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

Percentage Accurate: 65.2% → 99.8%

Time: 7.9s

Alternatives: 11

Speedup: 2.1×

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: 65.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -240000000000:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{elif}\;y \leq 370000:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{1 - x}{y} + \frac{x + -1}{y \cdot y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -240000000000.0)
   (+ (/ 1.0 y) x)
   (if (<= y 370000.0)
     (+ 1.0 (* y (/ (+ x -1.0) (+ y 1.0))))
     (+ x (+ (/ (- 1.0 x) y) (/ (+ x -1.0) (* y y)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -240000000000.0) {
		tmp = (1.0 / y) + x;
	} else if (y <= 370000.0) {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	} else {
		tmp = x + (((1.0 - x) / y) + ((x + -1.0) / (y * y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-240000000000.0d0)) then
        tmp = (1.0d0 / y) + x
    else if (y <= 370000.0d0) then
        tmp = 1.0d0 + (y * ((x + (-1.0d0)) / (y + 1.0d0)))
    else
        tmp = x + (((1.0d0 - x) / y) + ((x + (-1.0d0)) / (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -240000000000.0) {
		tmp = (1.0 / y) + x;
	} else if (y <= 370000.0) {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	} else {
		tmp = x + (((1.0 - x) / y) + ((x + -1.0) / (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -240000000000.0:
		tmp = (1.0 / y) + x
	elif y <= 370000.0:
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)))
	else:
		tmp = x + (((1.0 - x) / y) + ((x + -1.0) / (y * y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -240000000000.0)
		tmp = Float64(Float64(1.0 / y) + x);
	elseif (y <= 370000.0)
		tmp = Float64(1.0 + Float64(y * Float64(Float64(x + -1.0) / Float64(y + 1.0))));
	else
		tmp = Float64(x + Float64(Float64(Float64(1.0 - x) / y) + Float64(Float64(x + -1.0) / Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -240000000000.0)
		tmp = (1.0 / y) + x;
	elseif (y <= 370000.0)
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	else
		tmp = x + (((1.0 - x) / y) + ((x + -1.0) / (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -240000000000.0], N[(N[(1.0 / y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 370000.0], N[(1.0 + N[(y * N[(N[(x + -1.0), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] + N[(N[(x + -1.0), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.4e11

   1. Initial program 31.6%

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

   2. Taylor expanded in y around -inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. associate--l+ 100.0%

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

      3. mul-1-neg 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. neg-sub0 100.0%

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

      6. associate-+l- 100.0%

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

      7. neg-sub0 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. div-sub 100.0%

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

      11. unpow2 100.0%

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

      12. sub-neg 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in y around inf 100.0%

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

      1. +-commutative 100.0%

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

   10. Simplified 100.0%

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

   if -2.4e11 < y < 3.7e5

   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. associate-*l/ 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. associate--r- 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   if 3.7e5 < y

   1. Initial program 35.7%

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

   2. Taylor expanded in y around -inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. associate--l+ 100.0%

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

      3. mul-1-neg 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. neg-sub0 100.0%

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

      6. associate-+l- 100.0%

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

      7. neg-sub0 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. div-sub 100.0%

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

      11. unpow2 100.0%

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

      12. sub-neg 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -240000000000:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{elif}\;y \leq 370000:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{1 - x}{y} + \frac{x + -1}{y \cdot y}\right)\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -16500000000:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{elif}\;y \leq 5600000:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{1 - x}{y} + \frac{-1}{y \cdot y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -16500000000.0)
   (+ (/ 1.0 y) x)
   (if (<= y 5600000.0)
     (+ 1.0 (* y (/ (+ x -1.0) (+ y 1.0))))
     (+ x (+ (/ (- 1.0 x) y) (/ -1.0 (* y y)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -16500000000.0) {
		tmp = (1.0 / y) + x;
	} else if (y <= 5600000.0) {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	} else {
		tmp = x + (((1.0 - x) / y) + (-1.0 / (y * y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-16500000000.0d0)) then
        tmp = (1.0d0 / y) + x
    else if (y <= 5600000.0d0) then
        tmp = 1.0d0 + (y * ((x + (-1.0d0)) / (y + 1.0d0)))
    else
        tmp = x + (((1.0d0 - x) / y) + ((-1.0d0) / (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -16500000000.0) {
		tmp = (1.0 / y) + x;
	} else if (y <= 5600000.0) {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	} else {
		tmp = x + (((1.0 - x) / y) + (-1.0 / (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -16500000000.0:
		tmp = (1.0 / y) + x
	elif y <= 5600000.0:
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)))
	else:
		tmp = x + (((1.0 - x) / y) + (-1.0 / (y * y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -16500000000.0)
		tmp = Float64(Float64(1.0 / y) + x);
	elseif (y <= 5600000.0)
		tmp = Float64(1.0 + Float64(y * Float64(Float64(x + -1.0) / Float64(y + 1.0))));
	else
		tmp = Float64(x + Float64(Float64(Float64(1.0 - x) / y) + Float64(-1.0 / Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -16500000000.0)
		tmp = (1.0 / y) + x;
	elseif (y <= 5600000.0)
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	else
		tmp = x + (((1.0 - x) / y) + (-1.0 / (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -16500000000.0], N[(N[(1.0 / y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 5600000.0], N[(1.0 + N[(y * N[(N[(x + -1.0), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] + N[(-1.0 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.65e10

