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

Percentage Accurate: 65.5% → 99.9%

Time: 8.7s

Alternatives: 14

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: 65.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -190000 \lor \neg \left(y \leq 14000\right):\\ \;\;\;\;\left(x + \left(\frac{1 - x}{{y}^{3}} - \frac{x + -1}{y}\right)\right) + \frac{x + -1}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -190000.0) (not (<= y 14000.0)))
   (+
    (+ x (- (/ (- 1.0 x) (pow y 3.0)) (/ (+ x -1.0) y)))
    (/ (+ x -1.0) (* y y)))
   (+ 1.0 (/ (* y (+ x -1.0)) (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -190000.0) || !(y <= 14000.0)) {
		tmp = (x + (((1.0 - x) / pow(y, 3.0)) - ((x + -1.0) / y))) + ((x + -1.0) / (y * 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 <= (-190000.0d0)) .or. (.not. (y <= 14000.0d0))) then
        tmp = (x + (((1.0d0 - x) / (y ** 3.0d0)) - ((x + (-1.0d0)) / y))) + ((x + (-1.0d0)) / (y * 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 <= -190000.0) || !(y <= 14000.0)) {
		tmp = (x + (((1.0 - x) / Math.pow(y, 3.0)) - ((x + -1.0) / y))) + ((x + -1.0) / (y * y));
	} else {
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -190000.0) or not (y <= 14000.0):
		tmp = (x + (((1.0 - x) / math.pow(y, 3.0)) - ((x + -1.0) / y))) + ((x + -1.0) / (y * y))
	else:
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -190000.0) || !(y <= 14000.0))
		tmp = Float64(Float64(x + Float64(Float64(Float64(1.0 - x) / (y ^ 3.0)) - Float64(Float64(x + -1.0) / y))) + Float64(Float64(x + -1.0) / Float64(y * y)));
	else
		tmp = Float64(1.0 + Float64(Float64(y * Float64(x + -1.0)) / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -190000.0) || ~((y <= 14000.0)))
		tmp = (x + (((1.0 - x) / (y ^ 3.0)) - ((x + -1.0) / y))) + ((x + -1.0) / (y * 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, -190000.0], N[Not[LessEqual[y, 14000.0]], $MachinePrecision]], N[(N[(x + N[(N[(N[(1.0 - x), $MachinePrecision] / N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision] - N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x + -1.0), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.9e5 or 14000 < y

   1. Initial program 29.7%

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

      1. sub-neg 29.7%

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

      2. +-commutative 29.7%

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

      3. associate-/l* 57.2%

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

      4. distribute-neg-frac 57.2%

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

      5. *-lft-identity 57.2%

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

      6. associate-*l/ 57.2%

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

      7. fma-def 57.2%

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

      8. associate-/l* 57.2%

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

      9. *-lft-identity 57.2%

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

      10. +-commutative 57.2%

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

      11. neg-sub0 57.2%

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

      12. associate--r- 57.2%

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

      13. metadata-eval 57.2%

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

      14. +-commutative 57.2%

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

   3. Simplified 57.2%

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

   4. Taylor expanded in y around inf 100.0%

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

      1. +-commutative 100.0%

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

      2. associate--l+ 100.0%

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

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\left(x + \left(\frac{1 - x}{{y}^{3}} + \frac{1 - x}{y}\right)\right) + \frac{x + -1}{y \cdot y}} \]

   if -1.9e5 < y < 14000

   1. Initial program 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -190000 \lor \neg \left(y \leq 14000\right):\\ \;\;\;\;\left(x + \left(\frac{1 - x}{{y}^{3}} - \frac{x + -1}{y}\right)\right) + \frac{x + -1}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -210000 \lor \neg \left(y \leq 300000\right):\\ \;\;\;\;\frac{x + -1}{y \cdot y} + \left(x - \frac{x + -1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -210000.0) (not (<= y 300000.0)))
   (+ (/ (+ x -1.0) (* y y)) (- 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 <= -210000.0) || !(y <= 300000.0)) {
		tmp = ((x + -1.0) / (y * y)) + (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 <= (-210000.0d0)) .or. (.not. (y <= 300000.0d0))) then
        tmp = ((x + (-1.0d0)) / (y * y)) + (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 <= -210000.0) || !(y <= 300000.0)) {
		tmp = ((x + -1.0) / (y * y)) + (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 <= -210000.0) or not (y <= 300000.0):
		tmp = ((x + -1.0) / (y * y)) + (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 <= -210000.0) || !(y <= 300000.0))
		tmp = Float64(Float64(Float64(x + -1.0) / Float64(y * y)) + Float64(x - Float64(Float64(x + -1.0) / y)));
	else
		tmp = Float64(1.0 + Float64(Float64(y * Float64(x + -1.0)) / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -210000.0) || ~((y <= 300000.0)))
		tmp = ((x + -1.0) / (y * y)) + (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, -210000.0], N[Not[LessEqual[y, 300000.0]], $MachinePrecision]], N[(N[(N[(x + -1.0), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(x - N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.1e5 or 3e5 < y

   1. Initial program 29.1%

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

      1. sub-neg 29.1%

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

      2. +-commutative 29.1%

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

      3. associate-/l* 56.8%

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

      4. distribute-neg-frac 56.8%

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

      5. *-lft-identity 56.8%

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

      6. associate-*l/ 56.8%

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

      7. fma-def 56.8%

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

      8. associate-/l* 56.8%

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

      9. *-lft-identity 56.8%

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

      10. +-commutative 56.8%

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

      11. neg-sub0 56.8%

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

      12. associate--r- 56.8%

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

      13. metadata-eval 56.8%

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

      14. +-commutative 56.8%

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

   3. Simplified 56.8%

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

   4. Taylor expanded in y around inf 99.7%

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

      1. +-commutative 99.7%

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

      2. associate--l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. mul-1-neg 99.7%

