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

Percentage Accurate: 66.1% → 99.9%

Time: 10.7s

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: 66.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - x}{y}\\ \mathbf{if}\;y \leq -310000:\\ \;\;\;\;x - \left(\frac{x + -1}{y} + \frac{t\_0}{y}\right)\\ \mathbf{elif}\;y \leq 310000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x + -1}{y + 1}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(1 - x\right) - t\_0}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 x) y)))
   (if (<= y -310000.0)
     (- x (+ (/ (+ x -1.0) y) (/ t_0 y)))
     (if (<= y 310000.0)
       (fma y (/ (+ x -1.0) (+ y 1.0)) 1.0)
       (+ x (/ (- (- 1.0 x) t_0) y))))))

C

double code(double x, double y) {
	double t_0 = (1.0 - x) / y;
	double tmp;
	if (y <= -310000.0) {
		tmp = x - (((x + -1.0) / y) + (t_0 / y));
	} else if (y <= 310000.0) {
		tmp = fma(y, ((x + -1.0) / (y + 1.0)), 1.0);
	} else {
		tmp = x + (((1.0 - x) - t_0) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 - x) / y)
	tmp = 0.0
	if (y <= -310000.0)
		tmp = Float64(x - Float64(Float64(Float64(x + -1.0) / y) + Float64(t_0 / y)));
	elseif (y <= 310000.0)
		tmp = fma(y, Float64(Float64(x + -1.0) / Float64(y + 1.0)), 1.0);
	else
		tmp = Float64(x + Float64(Float64(Float64(1.0 - x) - t_0) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -310000.0], N[(x - N[(N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision] + N[(t$95$0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 310000.0], N[(y * N[(N[(x + -1.0), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(x + N[(N[(N[(1.0 - x), $MachinePrecision] - t$95$0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.1e5

   1. Initial program 28.1%

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

      1. associate-/l* 54.9%

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

      2. remove-double-neg 54.9%

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

      3. remove-double-neg 54.9%

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

      4. +-commutative 54.9%

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

   3. Simplified 54.9%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   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-mul-1 100.0%

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

      9. metadata-eval 100.0%

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

      10. distribute-lft-in 100.0%

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

      11. metadata-eval 100.0%

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

      12. sub-neg 100.0%

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

      13. associate-*r/ 100.0%

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

      14. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. unpow2 100.0%

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

      3. times-frac 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. associate-*l/ 100.0%

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

       2. *-un-lft-identity 100.0%

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

   11. Applied egg-rr 100.0%

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

   if -3.1e5 < y < 3.1e5

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-/l* 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. fma-define 100.0%

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

      7. distribute-frac-neg2 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. metadata-eval 100.0%

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

      11. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   if 3.1e5 < y

   1. Initial program 25.2%

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

      1. associate-/l* 54.6%

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

      2. remove-double-neg 54.6%

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

      3. remove-double-neg 54.6%

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

      4. +-commutative 54.6%

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

   3. Simplified 54.6%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   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-mul-1 100.0%

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

      9. metadata-eval 100.0%

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

      10. distribute-lft-in 100.0%

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

      11. metadata-eval 100.0%

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

      12. sub-neg 100.0%

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

      13. associate-*r/ 100.0%

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

      14. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. unpow2 100.0%

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

      3. times-frac 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. associate-*l/ 100.0%