   1. Initial program 31.6%

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

   2. Taylor expanded in y around -inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. associate--l+ 100.0%

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

      3. mul-1-neg 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. neg-sub0 100.0%

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

      6. associate-+l- 100.0%

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

      7. neg-sub0 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. div-sub 100.0%

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

      11. unpow2 100.0%

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

      12. sub-neg 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in y around inf 100.0%

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

      1. +-commutative 100.0%

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

   10. Simplified 100.0%

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

   if -1.65e10 < y < 5.6e6

   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. associate-*l/ 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. associate--r- 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   if 5.6e6 < y

   1. Initial program 34.8%

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

   2. Taylor expanded in y around -inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. associate--l+ 100.0%

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

      3. mul-1-neg 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. neg-sub0 100.0%

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

      6. associate-+l- 100.0%

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

      7. neg-sub0 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. div-sub 100.0%

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

      11. unpow2 100.0%

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

      12. sub-neg 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -16500000000:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{elif}\;y \leq 5600000:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{1 - x}{y} + \frac{-1}{y \cdot y}\right)\\ \end{array} \]

Alternative 3: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -16000000000 \lor \neg \left(y \leq 126000000\right):\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -16000000000.0) (not (<= y 126000000.0)))
   (+ (/ 1.0 y) x)
   (+ 1.0 (* y (/ (+ x -1.0) (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -16000000000.0) || !(y <= 126000000.0)) {
		tmp = (1.0 / y) + x;
	} else {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 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 <= (-16000000000.0d0)) .or. (.not. (y <= 126000000.0d0))) then
        tmp = (1.0d0 / y) + x
    else
        tmp = 1.0d0 + (y * ((x + (-1.0d0)) / (y + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -16000000000.0) || !(y <= 126000000.0)) {
		tmp = (1.0 / y) + x;
	} else {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -16000000000.0) or not (y <= 126000000.0):
		tmp = (1.0 / y) + x
	else:
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -16000000000.0) || !(y <= 126000000.0))
		tmp = Float64(Float64(1.0 / y) + x);
	else
		tmp = Float64(1.0 + Float64(y * Float64(Float64(x + -1.0) / Float64(y + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -16000000000.0) || ~((y <= 126000000.0)))
		tmp = (1.0 / y) + x;
	else
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -16000000000.0], N[Not[LessEqual[y, 126000000.0]], $MachinePrecision]], N[(N[(1.0 / y), $MachinePrecision] + x), $MachinePrecision], N[(1.0 + N[(y * N[(N[(x + -1.0), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6e10 or 1.26e8 < y

   1. Initial program 33.1%

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

   2. Taylor expanded in y around -inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. associate--l+ 100.0%

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

      3. mul-1-neg 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. neg-sub0 100.0%

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

      6. associate-+l- 100.0%

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

      7. neg-sub0 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. div-sub 100.0%

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

      11. unpow2 100.0%

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

      12. sub-neg 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around 0 99.9%

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

      1. unpow2 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in y around inf 99.4%

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

      1. +-commutative 99.4%

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

   10. Simplified 99.4%

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

   if -1.6e10 < y < 1.26e8

   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. associate-*l/ 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. associate--r- 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -16000000000 \lor \neg \left(y \leq 126000000\right):\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \end{array} \]