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

      5. unsub-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

      8. div-sub 99.7%

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

      9. sub-neg 99.7%

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

      10. metadata-eval 99.7%

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

      11. unpow2 99.7%

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

   6. Simplified 99.7%

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

   if -2.1e5 < y < 3e5

   1. Initial program 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -210000 \lor \neg \left(y \leq 300000\right):\\ \;\;\;\;\frac{x + -1}{y \cdot y} + \left(x - \frac{x + -1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \end{array} \]

Alternative 3: 73.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{x}{y}\\ \mathbf{if}\;y \leq -8.5 \cdot 10^{+119}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -4.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{-55}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ x y))))
   (if (<= y -8.5e+119)
     x
     (if (<= y -4.5e+88)
       (/ 1.0 y)
       (if (<= y -1.0)
         t_0
         (if (<= y -2.6e-55) (* y x) (if (<= y 1.0) (- 1.0 y) t_0)))))))

C

double code(double x, double y) {
	double t_0 = x - (x / y);
	double tmp;
	if (y <= -8.5e+119) {
		tmp = x;
	} else if (y <= -4.5e+88) {
		tmp = 1.0 / y;
	} else if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= -2.6e-55) {
		tmp = y * x;
	} else if (y <= 1.0) {
		tmp = 1.0 - y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - (x / y)
    if (y <= (-8.5d+119)) then
        tmp = x
    else if (y <= (-4.5d+88)) then
        tmp = 1.0d0 / y
    else if (y <= (-1.0d0)) then
        tmp = t_0
    else if (y <= (-2.6d-55)) then
        tmp = y * x
    else if (y <= 1.0d0) then
        tmp = 1.0d0 - y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x - (x / y);
	double tmp;
	if (y <= -8.5e+119) {
		tmp = x;
	} else if (y <= -4.5e+88) {
		tmp = 1.0 / y;
	} else if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= -2.6e-55) {
		tmp = y * x;
	} else if (y <= 1.0) {
		tmp = 1.0 - y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x - (x / y)
	tmp = 0
	if y <= -8.5e+119:
		tmp = x
	elif y <= -4.5e+88:
		tmp = 1.0 / y
	elif y <= -1.0:
		tmp = t_0
	elif y <= -2.6e-55:
		tmp = y * x
	elif y <= 1.0:
		tmp = 1.0 - y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x - Float64(x / y))
	tmp = 0.0
	if (y <= -8.5e+119)
		tmp = x;
	elseif (y <= -4.5e+88)
		tmp = Float64(1.0 / y);
	elseif (y <= -1.0)
		tmp = t_0;
	elseif (y <= -2.6e-55)
		tmp = Float64(y * x);
	elseif (y <= 1.0)
		tmp = Float64(1.0 - y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x - (x / y);
	tmp = 0.0;
	if (y <= -8.5e+119)
		tmp = x;
	elseif (y <= -4.5e+88)
		tmp = 1.0 / y;
	elseif (y <= -1.0)
		tmp = t_0;
	elseif (y <= -2.6e-55)
		tmp = y * x;
	elseif (y <= 1.0)
		tmp = 1.0 - y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x - N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8.5e+119], x, If[LessEqual[y, -4.5e+88], N[(1.0 / y), $MachinePrecision], If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, -2.6e-55], N[(y * x), $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 - y), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -8.49999999999999997e119

   1. Initial program 18.1%

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

      1. sub-neg 18.1%

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

      2. +-commutative 18.1%

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

      3. associate-/l* 57.3%

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

      4. distribute-neg-frac 57.3%

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

      5. *-lft-identity 57.3%

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

      6. associate-*l/ 57.3%

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

      7. fma-def 57.3%

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

      8. associate-/l* 57.3%

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

      9. *-lft-identity 57.3%

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

      10. +-commutative 57.3%

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

      11. neg-sub0 57.3%

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

      12. associate--r- 57.3%

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

      13. metadata-eval 57.3%

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

      14. +-commutative 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in y around inf 77.6%

