e+52) {
		tmp = 1.0 / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 0.75:
		tmp = 1.0 - y
	elif y <= 1.65e+52:
		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 <= 0.75)
		tmp = Float64(1.0 - y);
	elseif (y <= 1.65e+52)
		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 <= 0.75)
		tmp = 1.0 - y;
	elseif (y <= 1.65e+52)
		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, 0.75], N[(1.0 - y), $MachinePrecision], If[LessEqual[y, 1.65e+52], N[(1.0 / y), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 1.65e52 < y

   1. Initial program 24.5%

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

      1. associate-/l* 53.5%

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

      2. remove-double-neg 53.5%

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

      3. remove-double-neg 53.5%

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

      4. +-commutative 53.5%

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

   3. Simplified 53.5%

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

   5. Taylor expanded in y around inf 77.3%

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

   if -1 < y < 0.75

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 77.6%

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

      1. +-commutative 77.6%

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

   7. Simplified 77.6%

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

   8. Taylor expanded in y around 0 77.6%

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

      1. neg-mul-1 77.6%

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

      2. unsub-neg 77.6%

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

   10. Simplified 77.6%

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

   if 0.75 < y < 1.65e52

   1. Initial program 41.1%

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

      1. associate-/l* 41.1%

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

      2. remove-double-neg 41.1%

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

      3. remove-double-neg 41.1%

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

      4. +-commutative 41.1%

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

   3. Simplified 41.1%

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

   5. Taylor expanded in x around 0 27.3%

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

      1. +-commutative 27.3%

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

   7. Simplified 27.3%

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

   8. Taylor expanded in y around inf 68.1%

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

Alternative 6: 98.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 25.4%

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

      1. associate-/l* 52.8%

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

      2. remove-double-neg 52.8%

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

      3. remove-double-neg 52.8%

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

      4. +-commutative 52.8%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in y around inf 98.0%

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

      1. associate--l+ 98.0%

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

      2. div-sub 98.0%

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

   7. Simplified 98.0%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 99.4%

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

4. Final simplification 98.7%

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

Alternative 7: 86.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -10000 \lor \neg \left(y \leq 16200\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-1 - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -10000.0) (not (<= y 16200.0)))
   (+ x (/ (- 1.0 x) y))
   (/ -1.0 (- -1.0 y))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -10000.0) || !(y <= 16200.0)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = -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 <= (-10000.0d0)) .or. (.not. (y <= 16200.0d0))) then
        tmp = x + ((1.0d0 - x) / y)
    else
        tmp = (-1.0d0) / ((-1.0d0) - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -10000.0) || !(y <= 16200.0)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = -1.0 / (-1.0 - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -10000.0) or not (y <= 16200.0):
		tmp = x + ((1.0 - x) / y)
	else:
		tmp = -1.0 / (-1.0 - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -10000.0) || !(y <= 16200.0))
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	else
		tmp = Float64(-1.0 / Float64(-1.0 - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -10000.0) || ~((y <= 16200.0)))
		tmp = x + ((1.0 - x) / y);
	else
		tmp = -1.0 / (-1.0 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -10000.0], N[Not[LessEqual[y, 16200.0]], $MachinePrecision]], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1e4 or 16200 < y