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

       2. *-un-lft-identity 100.0%

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

       3. sub-div 100.0%

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

   11. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -310000:\\ \;\;\;\;x - \left(\frac{x + -1}{y} + \frac{\frac{1 - x}{y}}{y}\right)\\ \mathbf{elif}\;y \leq 310000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x + -1}{y + 1}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(1 - x\right) - \frac{1 - x}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -270000 \lor \neg \left(y \leq 310000\right):\\ \;\;\;\;x + \frac{\left(1 - x\right) - \frac{1 - x}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -270000.0) (not (<= y 310000.0)))
   (+ x (/ (- (- 1.0 x) (/ (- 1.0 x) y)) y))
   (+ 1.0 (/ (* y (- 1.0 x)) (- -1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -270000.0) || !(y <= 310000.0)) {
		tmp = x + (((1.0 - x) - ((1.0 - x) / y)) / y);
	} else {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-270000.0d0)) .or. (.not. (y <= 310000.0d0))) then
        tmp = x + (((1.0d0 - x) - ((1.0d0 - x) / y)) / y)
    else
        tmp = 1.0d0 + ((y * (1.0d0 - x)) / ((-1.0d0) - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -270000.0) || !(y <= 310000.0)) {
		tmp = x + (((1.0 - x) - ((1.0 - x) / y)) / y);
	} else {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -270000.0) or not (y <= 310000.0):
		tmp = x + (((1.0 - x) - ((1.0 - x) / y)) / y)
	else:
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -270000.0) || !(y <= 310000.0))
		tmp = Float64(x + Float64(Float64(Float64(1.0 - x) - Float64(Float64(1.0 - x) / y)) / y));
	else
		tmp = Float64(1.0 + Float64(Float64(y * Float64(1.0 - x)) / Float64(-1.0 - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -270000.0) || ~((y <= 310000.0)))
		tmp = x + (((1.0 - x) - ((1.0 - x) / y)) / y);
	else
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -270000.0], N[Not[LessEqual[y, 310000.0]], $MachinePrecision]], N[(x + N[(N[(N[(1.0 - x), $MachinePrecision] - N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.7e5 or 3.1e5 < y

   1. Initial program 26.6%

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

      1. associate-/l* 54.8%

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

      2. remove-double-neg 54.8%

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

      3. remove-double-neg 54.8%

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

      4. +-commutative 54.8%

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

   3. Simplified 54.8%

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

   5. Taylor expanded in y around inf 100.0%

      \[\leadsto \color{blue}{\left(x + \left(-1 \cdot \frac{1 + -1 \cdot x}{{y}^{2}} + \frac{1}{y}\right)\right) - \frac{x}{y}} \]
   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-mul-1 100.0%

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

      9. metadata-eval 100.0%

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

      10. distribute-lft-in 100.0%

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

      11. metadata-eval 100.0%

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

      12. sub-neg 100.0%

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

      13. associate-*r/ 100.0%

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

      14. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. unpow2 100.0%

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

      3. times-frac 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. associate-*l/ 100.0%

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

       2. *-un-lft-identity 100.0%

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

       3. sub-div 100.0%

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

   11. Applied egg-rr 100.0%

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

   if -2.7e5 < y < 3.1e5

   1. Initial program 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -270000 \lor \neg \left(y \leq 310000\right):\\ \;\;\;\;x + \frac{\left(1 - x\right) - \frac{1 - x}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - x}{y}\\ \mathbf{if}\;y \leq -270000:\\ \;\;\;\;x - \left(\frac{x + -1}{y} + \frac{t\_0}{y}\right)\\ \mathbf{elif}\;y \leq 310000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(1 - x\right) - t\_0}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 x) y)))
   (if (<= y -270000.0)
     (- x (+ (/ (+ x -1.0) y) (/ t_0 y)))
     (if (<= y 310000.0)
       (+ 1.0 (/ (* y (- 1.0 x)) (- -1.0 y)))
       (+ x (/ (- (- 1.0 x) t_0) y))))))