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

   if -8.49999999999999997e119 < y < -4.5e88

   1. Initial program 23.2%

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

      1. sub-neg 23.2%

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

      2. +-commutative 23.2%

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

      3. associate-/l* 23.2%

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

      4. distribute-neg-frac 23.2%

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

      5. *-lft-identity 23.2%

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

      6. associate-*l/ 23.2%

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

      7. fma-def 23.2%

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

      8. associate-/l* 23.2%

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

      9. *-lft-identity 23.2%

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

      10. +-commutative 23.2%

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

      11. neg-sub0 23.2%

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

      12. associate--r- 23.2%

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

      13. metadata-eval 23.2%

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

      14. +-commutative 23.2%

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

   3. Simplified 23.2%

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

   4. Taylor expanded in x around 0 3.8%

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

      1. mul-1-neg 3.8%

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

      2. sub-neg 3.8%

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

   6. Simplified 3.8%

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

   7. Taylor expanded in y around inf 80.4%

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

   if -4.5e88 < y < -1 or 1 < y

   1. Initial program 38.5%

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

      1. sub-neg 38.5%

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

      2. +-commutative 38.5%

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

      3. associate-/l* 62.6%

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

      4. distribute-neg-frac 62.6%

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

      5. *-lft-identity 62.6%

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

      6. associate-*l/ 62.6%

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

      7. fma-def 62.6%

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

      8. associate-/l* 62.6%

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

      9. *-lft-identity 62.6%

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

      10. +-commutative 62.6%

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

      11. neg-sub0 62.6%

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

      12. associate--r- 62.6%

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

      13. metadata-eval 62.6%

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

      14. +-commutative 62.6%

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

   3. Simplified 62.6%

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

   4. Taylor expanded in x around inf 52.4%

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

      1. associate-/l* 67.7%

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

   6. Simplified 67.7%

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

   7. Taylor expanded in y around inf 75.3%

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

      1. mul-1-neg 75.3%

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

      2. distribute-frac-neg 75.3%

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

      3. +-commutative 75.3%

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

      4. distribute-frac-neg 75.3%

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

      5. sub-neg 75.3%

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

   9. Simplified 75.3%

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

   if -1 < y < -2.5999999999999999e-55

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-/l* 99.3%

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

      4. distribute-neg-frac 99.3%

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

      5. *-lft-identity 99.3%

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

      6. associate-*l/ 99.7%

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

      7. fma-def 99.7%

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

      8. associate-/l* 99.9%

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

      9. *-lft-identity 99.9%

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

      10. +-commutative 99.9%

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

      11. neg-sub0 99.9%

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

      12. associate--r- 99.9%

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

      13. metadata-eval 99.9%

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

      14. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 73.5%

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

      1. associate-/l* 73.0%

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

   6. Simplified 73.0%

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

   7. Taylor expanded in y around 0 66.3%

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

   if -2.5999999999999999e-55 < y < 1

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-/l* 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. *-lft-identity 99.9%

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

      6. associate-*l/ 99.9%

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

      7. fma-def 99.9%

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

      8. associate-/l* 100.0%

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

      9. *-lft-identity 100.0%

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

      10. +-commutative 100.0%

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

      11. neg-sub0 100.0%

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

      12. associate--r- 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 71.3%

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

      1. mul-1-neg 71.3%

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

      2. sub-neg 71.3%

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

   6. Simplified 71.3%

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

   7. Taylor expanded in y around 0 71.3%

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

      1. neg-mul-1 71.3%

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

      2. unsub-neg 71.3%

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

   9. Simplified 71.3%

      \[\leadsto \color{blue}{1 - y} \]
3. Recombined 5 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+119}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -4.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;x - \frac{x}{y}\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{-55}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{y}\\ \end{array} \]

Alternative 4: 85.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+119}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -4.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq -1 \lor \neg \left(y \leq 1.1\right):\\ \;\;\;\;x - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(y \cdot x - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8.5e+119)
   x
   (if (<= y -4.5e+88)
     (/ 1.0 y)
     (if (or (<= y -1.0) (not (<= y 1.1)))
       (- x (/ x y))
       (+ 1.0 (- (* y x) y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8.5e+119) {
		tmp = x;
	} else if (y <= -4.5e+88) {
		tmp = 1.0 / y;
	} else if ((y <= -1.0) || !(y <= 1.1)) {
		tmp = x - (x / 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 <= (-8.5d+119)) then
        tmp = x
    else if (y <= (-4.5d+88)) then
        tmp = 1.0d0 / y
    else if ((y <= (-1.0d0)) .or. (.not. (y <= 1.1d0))) then
        tmp = x - (x / 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 <= -8.5e+119) {
		tmp = x;
	} else if (y <= -4.5e+88) {
		tmp = 1.0 / y;
	} else if ((y <= -1.0) || !(y <= 1.1)) {
		tmp = x - (x / y);
	} else {
		tmp = 1.0 + ((y * x) - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8.5e+119:
		tmp = x
	elif y <= -4.5e+88:
		tmp = 1.0 / y
	elif (y <= -1.0) or not (y <= 1.1):
		tmp = x - (x / y)
	else:
		tmp = 1.0 + ((y * x) - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8.5e+119)
		tmp = x;
	elseif (y <= -4.5e+88)
		tmp = Float64(1.0 / y);
	elseif ((y <= -1.0) || !(y <= 1.1))
		tmp = Float64(x - Float64(x / 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 <= -8.5e+119)
		tmp = x;
	elseif (y <= -4.5e+88)
		tmp = 1.0 / y;
	elseif ((y <= -1.0) || ~((y <= 1.1)))
		tmp = x - (x / y);
	else
		tmp = 1.0 + ((y * x) - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8.5e+119], x, If[LessEqual[y, -4.5e+88], N[(1.0 / y), $MachinePrecision], If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.1]], $MachinePrecision]], N[(x - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y * x), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -8.49999999999999997e119

   1. Initial program 18.1%

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

      1. sub-neg 18.1%

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

      2. +-commutative 18.1%

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

      3. associate-/l* 57.3%

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

      4. distribute-neg-frac 57.3%

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

      5. *-lft-identity 57.3%

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

      6. associate-*l/ 57.3%

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

      7. fma-def 57.3%

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

      8. associate-/l* 57.3%

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

      9. *-lft-identity 57.3%

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

      10. +-commutative 57.3%

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

      11. neg-sub0 57.3%

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

      12. associate--r- 57.3%

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

      13. metadata-eval 57.3%

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

      14. +-commutative 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in y around inf 77.6%