   1. Initial program 23.9%

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

      1. associate-/l* 51.9%

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

      2. remove-double-neg 51.9%

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

      3. remove-double-neg 51.9%

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

      4. +-commutative 51.9%

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

   3. Simplified 51.9%

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

   5. Taylor expanded in y around inf 99.5%

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

      1. associate--l+ 99.5%

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

      2. div-sub 99.5%

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

   7. Simplified 99.5%

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

   if -1e4 < y < 16200

   1. Initial program 99.8%

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

      1. associate-/l* 99.8%

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

      2. remove-double-neg 99.8%

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

      3. remove-double-neg 99.8%

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

      4. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 77.2%

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

      1. +-commutative 77.2%

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

   7. Simplified 77.2%

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

   8. Taylor expanded in y around inf 2.2%

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

      1. clear-num 2.2%

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

      2. inv-pow 2.2%

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

   10. Applied egg-rr 2.2%

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

       1. unpow-1 2.2%

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

       2. sub-neg 2.2%

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

       3. distribute-neg-frac 2.2%

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

       4. metadata-eval 2.2%

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

   12. Simplified 2.2%

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

   13. Taylor expanded in y around inf 77.2%

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

       1. +-commutative 77.2%

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

       2. distribute-lft-in 77.2%

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

       3. rgt-mult-inverse 77.4%

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

       4. *-rgt-identity 77.4%

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

   15. Simplified 77.4%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -10000 \lor \neg \left(y \leq 16200\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-1 - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -23500 \lor \neg \left(y \leq 2.9 \cdot 10^{+52}\right):\\ \;\;\;\;x - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-1 - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -23500.0) (not (<= y 2.9e+52)))
   (- x (/ x y))
   (/ -1.0 (- -1.0 y))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -23500.0) || !(y <= 2.9e+52)) {
		tmp = x - (x / y);
	} else {
		tmp = -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 <= (-23500.0d0)) .or. (.not. (y <= 2.9d+52))) then
        tmp = x - (x / y)
    else
        tmp = (-1.0d0) / ((-1.0d0) - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -23500.0) || !(y <= 2.9e+52)) {
		tmp = x - (x / y);
	} else {
		tmp = -1.0 / (-1.0 - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -23500.0) or not (y <= 2.9e+52):
		tmp = x - (x / y)
	else:
		tmp = -1.0 / (-1.0 - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -23500.0) || !(y <= 2.9e+52))
		tmp = Float64(x - Float64(x / y));
	else
		tmp = Float64(-1.0 / Float64(-1.0 - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -23500.0) || ~((y <= 2.9e+52)))
		tmp = x - (x / y);
	else
		tmp = -1.0 / (-1.0 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -23500.0], N[Not[LessEqual[y, 2.9e+52]], $MachinePrecision]], N[(x - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -23500 or 2.9e52 < y

   1. Initial program 24.5%

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

      1. sub-neg 24.5%

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

      2. +-commutative 24.5%

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

      3. *-commutative 24.5%

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

      4. associate-/l* 53.4%

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

      5. distribute-rgt-neg-in 53.4%

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

      6. fma-define 53.3%

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

      7. distribute-frac-neg2 53.3%

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

      8. +-commutative 53.3%

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

      9. distribute-neg-in 53.3%

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

      10. metadata-eval 53.3%

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

      11. unsub-neg 53.3%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in x around inf 49.5%

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

      1. *-commutative 49.5%

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

      2. *-lft-identity 49.5%

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

      3. times-frac 62.9%

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

      4. /-rgt-identity 62.9%

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

      5. +-commutative 62.9%

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

   7. Simplified 62.9%

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

   8. Taylor expanded in y around inf 78.0%

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

      1. neg-mul-1 78.0%

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

      2. unsub-neg 78.0%

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

   10. Simplified 78.0%

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

   if -23500 < y < 2.9e52

   1. Initial program 96.8%

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

      1. associate-/l* 96.8%

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

      2. remove-double-neg 96.8%

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

      3. remove-double-neg 96.8%

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

      4. +-commutative 96.8%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in x around 0 74.9%

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

      1. +-commutative 74.9%

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

   7. Simplified 74.9%

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

   8. Taylor expanded in y around inf 5.2%

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

      1. clear-num 5.2%

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

      2. inv-pow 5.2%

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

   10. Applied egg-rr 5.2%

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

       1. unpow-1 5.2%

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

       2. sub-neg 5.2%

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

       3. distribute-neg-frac 5.2%

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

       4. metadata-eval 5.2%

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

   12. Simplified 5.2%

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

   13. Taylor expanded in y around inf 77.9%

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

       1. +-commutative 77.9%

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

       2. distribute-lft-in 77.9%

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

       3. rgt-mult-inverse 78.1%

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

       4. *-rgt-identity 78.1%

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

   15. Simplified 78.1%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -23500 \lor \neg \left(y \leq 2.9 \cdot 10^{+52}\right):\\ \;\;\;\;x - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-1 - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \frac{y}{y + 1}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+52}:\\ \;\;\;\;\frac{-1}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.7e-9)
   (* x (/ y (+ y 1.0)))
   (if (<= y 2e+52) (/ -1.0 (- -1.0 y)) (- x (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.7e-9) {
		tmp = x * (y / (y + 1.0));
	} else if (y <= 2e+52) {
		tmp = -1.0 / (-1.0 - y);
	} else {
		tmp = x - (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 <= (-4.7d-9)) then
        tmp = x * (y / (y + 1.0d0))
    else if (y <= 2d+52) then
        tmp = (-1.0d0) / ((-1.0d0) - y)
    else
        tmp = x - (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.7e-9) {
		tmp = x * (y / (y + 1.0));
	} else if (y <= 2e+52) {
		tmp = -1.0 / (-1.0 - y);
	} else {
		tmp = x - (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.7e-9:
		tmp = x * (y / (y + 1.0))
	elif y <= 2e+52:
		tmp = -1.0 / (-1.0 - y)
	else:
		tmp = x - (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.7e-9)
		tmp = Float64(x * Float64(y / Float64(y + 1.0)));
	elseif (y <= 2e+52)
		tmp = Float64(-1.0 / Float64(-1.0 - y));
	else
		tmp = Float64(x - Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.7e-9)
		tmp = x * (y / (y + 1.0));
	elseif (y <= 2e+52)
		tmp = -1.0 / (-1.0 - y);
	else
		tmp = x - (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.7e-9], N[(x * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e+52], N[(-1.0 / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision], N[(x - N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.6999999999999999e-9

