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

Percentage Accurate: 65.8% → 99.9%

Time: 11.3s

Alternatives: 13

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: 65.8% 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, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2200000:\\ \;\;\;\;x - \frac{\frac{1}{y} + \left(x + -1\right)}{y}\\ \mathbf{elif}\;y \leq 320000:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{1 - x}{y} + \frac{x + -1}{y \cdot y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2200000.0)
   (- x (/ (+ (/ 1.0 y) (+ x -1.0)) y))
   (if (<= y 320000.0)
     (- 1.0 (/ (* y (- 1.0 x)) (+ y 1.0)))
     (+ x (+ (/ (- 1.0 x) y) (/ (+ x -1.0) (* y y)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2200000.0) {
		tmp = x - (((1.0 / y) + (x + -1.0)) / y);
	} else if (y <= 320000.0) {
		tmp = 1.0 - ((y * (1.0 - x)) / (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 <= (-2200000.0d0)) then
        tmp = x - (((1.0d0 / y) + (x + (-1.0d0))) / y)
    else if (y <= 320000.0d0) then
        tmp = 1.0d0 - ((y * (1.0d0 - x)) / (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 <= -2200000.0) {
		tmp = x - (((1.0 / y) + (x + -1.0)) / y);
	} else if (y <= 320000.0) {
		tmp = 1.0 - ((y * (1.0 - x)) / (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 <= -2200000.0:
		tmp = x - (((1.0 / y) + (x + -1.0)) / y)
	elif y <= 320000.0:
		tmp = 1.0 - ((y * (1.0 - x)) / (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 <= -2200000.0)
		tmp = Float64(x - Float64(Float64(Float64(1.0 / y) + Float64(x + -1.0)) / y));
	elseif (y <= 320000.0)
		tmp = Float64(1.0 - Float64(Float64(y * Float64(1.0 - x)) / 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 <= -2200000.0)
		tmp = x - (((1.0 / y) + (x + -1.0)) / y);
	elseif (y <= 320000.0)
		tmp = 1.0 - ((y * (1.0 - x)) / (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, -2200000.0], N[(x - N[(N[(N[(1.0 / y), $MachinePrecision] + N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 320000.0], N[(1.0 - N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $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 < -2.2e6

   1. Initial program 32.7%

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

   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. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. distribute-neg-frac 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-neg-in 100.0%

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

      9. metadata-eval 100.0%

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

      10. sub-neg 100.0%

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

      11. div-sub 100.0%

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

      12. sub-neg 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

      15. unpow2 100.0%

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

   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%

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

      2. associate-/r* 100.0%

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

   7. Simplified 100.0%

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

      1. add-sqr-sqrt 47.1%

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

      2. sqrt-unprod 99.4%

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

      3. associate-/l/ 99.4%

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

      4. associate-/l/ 99.4%

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

      5. frac-times 99.4%

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

      6. metadata-eval 99.4%

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

      7. metadata-eval 99.4%

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

      8. frac-times 99.4%

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

      9. sqrt-unprod 99.4%

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

      10. add-sqr-sqrt 99.4%

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

      11. associate-/r* 99.4%

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

      12. div-inv 99.4%

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

      13. metadata-eval 99.4%

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

      14. distribute-neg-frac 99.4%

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

      15. cancel-sign-sub-inv 99.4%

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

      16. div-inv 99.4%

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

      17. sub-div 99.4%

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

   9. Applied egg-rr 100.0%

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

   if -2.2e6 < y < 3.2e5

   1. Initial program 100.0%

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

   if 3.2e5 < y

   1. Initial program 31.7%

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

   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. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. distribute-neg-frac 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-neg-in 100.0%