C

double code(double x, double y) {
	double t_0 = (1.0 - x) / y;
	double tmp;
	if (y <= -270000.0) {
		tmp = x - (((x + -1.0) / y) + (t_0 / y));
	} else if (y <= 310000.0) {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	} else {
		tmp = x + (((1.0 - x) - t_0) / 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 <= (-270000.0d0)) then
        tmp = x - (((x + (-1.0d0)) / y) + (t_0 / y))
    else if (y <= 310000.0d0) then
        tmp = 1.0d0 + ((y * (1.0d0 - x)) / ((-1.0d0) - y))
    else
        tmp = x + (((1.0d0 - x) - t_0) / 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 <= -270000.0) {
		tmp = x - (((x + -1.0) / y) + (t_0 / y));
	} else if (y <= 310000.0) {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	} else {
		tmp = x + (((1.0 - x) - t_0) / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.0 - x) / y
	tmp = 0
	if y <= -270000.0:
		tmp = x - (((x + -1.0) / y) + (t_0 / y))
	elif y <= 310000.0:
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y))
	else:
		tmp = x + (((1.0 - x) - t_0) / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 - x) / y)
	tmp = 0.0
	if (y <= -270000.0)
		tmp = Float64(x - Float64(Float64(Float64(x + -1.0) / y) + Float64(t_0 / y)));
	elseif (y <= 310000.0)
		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) - t_0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (1.0 - x) / y;
	tmp = 0.0;
	if (y <= -270000.0)
		tmp = x - (((x + -1.0) / y) + (t_0 / y));
	elseif (y <= 310000.0)
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	else
		tmp = x + (((1.0 - x) - t_0) / 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, -270000.0], N[(x - N[(N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision] + N[(t$95$0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 310000.0], 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] - t$95$0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.7e5

   1. Initial program 28.1%

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

      1. associate-/l* 54.9%

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

      2. remove-double-neg 54.9%

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

      3. remove-double-neg 54.9%

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

      4. +-commutative 54.9%

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

   3. Simplified 54.9%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   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-mul-1 100.0%

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

      9. metadata-eval 100.0%

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

      10. distribute-lft-in 100.0%

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

      11. metadata-eval 100.0%

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

      12. sub-neg 100.0%

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

      13. associate-*r/ 100.0%

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

      14. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. unpow2 100.0%

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

      3. times-frac 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. associate-*l/ 100.0%

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

       2. *-un-lft-identity 100.0%

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

   11. Applied egg-rr 100.0%

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

   if -2.7e5 < y < 3.1e5

   1. Initial program 100.0%

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

   if 3.1e5 < y

   1. Initial program 25.2%

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

      1. associate-/l* 54.6%

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

      2. remove-double-neg 54.6%

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

      3. remove-double-neg 54.6%

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

      4. +-commutative 54.6%

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

   3. Simplified 54.6%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   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-mul-1 100.0%

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

      9. metadata-eval 100.0%

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

      10. distribute-lft-in 100.0%

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

      11. metadata-eval 100.0%

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

      12. sub-neg 100.0%

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

      13. associate-*r/ 100.0%

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

      14. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. unpow2 100.0%

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

      3. times-frac 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. associate-*l/ 100.0%

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

       2. *-un-lft-identity 100.0%

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

       3. sub-div 100.0%

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

   11. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -270000:\\ \;\;\;\;x - \left(\frac{x + -1}{y} + \frac{\frac{1 - x}{y}}{y}\right)\\ \mathbf{elif}\;y \leq 310000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(1 - x\right) - \frac{1 - x}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -22500000000 \lor \neg \left(y \leq 1750000000\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -22500000000.0) (not (<= y 1750000000.0)))
   (+ x (/ 1.0 y))
   (+ 1.0 (/ (* y (- 1.0 x)) (- -1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -22500000000.0) || !(y <= 1750000000.0)) {
		tmp = x + (1.0 / y);
	} else {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-22500000000.0d0)) .or. (.not. (y <= 1750000000.0d0))) then
        tmp = x + (1.0d0 / y)
    else
        tmp = 1.0d0 + ((y * (1.0d0 - x)) / ((-1.0d0) - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -22500000000.0) || !(y <= 1750000000.0)) {
		tmp = x + (1.0 / y);
	} else {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -22500000000.0) or not (y <= 1750000000.0):
		tmp = x + (1.0 / y)
	else:
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -22500000000.0) || !(y <= 1750000000.0))
		tmp = Float64(x + Float64(1.0 / y));
	else
		tmp = Float64(1.0 + Float64(Float64(y * Float64(1.0 - x)) / Float64(-1.0 - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -22500000000.0) || ~((y <= 1750000000.0)))
		tmp = x + (1.0 / y);
	else
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -22500000000.0], N[Not[LessEqual[y, 1750000000.0]], $MachinePrecision]], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.25e10 or 1.75e9 < y