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

   if -8.49999999999999997e119 < y < -4.5e88

   1. Initial program 23.2%

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

      1. sub-neg 23.2%

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

      2. +-commutative 23.2%

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

      3. associate-/l* 23.2%

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

      4. distribute-neg-frac 23.2%

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

      5. *-lft-identity 23.2%

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

      6. associate-*l/ 23.2%

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

      7. fma-def 23.2%

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

      8. associate-/l* 23.2%

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

      9. *-lft-identity 23.2%

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

      10. +-commutative 23.2%

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

      11. neg-sub0 23.2%

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

      12. associate--r- 23.2%

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

      13. metadata-eval 23.2%

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

      14. +-commutative 23.2%

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

   3. Simplified 23.2%

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

   4. Taylor expanded in x around 0 3.8%

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

      1. mul-1-neg 3.8%

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

      2. sub-neg 3.8%

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

   6. Simplified 3.8%

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

   7. Taylor expanded in y around inf 80.4%

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

   if -4.5e88 < y < -1 or 1.1000000000000001 < y

   1. Initial program 38.5%

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

      1. sub-neg 38.5%

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

      2. +-commutative 38.5%

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

      3. associate-/l* 62.6%

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

      4. distribute-neg-frac 62.6%

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

      5. *-lft-identity 62.6%

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

      6. associate-*l/ 62.6%

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

      7. fma-def 62.6%

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

      8. associate-/l* 62.6%

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

      9. *-lft-identity 62.6%

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

      10. +-commutative 62.6%

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

      11. neg-sub0 62.6%

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

      12. associate--r- 62.6%

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

      13. metadata-eval 62.6%

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

      14. +-commutative 62.6%

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

   3. Simplified 62.6%

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

   4. Taylor expanded in x around inf 52.4%

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

      1. associate-/l* 67.7%

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

   6. Simplified 67.7%

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

   7. Taylor expanded in y around inf 75.3%

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

      1. mul-1-neg 75.3%

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

      2. distribute-frac-neg 75.3%

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

      3. +-commutative 75.3%

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

      4. distribute-frac-neg 75.3%

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

      5. sub-neg 75.3%

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

   9. Simplified 75.3%

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

   if -1 < y < 1.1000000000000001

   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. distribute-neg-frac 100.0%

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

      3. neg-mul-1 100.0%

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

      4. associate-*l/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*l/ 100.0%

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

      7. associate-/r/ 100.0%

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

      8. metadata-eval 100.0%

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

      9. distribute-neg-frac 100.0%

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

      10. cancel-sign-sub-inv 100.0%

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

      11. associate-/r/ 99.9%

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

      12. associate-/r* 99.9%

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

      13. neg-mul-1 99.9%

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

      14. associate-/r/ 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. associate-/r/ 99.9%

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

      17. distribute-neg-frac 99.9%

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

      18. metadata-eval 99.9%

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

      19. associate-/r/ 100.0%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 97.9%

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

      1. sub-neg 97.9%

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

      2. distribute-lft-in 97.9%

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

      3. *-rgt-identity 97.9%

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

      4. distribute-rgt-neg-out 97.9%

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

      5. unsub-neg 97.9%

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

   6. Simplified 97.9%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+119}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -4.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq -1 \lor \neg \left(y \leq 1.1\right):\\ \;\;\;\;x - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(y \cdot x - y\right)\\ \end{array} \]

Alternative 5: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -180000000 \lor \neg \left(y \leq 170000000\right):\\ \;\;\;\;x - \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -180000000.0) (not (<= y 170000000.0)))
   (- 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 <= -180000000.0) || !(y <= 170000000.0)) {
		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 <= (-180000000.0d0)) .or. (.not. (y <= 170000000.0d0))) 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 <= -180000000.0) || !(y <= 170000000.0)) {
		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 <= -180000000.0) or not (y <= 170000000.0):
		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 <= -180000000.0) || !(y <= 170000000.0))
		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 <= -180000000.0) || ~((y <= 170000000.0)))
		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, -180000000.0], N[Not[LessEqual[y, 170000000.0]], $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 < -1.8e8 or 1.7e8 < y

   1. Initial program 28.4%

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

      1. sub-neg 28.4%

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

      2. +-commutative 28.4%

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

      3. associate-/l* 56.6%

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

      4. distribute-neg-frac 56.6%

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

      5. *-lft-identity 56.6%

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

      6. associate-*l/ 56.6%

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

      7. fma-def 56.6%

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

      8. associate-/l* 56.6%

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

      9. *-lft-identity 56.6%

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

      10. +-commutative 56.6%

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

      11. neg-sub0 56.6%

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

      12. associate--r- 56.6%

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

      13. metadata-eval 56.6%

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

      14. +-commutative 56.6%

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

   3. Simplified 56.6%

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

   4. Taylor expanded in y around inf 99.4%

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

      1. +-commutative 99.4%

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

      2. mul-1-neg 99.4%

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

      3. unsub-neg 99.4%

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

      4. sub-neg 99.4%

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

      5. metadata-eval 99.4%

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

   6. Simplified 99.4%

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

   if -1.8e8 < y < 1.7e8

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. distribute-neg-frac 99.6%

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

      3. neg-mul-1 99.6%

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

      4. associate-*l/ 99.5%

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

      5. metadata-eval 99.5%

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

      6. associate-*l/ 99.5%

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

      7. associate-/r/ 99.5%

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

      8. metadata-eval 99.5%

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

      9. distribute-neg-frac 99.5%

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

      10. cancel-sign-sub-inv 99.5%

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

      11. associate-/r/ 99.5%

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

      12. associate-/r* 99.5%

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

      13. neg-mul-1 99.5%

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

      14. associate-/r/ 99.5%

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

      15. distribute-rgt-neg-in 99.5%

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

      16. associate-/r/ 99.5%

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

      17. distribute-neg-frac 99.5%

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

      18. metadata-eval 99.5%

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

      19. associate-/r/ 99.5%

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

   3. Simplified 99.5%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -180000000 \lor \neg \left(y \leq 170000000\right):\\ \;\;\;\;x - \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \end{array} \]