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

      9. metadata-eval 100.0%

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

      10. sub-neg 100.0%

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

      11. div-sub 100.0%

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

      12. sub-neg 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

      15. unpow2 100.0%

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

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2200000:\\ \;\;\;\;x - \frac{\frac{1}{y} + \left(x + -1\right)}{y}\\ \mathbf{elif}\;y \leq 320000:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{1 - x}{y} + \frac{x + -1}{y \cdot y}\right)\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -122000000 \lor \neg \left(y \leq 400000000\right):\\ \;\;\;\;x - \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -122000000.0) (not (<= y 400000000.0)))
   (- x (/ (+ x -1.0) y))
   (+ 1.0 (* y (/ (+ x -1.0) (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -122000000.0) || !(y <= 400000000.0)) {
		tmp = x - ((x + -1.0) / y);
	} 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 <= (-122000000.0d0)) .or. (.not. (y <= 400000000.0d0))) then
        tmp = x - ((x + (-1.0d0)) / y)
    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 <= -122000000.0) || !(y <= 400000000.0)) {
		tmp = x - ((x + -1.0) / y);
	} else {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -122000000.0) or not (y <= 400000000.0):
		tmp = x - ((x + -1.0) / y)
	else:
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -122000000.0) || !(y <= 400000000.0))
		tmp = Float64(x - Float64(Float64(x + -1.0) / y));
	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 <= -122000000.0) || ~((y <= 400000000.0)))
		tmp = x - ((x + -1.0) / y);
	else
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -122000000.0], N[Not[LessEqual[y, 400000000.0]], $MachinePrecision]], N[(x - N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $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 < -1.22e8 or 4e8 < y

   1. Initial program 31.7%

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

   2. Taylor expanded in y around -inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. sub-neg 99.7%

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

      3. metadata-eval 99.7%

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

      4. distribute-neg-frac 99.7%

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

      5. +-commutative 99.7%

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

      6. distribute-neg-in 99.7%

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

      7. metadata-eval 99.7%

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

      8. sub-neg 99.7%

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

   4. Simplified 99.7%

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

   if -1.22e8 < y < 4e8

   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 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -122000000 \lor \neg \left(y \leq 400000000\right):\\ \;\;\;\;x - \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \end{array} \]

Alternative 3: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4100000:\\ \;\;\;\;x - \frac{\frac{1}{y} + \left(x + -1\right)}{y}\\ \mathbf{elif}\;y \leq 400000000:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x + -1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4100000.0)
   (- x (/ (+ (/ 1.0 y) (+ x -1.0)) y))
   (if (<= y 400000000.0)
     (+ 1.0 (* y (/ (+ x -1.0) (+ y 1.0))))
     (- x (/ (+ x -1.0) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4100000.0) {
		tmp = x - (((1.0 / y) + (x + -1.0)) / y);
	} else if (y <= 400000000.0) {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	} else {
		tmp = x - ((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 <= (-4100000.0d0)) then
        tmp = x - (((1.0d0 / y) + (x + (-1.0d0))) / y)
    else if (y <= 400000000.0d0) then
        tmp = 1.0d0 + (y * ((x + (-1.0d0)) / (y + 1.0d0)))
    else
        tmp = x - ((x + (-1.0d0)) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4100000.0) {
		tmp = x - (((1.0 / y) + (x + -1.0)) / y);
	} else if (y <= 400000000.0) {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	} else {
		tmp = x - ((x + -1.0) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4100000.0:
		tmp = x - (((1.0 / y) + (x + -1.0)) / y)
	elif y <= 400000000.0:
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)))
	else:
		tmp = x - ((x + -1.0) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4100000.0)
		tmp = Float64(x - Float64(Float64(Float64(1.0 / y) + Float64(x + -1.0)) / y));
	elseif (y <= 400000000.0)
		tmp = Float64(1.0 + Float64(y * Float64(Float64(x + -1.0) / Float64(y + 1.0))));
	else
		tmp = Float64(x - Float64(Float64(x + -1.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4100000.0)
		tmp = x - (((1.0 / y) + (x + -1.0)) / y);
	elseif (y <= 400000000.0)
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	else
		tmp = x - ((x + -1.0) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4100000.0], N[(x - N[(N[(N[(1.0 / y), $MachinePrecision] + N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 400000000.0], N[(1.0 + N[(y * N[(N[(x + -1.0), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.1e6

   1. Initial program 32.7%

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

   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. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. distribute-neg-frac 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-neg-in 100.0%