   1. Initial program 25.5%

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

      1. associate-/l* 54.1%

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

      2. remove-double-neg 54.1%

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

      3. remove-double-neg 54.1%

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

      4. +-commutative 54.1%

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

   3. Simplified 54.1%

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

   5. Taylor expanded in y around inf 99.2%

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

      1. associate--l+ 99.2%

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

      2. div-sub 99.2%

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

      3. sub-neg 99.2%

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

      4. +-commutative 99.2%

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

      5. neg-mul-1 99.2%

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

      6. metadata-eval 99.2%

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

      7. distribute-lft-in 99.2%

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

      8. metadata-eval 99.2%

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

      9. sub-neg 99.2%

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

      10. associate-*r/ 99.2%

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

      11. mul-1-neg 99.2%

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

      12. unsub-neg 99.2%

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

      13. sub-neg 99.2%

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

      14. metadata-eval 99.2%

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

   7. Simplified 99.2%

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

   8. Taylor expanded in x around 0 99.2%

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

   if -2.25e10 < y < 1.75e9

   1. Initial program 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -22500000000 \lor \neg \left(y \leq 1750000000\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -150000000:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 1750000000:\\ \;\;\;\;1 + \left(x + -1\right) \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -150000000.0)
   (+ x (/ (- 1.0 x) y))
   (if (<= y 1750000000.0)
     (+ 1.0 (* (+ x -1.0) (/ y (+ y 1.0))))
     (+ x (/ 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -150000000.0) {
		tmp = x + ((1.0 - x) / y);
	} else if (y <= 1750000000.0) {
		tmp = 1.0 + ((x + -1.0) * (y / (y + 1.0)));
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-150000000.0d0)) then
        tmp = x + ((1.0d0 - x) / y)
    else if (y <= 1750000000.0d0) then
        tmp = 1.0d0 + ((x + (-1.0d0)) * (y / (y + 1.0d0)))
    else
        tmp = x + (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -150000000.0) {
		tmp = x + ((1.0 - x) / y);
	} else if (y <= 1750000000.0) {
		tmp = 1.0 + ((x + -1.0) * (y / (y + 1.0)));
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -150000000.0:
		tmp = x + ((1.0 - x) / y)
	elif y <= 1750000000.0:
		tmp = 1.0 + ((x + -1.0) * (y / (y + 1.0)))
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -150000000.0)
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	elseif (y <= 1750000000.0)
		tmp = Float64(1.0 + Float64(Float64(x + -1.0) * Float64(y / Float64(y + 1.0))));
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -150000000.0)
		tmp = x + ((1.0 - x) / y);
	elseif (y <= 1750000000.0)
		tmp = 1.0 + ((x + -1.0) * (y / (y + 1.0)));
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -150000000.0], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1750000000.0], N[(1.0 + N[(N[(x + -1.0), $MachinePrecision] * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.5e8

   1. Initial program 27.0%

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

      1. associate-/l* 54.2%

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

      2. remove-double-neg 54.2%

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

      3. remove-double-neg 54.2%

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

      4. +-commutative 54.2%

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

   3. Simplified 54.2%

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

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. neg-mul-1 100.0%

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

      6. metadata-eval 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*r/ 100.0%

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

      11. mul-1-neg 100.0%

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

      12. unsub-neg 100.0%

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

      13. sub-neg 100.0%

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

      14. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   if -1.5e8 < y < 1.75e9