Alternative 6: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -18000000000 \lor \neg \left(y \leq 160000000\right):\\ \;\;\;\;x - \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -18000000000.0) (not (<= y 160000000.0)))
   (- 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 <= -18000000000.0) || !(y <= 160000000.0)) {
		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 <= (-18000000000.0d0)) .or. (.not. (y <= 160000000.0d0))) 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 <= -18000000000.0) || !(y <= 160000000.0)) {
		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 <= -18000000000.0) or not (y <= 160000000.0):
		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 <= -18000000000.0) || !(y <= 160000000.0))
		tmp = Float64(x - Float64(Float64(x + -1.0) / y));
	else
		tmp = Float64(1.0 + Float64(Float64(y * Float64(x + -1.0)) / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -18000000000.0) || ~((y <= 160000000.0)))
		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, -18000000000.0], N[Not[LessEqual[y, 160000000.0]], $MachinePrecision]], N[(x - N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8e10 or 1.6e8 < y

   1. Initial program 27.3%

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

      1. sub-neg 27.3%

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

      2. +-commutative 27.3%

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

      3. associate-/l* 55.8%

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

      4. distribute-neg-frac 55.8%

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

      5. *-lft-identity 55.8%

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

      6. associate-*l/ 55.8%

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

      7. fma-def 55.8%

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

      8. associate-/l* 55.9%

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

      9. *-lft-identity 55.9%

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

      10. +-commutative 55.9%

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

      11. neg-sub0 55.9%

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

      12. associate--r- 55.9%

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

      13. metadata-eval 55.9%

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

      14. +-commutative 55.9%

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

   3. Simplified 55.9%

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

   4. Taylor expanded in y around inf 99.4%

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

      1. +-commutative 99.4%

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

      2. mul-1-neg 99.4%

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

      3. unsub-neg 99.4%

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

      4. sub-neg 99.4%

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

      5. metadata-eval 99.4%

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

   6. Simplified 99.4%

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

   if -1.8e10 < y < 1.6e8

   1. Initial program 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -18000000000 \lor \neg \left(y \leq 160000000\right):\\ \;\;\;\;x - \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \end{array} \]

Alternative 7: 85.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+119}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -4.3 \cdot 10^{+88}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq -1 \lor \neg \left(y \leq 3\right):\\ \;\;\;\;x - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8.5e+119)
   x
   (if (<= y -4.3e+88)
     (/ 1.0 y)
     (if (or (<= y -1.0) (not (<= y 3.0))) (- x (/ x y)) (+ 1.0 (* y x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8.5e+119) {
		tmp = x;
	} else if (y <= -4.3e+88) {
		tmp = 1.0 / y;
	} else if ((y <= -1.0) || !(y <= 3.0)) {
		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 <= (-8.5d+119)) then
        tmp = x
    else if (y <= (-4.3d+88)) then
        tmp = 1.0d0 / y
    else if ((y <= (-1.0d0)) .or. (.not. (y <= 3.0d0))) 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 <= -8.5e+119) {
		tmp = x;
	} else if (y <= -4.3e+88) {
		tmp = 1.0 / y;
	} else if ((y <= -1.0) || !(y <= 3.0)) {
		tmp = x - (x / y);
	} else {
		tmp = 1.0 + (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8.5e+119:
		tmp = x
	elif y <= -4.3e+88:
		tmp = 1.0 / y
	elif (y <= -1.0) or not (y <= 3.0):
		tmp = x - (x / y)
	else:
		tmp = 1.0 + (y * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8.5e+119)
		tmp = x;
	elseif (y <= -4.3e+88)
		tmp = Float64(1.0 / y);
	elseif ((y <= -1.0) || !(y <= 3.0))
		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 <= -8.5e+119)
		tmp = x;
	elseif (y <= -4.3e+88)
		tmp = 1.0 / y;
	elseif ((y <= -1.0) || ~((y <= 3.0)))
		tmp = x - (x / y);
	else
		tmp = 1.0 + (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8.5e+119], x, If[LessEqual[y, -4.3e+88], N[(1.0 / y), $MachinePrecision], If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 3.0]], $MachinePrecision]], N[(x - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -8.49999999999999997e119

   1. Initial program 18.1%

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

      1. sub-neg 18.1%

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

      2. +-commutative 18.1%

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

      3. associate-/l* 57.3%

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

      4. distribute-neg-frac 57.3%

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

      5. *-lft-identity 57.3%

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

      6. associate-*l/ 57.3%

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

      7. fma-def 57.3%

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

      8. associate-/l* 57.3%

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

      9. *-lft-identity 57.3%

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

      10. +-commutative 57.3%

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

      11. neg-sub0 57.3%

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

      12. associate--r- 57.3%

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

      13. metadata-eval 57.3%

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

      14. +-commutative 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in y around inf 77.6%