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

      9. metadata-eval 100.0%

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

      10. sub-neg 100.0%

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

      11. div-sub 100.0%

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

      12. sub-neg 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

      15. unpow2 100.0%

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

   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%

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

      2. associate-/r* 100.0%

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

   7. Simplified 100.0%

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

      1. add-sqr-sqrt 47.1%

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

      2. sqrt-unprod 99.4%

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

      3. associate-/l/ 99.4%

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

      4. associate-/l/ 99.4%

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

      5. frac-times 99.4%

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

      6. metadata-eval 99.4%

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

      7. metadata-eval 99.4%

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

      8. frac-times 99.4%

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

      9. sqrt-unprod 99.4%

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

      10. add-sqr-sqrt 99.4%

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

      11. associate-/r* 99.4%

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

      12. div-inv 99.4%

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

      13. metadata-eval 99.4%

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

      14. distribute-neg-frac 99.4%

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

      15. cancel-sign-sub-inv 99.4%

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

      16. div-inv 99.4%

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

      17. sub-div 99.4%

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

   9. Applied egg-rr 100.0%

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

   if -4.1e6 < y < 4e8

   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 4e8 < y

   1. Initial program 30.5%

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

   2. Taylor expanded in y around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. metadata-eval 100.0%

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

      8. sub-neg 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4100000:\\ \;\;\;\;x - \frac{\frac{1}{y} + \left(x + -1\right)}{y}\\ \mathbf{elif}\;y \leq 400000000:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x + -1}{y}\\ \end{array} \]

Alternative 4: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9200000:\\ \;\;\;\;x - \frac{\frac{1}{y} + \left(x + -1\right)}{y}\\ \mathbf{elif}\;y \leq 255000000:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x + -1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9200000.0)
   (- x (/ (+ (/ 1.0 y) (+ x -1.0)) y))
   (if (<= y 255000000.0)
     (- 1.0 (/ (* y (- 1.0 x)) (+ y 1.0)))
     (- x (/ (+ x -1.0) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9200000.0) {
		tmp = x - (((1.0 / y) + (x + -1.0)) / y);
	} else if (y <= 255000000.0) {
		tmp = 1.0 - ((y * (1.0 - x)) / (y + 1.0));
	} else {
		tmp = x - ((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 <= (-9200000.0d0)) then
        tmp = x - (((1.0d0 / y) + (x + (-1.0d0))) / y)
    else if (y <= 255000000.0d0) then
        tmp = 1.0d0 - ((y * (1.0d0 - x)) / (y + 1.0d0))
    else
        tmp = x - ((x + (-1.0d0)) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -9200000.0) {
		tmp = x - (((1.0 / y) + (x + -1.0)) / y);
	} else if (y <= 255000000.0) {
		tmp = 1.0 - ((y * (1.0 - x)) / (y + 1.0));
	} else {
		tmp = x - ((x + -1.0) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -9200000.0:
		tmp = x - (((1.0 / y) + (x + -1.0)) / y)
	elif y <= 255000000.0:
		tmp = 1.0 - ((y * (1.0 - x)) / (y + 1.0))
	else:
		tmp = x - ((x + -1.0) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9200000.0)
		tmp = Float64(x - Float64(Float64(Float64(1.0 / y) + Float64(x + -1.0)) / y));
	elseif (y <= 255000000.0)
		tmp = Float64(1.0 - Float64(Float64(y * Float64(1.0 - x)) / Float64(y + 1.0)));
	else
		tmp = Float64(x - Float64(Float64(x + -1.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -9200000.0)
		tmp = x - (((1.0 / y) + (x + -1.0)) / y);
	elseif (y <= 255000000.0)
		tmp = 1.0 - ((y * (1.0 - x)) / (y + 1.0));
	else
		tmp = x - ((x + -1.0) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9200000.0], N[(x - N[(N[(N[(1.0 / y), $MachinePrecision] + N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 255000000.0], N[(1.0 - N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.2e6

   1. Initial program 32.7%

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

   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. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. distribute-neg-frac 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-neg-in 100.0%