   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

   if 1.75e9 < y

   1. Initial program 25.2%

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

      1. associate-/l* 54.6%

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

      2. remove-double-neg 54.6%

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

      3. remove-double-neg 54.6%

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

      4. +-commutative 54.6%

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

   3. Simplified 54.6%

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

   5. Taylor expanded in y around inf 98.4%

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

      1. associate--l+ 98.4%

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

      2. div-sub 98.4%

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

      3. sub-neg 98.4%

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

      4. +-commutative 98.4%

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

      5. neg-mul-1 98.4%

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

      6. metadata-eval 98.4%

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

      7. distribute-lft-in 98.4%

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

      8. metadata-eval 98.4%

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

      9. sub-neg 98.4%

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

      10. associate-*r/ 98.4%

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

      11. mul-1-neg 98.4%

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

      12. unsub-neg 98.4%

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

      13. sub-neg 98.4%

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

      14. metadata-eval 98.4%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in x around 0 98.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -150000000:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 1750000000:\\ \;\;\;\;1 + \left(x + -1\right) \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+39}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 1.0) 1.0 (if (<= y 3.6e+39) (/ 1.0 y) x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 1.0) {
		tmp = 1.0;
	} else if (y <= 3.6e+39) {
		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 <= 1.0d0) then
        tmp = 1.0d0
    else if (y <= 3.6d+39) 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 <= 1.0) {
		tmp = 1.0;
	} else if (y <= 3.6e+39) {
		tmp = 1.0 / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 1.0:
		tmp = 1.0
	elif y <= 3.6e+39:
		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 <= 1.0)
		tmp = 1.0;
	elseif (y <= 3.6e+39)
		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 <= 1.0)
		tmp = 1.0;
	elseif (y <= 3.6e+39)
		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, 1.0], 1.0, If[LessEqual[y, 3.6e+39], N[(1.0 / y), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 3.59999999999999984e39 < y

   1. Initial program 26.8%

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

      1. associate-/l* 56.6%

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

      2. remove-double-neg 56.6%

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

      3. remove-double-neg 56.6%

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

      4. +-commutative 56.6%

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

   3. Simplified 56.6%

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

   5. Taylor expanded in y around inf 80.7%

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

   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 72.7%

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

   if 1 < y < 3.59999999999999984e39

   1. Initial program 24.4%

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

      1. associate-/l* 33.1%

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

      2. remove-double-neg 33.1%

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

      3. remove-double-neg 33.1%

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

      4. +-commutative 33.1%

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

   3. Simplified 33.1%

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

   5. Taylor expanded in y around inf 89.8%

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

      1. associate--l+ 89.8%

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

      2. div-sub 89.8%

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

      3. sub-neg 89.8%

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

      4. +-commutative 89.8%

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

      5. neg-mul-1 89.8%

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

      6. metadata-eval 89.8%

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

      7. distribute-lft-in 89.8%

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

      8. metadata-eval 89.8%

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

      9. sub-neg 89.8%

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

      10. associate-*r/ 89.8%

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

      11. mul-1-neg 89.8%

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

      12. unsub-neg 89.8%

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

      13. sub-neg 89.8%

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

      14. metadata-eval 89.8%

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

   7. Simplified 89.8%

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

   8. Taylor expanded in x around 0 89.8%

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

   9. Taylor expanded in x around 0 79.9%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+39}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.85\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 0.85)))
   (+ x (/ 1.0 y))
   (+ 1.0 (* y (+ x -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.85)) {
		tmp = x + (1.0 / y);
	} else {
		tmp = 1.0 + (y * (x + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.0d0)) .or. (.not. (y <= 0.85d0))) then
        tmp = x + (1.0d0 / y)
    else
        tmp = 1.0d0 + (y * (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.85)) {
		tmp = x + (1.0 / y);
	} else {
		tmp = 1.0 + (y * (x + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 0.85):
		tmp = x + (1.0 / y)
	else:
		tmp = 1.0 + (y * (x + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 0.85))
		tmp = Float64(x + Float64(1.0 / y));
	else
		tmp = Float64(1.0 + Float64(y * Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 0.85)))
		tmp = x + (1.0 / y);
	else
		tmp = 1.0 + (y * (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 0.85]], $MachinePrecision]], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.849999999999999978 < y