Alternative 4: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -58000000000:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{elif}\;y \leq 126000000:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{1}{y} + \frac{-1}{y \cdot y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -58000000000.0)
   (+ (/ 1.0 y) x)
   (if (<= y 126000000.0)
     (+ 1.0 (* y (/ (+ x -1.0) (+ y 1.0))))
     (+ x (+ (/ 1.0 y) (/ -1.0 (* y y)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -58000000000.0) {
		tmp = (1.0 / y) + x;
	} else if (y <= 126000000.0) {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	} else {
		tmp = x + ((1.0 / y) + (-1.0 / (y * y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-58000000000.0d0)) then
        tmp = (1.0d0 / y) + x
    else if (y <= 126000000.0d0) then
        tmp = 1.0d0 + (y * ((x + (-1.0d0)) / (y + 1.0d0)))
    else
        tmp = x + ((1.0d0 / y) + ((-1.0d0) / (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -58000000000.0) {
		tmp = (1.0 / y) + x;
	} else if (y <= 126000000.0) {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	} else {
		tmp = x + ((1.0 / y) + (-1.0 / (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -58000000000.0:
		tmp = (1.0 / y) + x
	elif y <= 126000000.0:
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)))
	else:
		tmp = x + ((1.0 / y) + (-1.0 / (y * y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -58000000000.0)
		tmp = Float64(Float64(1.0 / y) + x);
	elseif (y <= 126000000.0)
		tmp = Float64(1.0 + Float64(y * Float64(Float64(x + -1.0) / Float64(y + 1.0))));
	else
		tmp = Float64(x + Float64(Float64(1.0 / y) + Float64(-1.0 / Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -58000000000.0)
		tmp = (1.0 / y) + x;
	elseif (y <= 126000000.0)
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	else
		tmp = x + ((1.0 / y) + (-1.0 / (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -58000000000.0], N[(N[(1.0 / y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 126000000.0], N[(1.0 + N[(y * N[(N[(x + -1.0), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(1.0 / y), $MachinePrecision] + N[(-1.0 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.8e10

   1. Initial program 31.6%

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

   2. Taylor expanded in y around -inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. associate--l+ 100.0%

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

      3. mul-1-neg 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. neg-sub0 100.0%

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

      6. associate-+l- 100.0%

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

      7. neg-sub0 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. div-sub 100.0%

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

      11. unpow2 100.0%

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

      12. sub-neg 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in y around inf 100.0%

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

      1. +-commutative 100.0%

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

   10. Simplified 100.0%

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

   if -5.8e10 < y < 1.26e8

   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. associate-*l/ 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. associate--r- 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   if 1.26e8 < y

   1. Initial program 34.8%

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

   2. Taylor expanded in y around -inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. associate--l+ 100.0%

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

      3. mul-1-neg 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. neg-sub0 100.0%

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

      6. associate-+l- 100.0%

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

      7. neg-sub0 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. div-sub 100.0%

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

      11. unpow2 100.0%

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

      12. sub-neg 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around 0 99.8%

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

      1. unpow2 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -58000000000:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{elif}\;y \leq 126000000:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{1}{y} + \frac{-1}{y \cdot y}\right)\\ \end{array} \]

Alternative 5: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{elif}\;y \leq 0.82:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   (+ (/ 1.0 y) x)
   (if (<= y 0.82) (- 1.0 y) (+ x (/ (- 1.0 x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = (1.0 / y) + x;
	} else if (y <= 0.82) {
		tmp = 1.0 - y;
	} else {
		tmp = x + ((1.0 - x) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = (1.0d0 / y) + x
    else if (y <= 0.82d0) then
        tmp = 1.0d0 - y
    else
        tmp = x + ((1.0d0 - x) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = (1.0 / y) + x;
	} else if (y <= 0.82) {
		tmp = 1.0 - y;
	} else {
		tmp = x + ((1.0 - x) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = (1.0 / y) + x
	elif y <= 0.82:
		tmp = 1.0 - y
	else:
		tmp = x + ((1.0 - x) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(Float64(1.0 / y) + x);
	elseif (y <= 0.82)
		tmp = Float64(1.0 - y);
	else
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = (1.0 / y) + x;
	elseif (y <= 0.82)
		tmp = 1.0 - y;
	else
		tmp = x + ((1.0 - x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], N[(N[(1.0 / y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 0.82], N[(1.0 - y), $MachinePrecision], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 31.6%

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

   2. Taylor expanded in y around -inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. associate--l+ 100.0%

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

      3. mul-1-neg 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. neg-sub0 100.0%