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

      9. metadata-eval 100.0%

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

      10. sub-neg 100.0%

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

      11. div-sub 100.0%

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

      12. sub-neg 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

      15. unpow2 100.0%

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

   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%

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

      2. associate-/r* 100.0%

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

   7. Simplified 100.0%

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

      1. add-sqr-sqrt 47.1%

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

      2. sqrt-unprod 99.4%

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

      3. associate-/l/ 99.4%

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

      4. associate-/l/ 99.4%

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

      5. frac-times 99.4%

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

      6. metadata-eval 99.4%

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

      7. metadata-eval 99.4%

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

      8. frac-times 99.4%

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

      9. sqrt-unprod 99.4%

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

      10. add-sqr-sqrt 99.4%

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

      11. associate-/r* 99.4%

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

      12. div-inv 99.4%

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

      13. metadata-eval 99.4%

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

      14. distribute-neg-frac 99.4%

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

      15. cancel-sign-sub-inv 99.4%

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

      16. div-inv 99.4%

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

      17. sub-div 99.4%

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

   9. Applied egg-rr 100.0%

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

   if -9.2e6 < y < 2.55e8

   1. Initial program 100.0%

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

   if 2.55e8 < y

   1. Initial program 30.5%

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

   2. Taylor expanded in y around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. metadata-eval 100.0%

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

      8. sub-neg 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9200000:\\ \;\;\;\;x - \frac{\frac{1}{y} + \left(x + -1\right)}{y}\\ \mathbf{elif}\;y \leq 255000000:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x + -1}{y}\\ \end{array} \]

Alternative 5: 98.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1960000 \lor \neg \left(y \leq 14500000\right):\\ \;\;\;\;x - \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1960000.0) (not (<= y 14500000.0)))
   (- x (/ (+ x -1.0) y))
   (+ 1.0 (* y (/ x (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1960000.0) || !(y <= 14500000.0)) {
		tmp = x - ((x + -1.0) / y);
	} else {
		tmp = 1.0 + (y * (x / (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 <= (-1960000.0d0)) .or. (.not. (y <= 14500000.0d0))) then
        tmp = x - ((x + (-1.0d0)) / y)
    else
        tmp = 1.0d0 + (y * (x / (y + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1960000.0) || !(y <= 14500000.0)) {
		tmp = x - ((x + -1.0) / y);
	} else {
		tmp = 1.0 + (y * (x / (y + 1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1960000.0) or not (y <= 14500000.0):
		tmp = x - ((x + -1.0) / y)
	else:
		tmp = 1.0 + (y * (x / (y + 1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1960000.0) || !(y <= 14500000.0))
		tmp = Float64(x - Float64(Float64(x + -1.0) / y));
	else
		tmp = Float64(1.0 + Float64(y * Float64(x / Float64(y + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1960000.0) || ~((y <= 14500000.0)))
		tmp = x - ((x + -1.0) / y);
	else
		tmp = 1.0 + (y * (x / (y + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1960000.0], N[Not[LessEqual[y, 14500000.0]], $MachinePrecision]], N[(x - N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.96e6 or 1.45e7 < y

   1. Initial program 31.7%

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

   2. Taylor expanded in y around -inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. sub-neg 99.7%

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

      3. metadata-eval 99.7%

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

      4. distribute-neg-frac 99.7%

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

      5. +-commutative 99.7%

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

      6. distribute-neg-in 99.7%

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

      7. metadata-eval 99.7%

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

      8. sub-neg 99.7%

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

   4. Simplified 99.7%

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

   if -1.96e6 < y < 1.45e7

   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 x around inf 98.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1960000 \lor \neg \left(y \leq 14500000\right):\\ \;\;\;\;x - \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x}{y + 1}\\ \end{array} \]

Alternative 6: 86.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{-9} \lor \neg \left(y \leq 0.00135\right):\\ \;\;\;\;x \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4.3e-9) (not (<= y 0.00135)))
   (* x (/ y (+ y 1.0)))
   (+ 1.0 (* y x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4.3e-9) || !(y <= 0.00135)) {
		tmp = x * (y / (y + 1.0));
	} 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 <= (-4.3d-9)) .or. (.not. (y <= 0.00135d0))) then
        tmp = x * (y / (y + 1.0d0))
    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 <= -4.3e-9) || !(y <= 0.00135)) {
		tmp = x * (y / (y + 1.0));
	} else {
		tmp = 1.0 + (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4.3e-9) or not (y <= 0.00135):
		tmp = x * (y / (y + 1.0))
	else:
		tmp = 1.0 + (y * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4.3e-9) || !(y <= 0.00135))
		tmp = Float64(x * Float64(y / Float64(y + 1.0)));
	else
		tmp = Float64(1.0 + Float64(y * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -4.3e-9) || ~((y <= 0.00135)))
		tmp = x * (y / (y + 1.0));
	else
		tmp = 1.0 + (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4.3e-9], N[Not[LessEqual[y, 0.00135]], $MachinePrecision]], N[(x * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.29999999999999963e-9 or 0.0013500000000000001 < y