   1. Initial program 26.6%

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

      1. associate-/l* 54.8%

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

      2. remove-double-neg 54.8%

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

      3. remove-double-neg 54.8%

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

      4. +-commutative 54.8%

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

   3. Simplified 54.8%

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

   5. Taylor expanded in y around inf 99.0%

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

      1. associate--l+ 99.0%

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

      2. div-sub 99.0%

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

      3. sub-neg 99.0%

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

      4. +-commutative 99.0%

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

      5. neg-mul-1 99.0%

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

      6. metadata-eval 99.0%

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

      7. distribute-lft-in 99.0%

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

      8. metadata-eval 99.0%

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

      9. sub-neg 99.0%

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

      10. associate-*r/ 99.0%

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

      11. mul-1-neg 99.0%

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

      12. unsub-neg 99.0%

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

      13. sub-neg 99.0%

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

      14. metadata-eval 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in x around 0 98.5%

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

   if -1 < y < 0.849999999999999978

   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 99.2%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.85\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(x + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 0.85:\\ \;\;\;\;1 + y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   (+ x (/ (- 1.0 x) y))
   (if (<= y 0.85) (+ 1.0 (* y (+ x -1.0))) (+ x (/ 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x + ((1.0 - x) / y);
	} else if (y <= 0.85) {
		tmp = 1.0 + (y * (x + -1.0));
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x + ((1.0d0 - x) / y)
    else if (y <= 0.85d0) then
        tmp = 1.0d0 + (y * (x + (-1.0d0)))
    else
        tmp = x + (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x + ((1.0 - x) / y);
	} else if (y <= 0.85) {
		tmp = 1.0 + (y * (x + -1.0));
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x + ((1.0 - x) / y)
	elif y <= 0.85:
		tmp = 1.0 + (y * (x + -1.0))
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	elseif (y <= 0.85)
		tmp = Float64(1.0 + Float64(y * Float64(x + -1.0)));
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x + ((1.0 - x) / y);
	elseif (y <= 0.85)
		tmp = 1.0 + (y * (x + -1.0));
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.85], N[(1.0 + N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 28.1%

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

      1. associate-/l* 54.9%

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

      2. remove-double-neg 54.9%

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

      3. remove-double-neg 54.9%

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

      4. +-commutative 54.9%

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

   3. Simplified 54.9%

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

   5. Taylor expanded in y around inf 99.6%

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

      1. associate--l+ 99.6%

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

      2. div-sub 99.6%

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

      3. sub-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. neg-mul-1 99.6%

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

      6. metadata-eval 99.6%

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

      7. distribute-lft-in 99.6%

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

      8. metadata-eval 99.6%

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

      9. sub-neg 99.6%

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

      10. associate-*r/ 99.6%

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

      11. mul-1-neg 99.6%

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

      12. unsub-neg 99.6%

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

      13. sub-neg 99.6%

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

      14. metadata-eval 99.6%

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

   7. Simplified 99.6%

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

   if -1 < y < 0.849999999999999978

   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 99.2%

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

   if 0.849999999999999978 < y

   1. Initial program 25.2%

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

      1. associate-/l* 54.6%

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

      2. remove-double-neg 54.6%

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

      3. remove-double-neg 54.6%

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

      4. +-commutative 54.6%

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

   3. Simplified 54.6%

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

   5. Taylor expanded in y around inf 98.4%

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

      1. associate--l+ 98.4%

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

      2. div-sub 98.4%

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

      3. sub-neg 98.4%

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

      4. +-commutative 98.4%

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

      5. neg-mul-1 98.4%

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

      6. metadata-eval 98.4%

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

      7. distribute-lft-in 98.4%

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

      8. metadata-eval 98.4%

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

      9. sub-neg 98.4%

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

      10. associate-*r/ 98.4%

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

      11. mul-1-neg 98.4%

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

      12. unsub-neg 98.4%

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

      13. sub-neg 98.4%

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

      14. metadata-eval 98.4%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in x around 0 98.4%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 0.85:\\ \;\;\;\;1 + y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 6.6 \cdot 10^{-5}\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 6.6e-5))) (+ x (/ 1.0 y)) 1.0))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 6.6e-5)) {
		tmp = x + (1.0 / y);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.0d0)) .or. (.not. (y <= 6.6d-5))) then
        tmp = x + (1.0d0 / y)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 6.6e-5)) {
		tmp = x + (1.0 / y);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 6.6e-5):
		tmp = x + (1.0 / y)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 6.6e-5))
		tmp = Float64(x + Float64(1.0 / y));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 6.6e-5)))
		tmp = x + (1.0 / y);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 6.6e-5]], $MachinePrecision]], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 6.6000000000000005e-5 < y