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

      6. associate-+l- 100.0%

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

      7. neg-sub0 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. div-sub 100.0%

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

      11. unpow2 100.0%

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

      12. sub-neg 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in y around inf 100.0%

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

      1. +-commutative 100.0%

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

   10. Simplified 100.0%

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

   if -1 < y < 0.819999999999999951

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 79.6%

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

   3. Taylor expanded in y around 0 79.6%

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

      1. neg-mul-1 79.6%

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

      2. unsub-neg 79.6%

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

   5. Simplified 79.6%

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

   if 0.819999999999999951 < y

   1. Initial program 36.6%

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

   2. Taylor expanded in y around -inf 98.1%

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

      1. mul-1-neg 98.1%

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

      2. distribute-neg-frac 98.1%

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

      3. neg-sub0 98.1%

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

      4. associate-+l- 98.1%

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

      5. neg-sub0 98.1%

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

      6. +-commutative 98.1%

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

      7. sub-neg 98.1%

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

   4. Simplified 98.1%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{elif}\;y \leq 0.82:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]

Alternative 6: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 31.6%

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

   2. Taylor expanded in y around -inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. associate--l+ 100.0%

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

      3. mul-1-neg 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. neg-sub0 100.0%

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

      6. associate-+l- 100.0%

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

      7. neg-sub0 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. div-sub 100.0%

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

      11. unpow2 100.0%

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

      12. sub-neg 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in y around inf 100.0%

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

      1. +-commutative 100.0%

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

   10. Simplified 100.0%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

      1. sub-neg 100.0%

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

      2. distribute-rgt-in 100.0%

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

      3. distribute-lft-neg-out 100.0%

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

      4. unsub-neg 100.0%

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

      5. *-lft-identity 100.0%

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

      6. *-commutative 100.0%

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

   4. Simplified 100.0%

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

   if 1 < y

   1. Initial program 36.6%

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

   2. Taylor expanded in y around -inf 98.1%

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

      1. mul-1-neg 98.1%

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

      2. distribute-neg-frac 98.1%

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

      3. neg-sub0 98.1%

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

      4. associate-+l- 98.1%

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

      5. neg-sub0 98.1%

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

      6. +-commutative 98.1%

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

      7. sub-neg 98.1%

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

   4. Simplified 98.1%

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

4. Final simplification 99.5%

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

Alternative 7: 73.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+24}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   x
   (if (<= y 0.75) (- 1.0 y) (if (<= y 1.35e+24) (/ 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 <= 1.35e+24) {
		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 <= 1.35d+24) 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 <= 1.35e+24) {
		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 <= 1.35e+24:
		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 <= 1.35e+24)
		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 <= 1.35e+24)
		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, 1.35e+24], N[(1.0 / y), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 1.35e24 < y

   1. Initial program 33.5%

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

      1. sub-neg 33.5%

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

      2. associate-*l/ 60.7%

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

      3. distribute-lft-neg-in 60.7%

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

      4. distribute-frac-neg 60.7%

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

      5. neg-sub0 60.7%

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

      6. associate--r- 60.7%

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

      7. metadata-eval 60.7%

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

      8. +-commutative 60.7%

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

      9. +-commutative 60.7%

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

   3. Simplified 60.7%

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

   4. Taylor expanded in y around inf 78.3%

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

   if -1 < y < 0.75

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 79.6%

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

   3. Taylor expanded in y around 0 79.6%

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

      1. neg-mul-1 79.6%

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

      2. unsub-neg 79.6%

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

   5. Simplified 79.6%

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

   if 0.75 < y < 1.35e24

   1. Initial program 43.1%

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

   2. Taylor expanded in y around -inf 83.7%

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

      1. mul-1-neg 83.7%

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

      2. distribute-neg-frac 83.7%

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

      3. neg-sub0 83.7%

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

      4. associate-+l- 83.7%

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

      5. neg-sub0 83.7%

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

      6. +-commutative 83.7%

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

      7. sub-neg 83.7%

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

   4. Simplified 83.7%

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

   5. Taylor expanded in x around 0 64.5%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+24}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8: 85.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.088\right):\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 0.088))) (+ (/ 1.0 y) x) (- 1.0 y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.088)) {
		tmp = (1.0 / y) + x;
	} else {
		tmp = 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)) .or. (.not. (y <= 0.088d0))) then
        tmp = (1.0d0 / y) + x
    else
        tmp = 1.0d0 - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.088)) {
		tmp = (1.0 / y) + x;
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 0.088):
		tmp = (1.0 / y) + x
	else:
		tmp = 1.0 - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 0.088))
		tmp = Float64(Float64(1.0 / y) + x);
	else
		tmp = Float64(1.0 - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 0.088)))
		tmp = (1.0 / y) + x;
	else
		tmp = 1.0 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 0.088]], $MachinePrecision]], N[(N[(1.0 / y), $MachinePrecision] + x), $MachinePrecision], N[(1.0 - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.087999999999999995 < y