   1. Initial program 34.8%

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

      1. sub-neg 34.8%

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

      2. associate-*l/ 53.6%

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

      3. distribute-lft-neg-in 53.6%

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

      4. distribute-frac-neg 53.6%

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

      5. neg-sub0 53.6%

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

      6. associate--r- 53.6%

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

      7. metadata-eval 53.6%

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

      8. +-commutative 53.6%

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

      9. +-commutative 53.6%

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

   3. Simplified 53.6%

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

   4. Taylor expanded in x around inf 58.3%

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

      1. associate-*r/ 77.3%

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

      2. *-commutative 77.3%

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

   6. Simplified 77.3%

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

   if -4.29999999999999963e-9 < y < 0.0013500000000000001

   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 x around inf 98.8%

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

   5. Taylor expanded in y around 0 98.4%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{-9} \lor \neg \left(y \leq 0.00135\right):\\ \;\;\;\;x \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot x\\ \end{array} \]

Alternative 7: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.2\right):\\ \;\;\;\;x - \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.2))) (- x (/ (+ x -1.0) y)) (+ 1.0 (* y x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.2)) {
		tmp = x - ((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 <= 1.2d0))) then
        tmp = x - ((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 <= 1.2)) {
		tmp = x - ((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 <= 1.2):
		tmp = x - ((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 <= 1.2))
		tmp = Float64(x - Float64(Float64(x + -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 <= 1.2)))
		tmp = x - ((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, 1.2]], $MachinePrecision]], N[(x - N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.19999999999999996 < y

   1. Initial program 33.8%

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

   2. Taylor expanded in y around -inf 99.0%

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

      1. mul-1-neg 99.0%

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

      2. sub-neg 99.0%

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

      3. metadata-eval 99.0%

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

      4. distribute-neg-frac 99.0%

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

      5. +-commutative 99.0%

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

      6. distribute-neg-in 99.0%

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

      7. metadata-eval 99.0%

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

      8. sub-neg 99.0%

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

   4. Simplified 99.0%

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

   if -1 < y < 1.19999999999999996

   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 x around inf 98.8%

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

   5. Taylor expanded in y around 0 97.5%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.2\right):\\ \;\;\;\;x - \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot x\\ \end{array} \]

Alternative 8: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 33.8%

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

   2. Taylor expanded in y around -inf 99.0%

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

      1. mul-1-neg 99.0%

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

      2. sub-neg 99.0%

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

      3. metadata-eval 99.0%

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

      4. distribute-neg-frac 99.0%

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

      5. +-commutative 99.0%

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

      6. distribute-neg-in 99.0%

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

      7. metadata-eval 99.0%

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

      8. sub-neg 99.0%

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

   4. Simplified 99.0%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 98.1%

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

      1. sub-neg 98.1%

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

      2. distribute-rgt-in 98.1%

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

      3. distribute-lft-neg-out 98.1%

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

      4. unsub-neg 98.1%

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

      5. *-lft-identity 98.1%

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

      6. *-commutative 98.1%

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

   4. Simplified 98.1%

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

4. Final simplification 98.6%

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

Alternative 9: 86.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 11.4\right):\\ \;\;\;\;x - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 11.4))) (- x (/ x y)) (+ 1.0 (* y x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 11.4)) {
		tmp = x - (x / 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 <= 11.4d0))) then
        tmp = x - (x / 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 <= 11.4)) {
		tmp = x - (x / y);
	} else {
		tmp = 1.0 + (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 11.4):
		tmp = x - (x / y)
	else:
		tmp = 1.0 + (y * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 11.4))
		tmp = Float64(x - Float64(x / 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 <= 11.4)))
		tmp = x - (x / 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, 11.4]], $MachinePrecision]], N[(x - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 11.4000000000000004 < y

   1. Initial program 33.8%

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

      1. sub-neg 33.8%

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

      2. associate-*l/ 52.9%

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

      3. distribute-lft-neg-in 52.9%

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

      4. distribute-frac-neg 52.9%

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

      5. neg-sub0 52.9%

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

      6. associate--r- 52.9%

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

      7. metadata-eval 52.9%

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

      8. +-commutative 52.9%

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

      9. +-commutative 52.9%

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

   3. Simplified 52.9%

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

   4. Taylor expanded in x around inf 57.7%

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

      1. associate-*r/ 76.9%

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

      2. *-commutative 76.9%

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

   6. Simplified 76.9%

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

   7. Taylor expanded in y around inf 76.3%

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

      1. mul-1-neg 76.3%

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

      2. unsub-neg 76.3%

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

   9. Simplified 76.3%

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

   if -1 < y < 11.4000000000000004

   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 x around inf 98.8%

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

   5. Taylor expanded in y around 0 97.5%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 11.4\right):\\ \;\;\;\;x - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot x\\ \end{array} \]