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

   if -8.49999999999999997e119 < y < -4.29999999999999974e88

   1. Initial program 23.2%

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

      1. sub-neg 23.2%

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

      2. +-commutative 23.2%

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

      3. associate-/l* 23.2%

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

      4. distribute-neg-frac 23.2%

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

      5. *-lft-identity 23.2%

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

      6. associate-*l/ 23.2%

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

      7. fma-def 23.2%

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

      8. associate-/l* 23.2%

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

      9. *-lft-identity 23.2%

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

      10. +-commutative 23.2%

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

      11. neg-sub0 23.2%

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

      12. associate--r- 23.2%

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

      13. metadata-eval 23.2%

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

      14. +-commutative 23.2%

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

   3. Simplified 23.2%

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

   4. Taylor expanded in x around 0 3.8%

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

      1. mul-1-neg 3.8%

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

      2. sub-neg 3.8%

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

   6. Simplified 3.8%

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

   7. Taylor expanded in y around inf 80.4%

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

   if -4.29999999999999974e88 < y < -1 or 3 < y

   1. Initial program 38.5%

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

      1. sub-neg 38.5%

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

      2. +-commutative 38.5%

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

      3. associate-/l* 62.6%

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

      4. distribute-neg-frac 62.6%

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

      5. *-lft-identity 62.6%

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

      6. associate-*l/ 62.6%

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

      7. fma-def 62.6%

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

      8. associate-/l* 62.6%

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

      9. *-lft-identity 62.6%

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

      10. +-commutative 62.6%

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

      11. neg-sub0 62.6%

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

      12. associate--r- 62.6%

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

      13. metadata-eval 62.6%

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

      14. +-commutative 62.6%

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

   3. Simplified 62.6%

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

   4. Taylor expanded in x around inf 52.4%

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

      1. associate-/l* 67.7%

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

   6. Simplified 67.7%

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

   7. Taylor expanded in y around inf 75.3%

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

      1. mul-1-neg 75.3%

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

      2. distribute-frac-neg 75.3%

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

      3. +-commutative 75.3%

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

      4. distribute-frac-neg 75.3%

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

      5. sub-neg 75.3%

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

   9. Simplified 75.3%

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

   if -1 < y < 3

   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. distribute-neg-frac 100.0%

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

      3. neg-mul-1 100.0%

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

      4. associate-*l/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*l/ 100.0%

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

      7. associate-/r/ 100.0%

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

      8. metadata-eval 100.0%

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

      9. distribute-neg-frac 100.0%

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

      10. cancel-sign-sub-inv 100.0%

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

      11. associate-/r/ 99.9%

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

      12. associate-/r* 99.9%

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

      13. neg-mul-1 99.9%

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

      14. associate-/r/ 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. associate-/r/ 99.9%

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

      17. distribute-neg-frac 99.9%

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

      18. metadata-eval 99.9%

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

      19. associate-/r/ 100.0%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 97.9%

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

      1. sub-neg 97.9%

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

      2. distribute-lft-in 97.9%

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

      3. *-rgt-identity 97.9%

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

      4. distribute-rgt-neg-out 97.9%

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

      5. unsub-neg 97.9%

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

   6. Simplified 97.9%

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

   7. Taylor expanded in x around inf 97.4%

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

      1. mul-1-neg 97.4%

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

      2. distribute-rgt-neg-in 97.4%

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

   9. Simplified 97.4%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+119}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -4.3 \cdot 10^{+88}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq -1 \lor \neg \left(y \leq 3\right):\\ \;\;\;\;x - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot x\\ \end{array} \]

Alternative 8: 72.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+119}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq -3.05 \cdot 10^{-56}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.045:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8.5e+119)
   x
   (if (<= y -1.0)
     (/ 1.0 y)
     (if (<= y -3.05e-56) (* y x) (if (<= y 0.045) (- 1.0 y) x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8.5e+119) {
		tmp = x;
	} else if (y <= -1.0) {
		tmp = 1.0 / y;
	} else if (y <= -3.05e-56) {
		tmp = y * x;
	} else if (y <= 0.045) {
		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 <= (-8.5d+119)) then
        tmp = x
    else if (y <= (-1.0d0)) then
        tmp = 1.0d0 / y
    else if (y <= (-3.05d-56)) then
        tmp = y * x
    else if (y <= 0.045d0) 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 <= -8.5e+119) {
		tmp = x;
	} else if (y <= -1.0) {
		tmp = 1.0 / y;
	} else if (y <= -3.05e-56) {
		tmp = y * x;
	} else if (y <= 0.045) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8.5e+119:
		tmp = x
	elif y <= -1.0:
		tmp = 1.0 / y
	elif y <= -3.05e-56:
		tmp = y * x
	elif y <= 0.045:
		tmp = 1.0 - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8.5e+119)
		tmp = x;
	elseif (y <= -1.0)
		tmp = Float64(1.0 / y);
	elseif (y <= -3.05e-56)
		tmp = Float64(y * x);
	elseif (y <= 0.045)
		tmp = Float64(1.0 - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8.5e+119)
		tmp = x;
	elseif (y <= -1.0)
		tmp = 1.0 / y;
	elseif (y <= -3.05e-56)
		tmp = y * x;
	elseif (y <= 0.045)
		tmp = 1.0 - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8.5e+119], x, If[LessEqual[y, -1.0], N[(1.0 / y), $MachinePrecision], If[LessEqual[y, -3.05e-56], N[(y * x), $MachinePrecision], If[LessEqual[y, 0.045], N[(1.0 - y), $MachinePrecision], x]]]]