   1. Initial program 34.0%

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

   2. Taylor expanded in y around -inf 99.8%

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

      1. associate--l+ 99.8%

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

      2. associate--l+ 99.8%

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

      3. mul-1-neg 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. neg-sub0 99.8%

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

      6. associate-+l- 99.8%

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

      7. neg-sub0 99.8%

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

      8. +-commutative 99.8%

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

      9. sub-neg 99.8%

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

      10. div-sub 99.8%

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

      11. unpow2 99.8%

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

      12. sub-neg 99.8%

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

      13. metadata-eval 99.8%

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

      14. +-commutative 99.8%

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

   4. Simplified 99.8%

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

   5. Taylor expanded in x around 0 99.1%

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

      1. unpow2 99.1%

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

   7. Simplified 99.1%

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

   8. Taylor expanded in y around inf 98.7%

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

      1. +-commutative 98.7%

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

   10. Simplified 98.7%

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

   if -1 < y < 0.087999999999999995

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 79.6%

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

   3. Taylor expanded in y around 0 79.6%

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

      1. neg-mul-1 79.6%

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

      2. unsub-neg 79.6%

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

   5. Simplified 79.6%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.088\right):\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]

Alternative 9: 73.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.98:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 0.98) (- 1.0 y) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.98) {
		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 <= 0.98d0) 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 <= 0.98) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 0.98:
		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 <= 0.98)
		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 <= 0.98)
		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, 0.98], N[(1.0 - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.97999999999999998 < y

   1. Initial program 34.0%

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

      1. sub-neg 34.0%

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

      2. associate-*l/ 59.7%

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

      3. distribute-lft-neg-in 59.7%

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

      4. distribute-frac-neg 59.7%

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

      5. neg-sub0 59.7%

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

      6. associate--r- 59.7%

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

      7. metadata-eval 59.7%

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

      8. +-commutative 59.7%

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

      9. +-commutative 59.7%

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

   3. Simplified 59.7%

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

   4. Taylor expanded in y around inf 74.7%

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

   if -1 < y < 0.97999999999999998

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 79.6%

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

   3. Taylor expanded in y around 0 79.6%

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

      1. neg-mul-1 79.6%

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

      2. unsub-neg 79.6%

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

   5. Simplified 79.6%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.98:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10: 72.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.215:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1.0) x (if (<= y 0.215) 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.215) {
		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 <= 0.215d0) 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 <= 0.215) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 0.215:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 0.215)
		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 <= 0.215)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 0.215], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.214999999999999997 < y

   1. Initial program 34.0%

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

      1. sub-neg 34.0%

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

      2. associate-*l/ 59.7%

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

      3. distribute-lft-neg-in 59.7%

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

      4. distribute-frac-neg 59.7%

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

      5. neg-sub0 59.7%

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

      6. associate--r- 59.7%

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

      7. metadata-eval 59.7%

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

      8. +-commutative 59.7%

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

      9. +-commutative 59.7%

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

   3. Simplified 59.7%

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

   4. Taylor expanded in y around inf 74.7%

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

   if -1 < y < 0.214999999999999997

   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. associate-*l/ 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. associate--r- 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 79.0%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.215:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11: 37.9% accurate, 11.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 62.9%

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

   1. sub-neg 62.9%

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

   2. associate-*l/ 77.3%

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

   3. distribute-lft-neg-in 77.3%

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

   4. distribute-frac-neg 77.3%

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

   5. neg-sub0 77.3%

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

   6. associate--r- 77.3%

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

   7. metadata-eval 77.3%

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

   8. +-commutative 77.3%

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

   9. +-commutative 77.3%

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

3. Simplified 77.3%

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

4. Taylor expanded in y around 0 36.7%

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

5. Final simplification 36.7%

   \[\leadsto 1 \]

Developer target: 99.6% accurate, 0.7× 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 2023293 
(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))))