   1. Initial program 27.2%

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

      1. associate-/l* 55.1%

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

      2. remove-double-neg 55.1%

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

      3. remove-double-neg 55.1%

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

      4. +-commutative 55.1%

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

   3. Simplified 55.1%

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

   5. Taylor expanded in y around inf 98.3%

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

      1. associate--l+ 98.3%

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

      2. div-sub 98.3%

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

      3. sub-neg 98.3%

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

      4. +-commutative 98.3%

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

      5. neg-mul-1 98.3%

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

      6. metadata-eval 98.3%

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

      7. distribute-lft-in 98.3%

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

      8. metadata-eval 98.3%

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

      9. sub-neg 98.3%

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

      10. associate-*r/ 98.3%

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

      11. mul-1-neg 98.3%

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

      12. unsub-neg 98.3%

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

      13. sub-neg 98.3%

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

      14. metadata-eval 98.3%

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

   7. Simplified 98.3%

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

   8. Taylor expanded in x around 0 97.9%

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

   if -1 < y < 6.6000000000000005e-5

   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 73.3%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 6.6 \cdot 10^{-5}\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 26.6%

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

      1. associate-/l* 54.8%

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

      2. remove-double-neg 54.8%

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

      3. remove-double-neg 54.8%

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

      4. +-commutative 54.8%

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

   3. Simplified 54.8%

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

   5. Taylor expanded in y around inf 99.0%

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

      1. associate--l+ 99.0%

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

      2. div-sub 99.0%

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

      3. sub-neg 99.0%

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

      4. +-commutative 99.0%

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

      5. neg-mul-1 99.0%

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

      6. metadata-eval 99.0%

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

      7. distribute-lft-in 99.0%

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

      8. metadata-eval 99.0%

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

      9. sub-neg 99.0%

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

      10. associate-*r/ 99.0%

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

      11. mul-1-neg 99.0%

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

      12. unsub-neg 99.0%

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

      13. sub-neg 99.0%

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

      14. metadata-eval 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in x around 0 98.5%

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

   if -1 < y < 1

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. 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 99.2%

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

   6. Taylor expanded in x around inf 98.8%

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

      1. mul-1-neg 98.8%

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

      2. distribute-lft-neg-out 98.8%

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

      3. *-commutative 98.8%

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

   8. Simplified 98.8%

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

4. Final simplification 98.6%

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

Alternative 11: 74.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 44000000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 44000000000000.0) 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 44000000000000.0) {
		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 <= 44000000000000.0d0) 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 <= 44000000000000.0) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 44000000000000.0:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 44000000000000.0)
		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 <= 44000000000000.0)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 44000000000000.0], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 4.4e13 < y

   1. Initial program 26.0%

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

      1. associate-/l* 55.2%

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

      2. remove-double-neg 55.2%

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

      3. remove-double-neg 55.2%

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

      4. +-commutative 55.2%

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

   3. Simplified 55.2%

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

   5. Taylor expanded in y around inf 78.4%

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

   if -1 < y < 4.4e13

   1. Initial program 97.8%

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

      1. associate-/l* 97.8%

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

      2. remove-double-neg 97.8%

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

      3. remove-double-neg 97.8%

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

      4. +-commutative 97.8%

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

   3. Simplified 97.8%

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

   5. Taylor expanded in y around 0 70.3%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 44000000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 38.7% 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 62.7%

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

   1. associate-/l* 77.0%

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

   2. remove-double-neg 77.0%

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

   3. remove-double-neg 77.0%

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

   4. +-commutative 77.0%

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

3. Simplified 77.0%

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

5. Taylor expanded in y around 0 37.8%

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

6. Final simplification 37.8%

   \[\leadsto 1 \]
7. Add Preprocessing

Developer target: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024059 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1.0 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))) (- (/ 1.0 y) (- (/ x y) x))))

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