Derivation

1. Split input into 4 regimes

2. if y < -8.49999999999999997e119 or 0.044999999999999998 < y

   1. Initial program 28.5%

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

      1. sub-neg 28.5%

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

      2. +-commutative 28.5%

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

      3. associate-/l* 60.6%

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

      4. distribute-neg-frac 60.6%

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

      5. *-lft-identity 60.6%

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

      6. associate-*l/ 60.6%

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

      7. fma-def 60.6%

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

      8. associate-/l* 60.6%

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

      9. *-lft-identity 60.6%

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

      10. +-commutative 60.6%

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

      11. neg-sub0 60.6%

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

      12. associate--r- 60.6%

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

      13. metadata-eval 60.6%

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

      14. +-commutative 60.6%

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

   3. Simplified 60.6%

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

   4. Taylor expanded in y around inf 77.5%

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

   if -8.49999999999999997e119 < y < -1

   1. Initial program 45.7%

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

      1. sub-neg 45.7%

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

      2. +-commutative 45.7%

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

      3. associate-/l* 49.5%

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

      4. distribute-neg-frac 49.5%

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

      5. *-lft-identity 49.5%

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

      6. associate-*l/ 49.5%

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

      7. fma-def 49.5%

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

      8. associate-/l* 49.5%

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

      9. *-lft-identity 49.5%

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

      10. +-commutative 49.5%

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

      11. neg-sub0 49.5%

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

      12. associate--r- 49.5%

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

      13. metadata-eval 49.5%

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

      14. +-commutative 49.5%

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

   3. Simplified 49.5%

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

   4. Taylor expanded in x around 0 6.7%

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

      1. mul-1-neg 6.7%

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

      2. sub-neg 6.7%

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

   6. Simplified 6.7%

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

   7. Taylor expanded in y around inf 55.4%

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

   if -1 < y < -3.0499999999999999e-56

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-/l* 99.3%

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

      4. distribute-neg-frac 99.3%

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

      5. *-lft-identity 99.3%

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

      6. associate-*l/ 99.7%

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

      7. fma-def 99.7%

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

      8. associate-/l* 99.9%

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

      9. *-lft-identity 99.9%

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

      10. +-commutative 99.9%

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

      11. neg-sub0 99.9%

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

      12. associate--r- 99.9%

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

      13. metadata-eval 99.9%

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

      14. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 73.5%

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

      1. associate-/l* 73.0%

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

   6. Simplified 73.0%

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

   7. Taylor expanded in y around 0 66.3%

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

   if -3.0499999999999999e-56 < y < 0.044999999999999998

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-/l* 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. *-lft-identity 99.9%

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

      6. associate-*l/ 99.9%

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

      7. fma-def 99.9%

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

      8. associate-/l* 100.0%

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

      9. *-lft-identity 100.0%

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

      10. +-commutative 100.0%

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

      11. neg-sub0 100.0%

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

      12. associate--r- 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 72.5%

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

      1. mul-1-neg 72.5%

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

      2. sub-neg 72.5%

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

   6. Simplified 72.5%

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

   7. Taylor expanded in y around 0 72.5%

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

      1. neg-mul-1 72.5%

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

      2. unsub-neg 72.5%

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

   9. Simplified 72.5%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+119}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq -3.05 \cdot 10^{-56}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.045:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9: 98.5% 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 30.8%

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

      1. sub-neg 30.8%

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

      2. +-commutative 30.8%

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

      3. associate-/l* 57.8%

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

      4. distribute-neg-frac 57.8%

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

      5. *-lft-identity 57.8%

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

      6. associate-*l/ 57.8%

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

      7. fma-def 57.8%

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

      8. associate-/l* 57.9%

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

      9. *-lft-identity 57.9%

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

      10. +-commutative 57.9%

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

      11. neg-sub0 57.9%

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

      12. associate--r- 57.9%

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

      13. metadata-eval 57.9%

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

      14. +-commutative 57.9%

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

   3. Simplified 57.9%

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

   4. Taylor expanded in y around inf 97.7%

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

      1. +-commutative 97.7%

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

      2. mul-1-neg 97.7%

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

      3. unsub-neg 97.7%

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

      4. sub-neg 97.7%

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

      5. metadata-eval 97.7%

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

   6. Simplified 97.7%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. neg-mul-1 100.0%

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

      4. associate-*l/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*l/ 100.0%

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

      7. associate-/r/ 100.0%

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

      8. metadata-eval 100.0%

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

      9. distribute-neg-frac 100.0%

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

      10. cancel-sign-sub-inv 100.0%

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

      11. associate-/r/ 99.9%

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

      12. associate-/r* 99.9%

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

      13. neg-mul-1 99.9%

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

      14. associate-/r/ 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. associate-/r/ 99.9%

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

      17. distribute-neg-frac 99.9%

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

      18. metadata-eval 99.9%

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

      19. associate-/r/ 100.0%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 97.9%

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

      1. sub-neg 97.9%

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

      2. distribute-lft-in 97.9%

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

      3. *-rgt-identity 97.9%

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

      4. distribute-rgt-neg-out 97.9%

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

      5. unsub-neg 97.9%

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

   6. Simplified 97.9%

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

4. Final simplification 97.8%

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

Alternative 10: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 28.6%

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

      1. sub-neg 28.6%

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

      2. distribute-neg-frac 28.6%

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

      3. neg-mul-1 28.6%

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

      4. associate-*l/ 28.0%

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

      5. metadata-eval 28.0%

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

      6. associate-*l/ 28.0%

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

      7. associate-/r/ 28.0%

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

      8. metadata-eval 28.0%

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

      9. distribute-neg-frac 28.0%

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

      10. cancel-sign-sub-inv 28.0%

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

      11. associate-/r/ 28.5%

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

      12. associate-/r* 28.5%

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

      13. neg-mul-1 28.5%

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

      14. associate-/r/ 28.0%

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

      15. distribute-rgt-neg-in 28.0%

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

      16. associate-/r/ 28.5%

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

      17. distribute-neg-frac 28.5%

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

      18. metadata-eval 28.5%

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

      19. associate-/r/ 28.0%

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

   3. Simplified 53.7%

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

   4. Taylor expanded in y around inf 97.7%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. neg-mul-1 100.0%

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

      4. associate-*l/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*l/ 100.0%

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

      7. associate-/r/ 100.0%

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

      8. metadata-eval 100.0%

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

      9. distribute-neg-frac 100.0%

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

      10. cancel-sign-sub-inv 100.0%

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

      11. associate-/r/ 99.9%

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

      12. associate-/r* 99.9%

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

      13. neg-mul-1 99.9%

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

      14. associate-/r/ 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. associate-/r/ 99.9%