   1. Initial program 23.2%

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

      1. sub-neg 23.2%

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

      2. +-commutative 23.2%

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

      3. *-commutative 23.2%

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

      4. associate-/l* 53.8%

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

      5. distribute-rgt-neg-in 53.8%

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

      6. fma-define 53.7%

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

      7. distribute-frac-neg2 53.7%

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

      8. +-commutative 53.7%

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

      9. distribute-neg-in 53.7%

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

      10. metadata-eval 53.7%

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

      11. unsub-neg 53.7%

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

   3. Simplified 53.7%

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

   5. Taylor expanded in x around inf 45.5%

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

      1. associate-/l* 76.3%

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

      2. +-commutative 76.3%

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

   7. Simplified 76.3%

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

   if -4.6999999999999999e-9 < y < 2e52

   1. Initial program 96.8%

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

      1. associate-/l* 96.8%

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

      2. remove-double-neg 96.8%

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

      3. remove-double-neg 96.8%

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

      4. +-commutative 96.8%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in x around 0 74.9%

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

      1. +-commutative 74.9%

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

   7. Simplified 74.9%

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

   8. Taylor expanded in y around inf 5.2%

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

      1. clear-num 5.2%

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

      2. inv-pow 5.2%

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

   10. Applied egg-rr 5.2%

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

       1. unpow-1 5.2%

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

       2. sub-neg 5.2%

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

       3. distribute-neg-frac 5.2%

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

       4. metadata-eval 5.2%

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

   12. Simplified 5.2%

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

   13. Taylor expanded in y around inf 77.9%

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

       1. +-commutative 77.9%

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

       2. distribute-lft-in 77.9%

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

       3. rgt-mult-inverse 78.1%

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

       4. *-rgt-identity 78.1%

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

   15. Simplified 78.1%

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

   if 2e52 < y

   1. Initial program 26.2%

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

      1. sub-neg 26.2%

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

      2. +-commutative 26.2%

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

      3. *-commutative 26.2%

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

      4. associate-/l* 52.9%

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

      5. distribute-rgt-neg-in 52.9%

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

      6. fma-define 52.9%

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

      7. distribute-frac-neg2 52.9%

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

      8. +-commutative 52.9%

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

      9. distribute-neg-in 52.9%

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

      10. metadata-eval 52.9%

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

      11. unsub-neg 52.9%

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

   3. Simplified 52.9%

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

   5. Taylor expanded in x around inf 54.3%

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

      1. *-commutative 54.3%

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

      2. *-lft-identity 54.3%

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

      3. times-frac 62.5%

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

      4. /-rgt-identity 62.5%

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

      5. +-commutative 62.5%

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

   7. Simplified 62.5%

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

   8. Taylor expanded in y around inf 81.0%

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

      1. neg-mul-1 81.0%

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

      2. unsub-neg 81.0%

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

   10. Simplified 81.0%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \frac{y}{y + 1}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+52}:\\ \;\;\;\;\frac{-1}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.74:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 0.74) (- 1.0 y) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.74) {
		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 <= 0.74d0) 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 <= 0.74) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 0.74:
		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 <= 0.74)
		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 <= 0.74)
		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, 0.74], N[(1.0 - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.73999999999999999 < y

   1. Initial program 25.4%

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

      1. associate-/l* 52.8%

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

      2. remove-double-neg 52.8%

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

      3. remove-double-neg 52.8%

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

      4. +-commutative 52.8%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in y around inf 73.6%

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

   if -1 < y < 0.73999999999999999

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 77.6%

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

      1. +-commutative 77.6%

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

   7. Simplified 77.6%

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

   8. Taylor expanded in y around 0 77.6%

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

      1. neg-mul-1 77.6%

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

      2. unsub-neg 77.6%

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

   10. Simplified 77.6%

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

Alternative 11: 71.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+52}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 1.65e+52) 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 1.65e+52) {
		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 <= 1.65d+52) 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 <= 1.65e+52) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 1.65e+52:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 1.65e+52)
		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 <= 1.65e+52)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 1.65e+52], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.65e52 < y

   1. Initial program 24.5%

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

      1. associate-/l* 53.5%

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

      2. remove-double-neg 53.5%

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

      3. remove-double-neg 53.5%

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

      4. +-commutative 53.5%

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

   3. Simplified 53.5%

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

   5. Taylor expanded in y around inf 77.3%

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

   if -1 < y < 1.65e52

   1. Initial program 96.8%

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

      1. associate-/l* 96.8%

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

      2. remove-double-neg 96.8%

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

      3. remove-double-neg 96.8%

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

      4. +-commutative 96.8%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in x around 0 74.9%

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

      1. +-commutative 74.9%

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

   7. Simplified 74.9%

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

   8. Taylor expanded in y around 0 73.2%

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

Alternative 12: 38.2% 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 60.9%

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

   1. associate-/l* 75.3%

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

   2. remove-double-neg 75.3%

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

   3. remove-double-neg 75.3%

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

   4. +-commutative 75.3%

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

3. Simplified 75.3%

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

5. Taylor expanded in x around 0 39.5%

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

   1. +-commutative 39.5%

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

7. Simplified 39.5%

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

8. Taylor expanded in y around 0 38.7%

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

Developer Target 1: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024135 
(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))))