Alternative 10: 86.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 27:\\ \;\;\;\;1 + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 27.0) (+ 1.0 (* y x)) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 27.0) {
		tmp = 1.0 + (y * x);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x
    else if (y <= 27.0d0) then
        tmp = 1.0d0 + (y * x)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 27.0) {
		tmp = 1.0 + (y * x);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 27.0:
		tmp = 1.0 + (y * x)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 27.0)
		tmp = Float64(1.0 + Float64(y * x));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 27.0)
		tmp = 1.0 + (y * x);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 27.0], N[(1.0 + N[(y * x), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 27 < y

   1. Initial program 33.8%

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

      1. sub-neg 33.8%

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

      2. associate-*l/ 52.9%

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

      3. distribute-lft-neg-in 52.9%

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

      4. distribute-frac-neg 52.9%

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

      5. neg-sub0 52.9%

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

      6. associate--r- 52.9%

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

      7. metadata-eval 52.9%

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

      8. +-commutative 52.9%

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

      9. +-commutative 52.9%

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

   3. Simplified 52.9%

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

   4. Taylor expanded in y around inf 75.1%

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

   if -1 < y < 27

   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 x around inf 98.8%

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

   5. Taylor expanded in y around 0 97.5%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 27:\\ \;\;\;\;1 + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11: 74.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.0024:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 0.0024) (- 1.0 y) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.0024) {
		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 <= 0.0024d0) 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 <= 0.0024) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 0.0024:
		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 <= 0.0024)
		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 <= 0.0024)
		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, 0.0024], N[(1.0 - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.00239999999999999979 < y

   1. Initial program 34.3%

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

      1. sub-neg 34.3%

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

      2. associate-*l/ 53.3%

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

      3. distribute-lft-neg-in 53.3%

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

      4. distribute-frac-neg 53.3%

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

      5. neg-sub0 53.3%

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

      6. associate--r- 53.3%

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

      7. metadata-eval 53.3%

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

      8. +-commutative 53.3%

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

      9. +-commutative 53.3%

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

   3. Simplified 53.3%

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

   4. Taylor expanded in y around inf 74.6%

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

   if -1 < y < 0.00239999999999999979

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 77.1%

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

   3. Taylor expanded in y around 0 76.5%

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

      1. neg-mul-1 76.5%

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

      2. unsub-neg 76.5%

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

   5. Simplified 76.5%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.0024:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12: 73.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.0024:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1.0) x (if (<= y 0.0024) 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.0024) {
		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 <= 0.0024d0) 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 <= 0.0024) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 0.0024:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 0.0024)
		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 <= 0.0024)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 0.0024], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.00239999999999999979 < y

   1. Initial program 34.3%

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

      1. sub-neg 34.3%

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

      2. associate-*l/ 53.3%

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

      3. distribute-lft-neg-in 53.3%

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

      4. distribute-frac-neg 53.3%

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

      5. neg-sub0 53.3%

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

      6. associate--r- 53.3%

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

      7. metadata-eval 53.3%

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

      8. +-commutative 53.3%

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

      9. +-commutative 53.3%

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

   3. Simplified 53.3%

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

   4. Taylor expanded in y around inf 74.6%

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

   if -1 < y < 0.00239999999999999979

   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 75.9%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.0024:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13: 38.3% 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 66.4%

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

   1. sub-neg 66.4%

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

   2. associate-*l/ 76.1%

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

   3. distribute-lft-neg-in 76.1%

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

   4. distribute-frac-neg 76.1%

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

   5. neg-sub0 76.1%

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

   6. associate--r- 76.1%

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

   7. metadata-eval 76.1%

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

   8. +-commutative 76.1%

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

   9. +-commutative 76.1%

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

3. Simplified 76.1%

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

4. Taylor expanded in y around 0 38.9%

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

5. Final simplification 38.9%

   \[\leadsto 1 \]

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 2023285 
(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))))