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

      17. distribute-neg-frac 99.9%

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

      18. metadata-eval 99.9%

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

      19. associate-/r/ 100.0%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 97.9%

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

      1. sub-neg 97.9%

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

      2. distribute-lft-in 97.9%

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

      3. *-rgt-identity 97.9%

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

      4. distribute-rgt-neg-out 97.9%

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

      5. unsub-neg 97.9%

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

   6. Simplified 97.9%

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

   if 1 < y

   1. Initial program 33.2%

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

      1. sub-neg 33.2%

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

      2. +-commutative 33.2%

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

      3. associate-/l* 61.5%

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

      4. distribute-neg-frac 61.5%

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

      5. *-lft-identity 61.5%

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

      6. associate-*l/ 61.5%

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

      7. fma-def 61.5%

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

      8. associate-/l* 61.6%

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

      9. *-lft-identity 61.6%

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

      10. +-commutative 61.6%

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

      11. neg-sub0 61.6%

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

      12. associate--r- 61.6%

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

      13. metadata-eval 61.6%

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

      14. +-commutative 61.6%

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

   3. Simplified 61.6%

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

   4. Taylor expanded in y around inf 97.6%

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

      1. +-commutative 97.6%

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

      2. mul-1-neg 97.6%

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

      3. unsub-neg 97.6%

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

      4. sub-neg 97.6%

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

      5. metadata-eval 97.6%

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

   6. Simplified 97.6%

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

4. Final simplification 97.8%

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

Alternative 11: 74.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.22 \cdot 10^{-52}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.039:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   x
   (if (<= y -1.22e-52) (* y x) (if (<= y 0.039) (- 1.0 y) x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= -1.22e-52) {
		tmp = y * x;
	} else if (y <= 0.039) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x
    else if (y <= (-1.22d-52)) then
        tmp = y * x
    else if (y <= 0.039d0) then
        tmp = 1.0d0 - y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= -1.22e-52) {
		tmp = y * x;
	} else if (y <= 0.039) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= -1.22e-52:
		tmp = y * x
	elif y <= 0.039:
		tmp = 1.0 - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= -1.22e-52)
		tmp = Float64(y * x);
	elseif (y <= 0.039)
		tmp = Float64(1.0 - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x;
	elseif (y <= -1.22e-52)
		tmp = y * x;
	elseif (y <= 0.039)
		tmp = 1.0 - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, -1.22e-52], N[(y * x), $MachinePrecision], If[LessEqual[y, 0.039], N[(1.0 - y), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 0.0389999999999999999 < y

   1. Initial program 31.9%

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

      1. sub-neg 31.9%

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

      2. +-commutative 31.9%

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

      3. associate-/l* 58.5%

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

      4. distribute-neg-frac 58.5%

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

      5. *-lft-identity 58.5%

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

      6. associate-*l/ 58.4%

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

      7. fma-def 58.4%

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

      8. associate-/l* 58.5%

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

      9. *-lft-identity 58.5%

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

      10. +-commutative 58.5%

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

      11. neg-sub0 58.5%

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

      12. associate--r- 58.5%

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

      13. metadata-eval 58.5%

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

      14. +-commutative 58.5%

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

   3. Simplified 58.5%

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

   4. Taylor expanded in y around inf 70.2%

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

   if -1 < y < -1.22e-52

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-/l* 99.3%

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

      4. distribute-neg-frac 99.3%

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

      5. *-lft-identity 99.3%

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

      6. associate-*l/ 99.7%

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

      7. fma-def 99.7%

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

      8. associate-/l* 99.9%

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

      9. *-lft-identity 99.9%

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

      10. +-commutative 99.9%

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

      11. neg-sub0 99.9%

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

      12. associate--r- 99.9%

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

      13. metadata-eval 99.9%

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

      14. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 73.5%

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

      1. associate-/l* 73.0%

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

   6. Simplified 73.0%

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

   7. Taylor expanded in y around 0 66.3%

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

   if -1.22e-52 < y < 0.0389999999999999999

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-/l* 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. *-lft-identity 99.9%

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

      6. associate-*l/ 99.9%

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

      7. fma-def 99.9%

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

      8. associate-/l* 100.0%

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

      9. *-lft-identity 100.0%

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

      10. +-commutative 100.0%

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

      11. neg-sub0 100.0%

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

      12. associate--r- 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 72.5%

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

      1. mul-1-neg 72.5%

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

      2. sub-neg 72.5%

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

   6. Simplified 72.5%

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

   7. Taylor expanded in y around 0 72.5%

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

      1. neg-mul-1 72.5%

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

      2. unsub-neg 72.5%

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

   9. Simplified 72.5%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.22 \cdot 10^{-52}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.039:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12: 74.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-58}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.0145:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y -8.5e-58) (* y x) (if (<= y 0.0145) 1.0 x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= -8.5e-58) {
		tmp = y * x;
	} else if (y <= 0.0145) {
		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 <= (-8.5d-58)) then
        tmp = y * x
    else if (y <= 0.0145d0) 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 <= -8.5e-58) {
		tmp = y * x;
	} else if (y <= 0.0145) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= -8.5e-58:
		tmp = y * x
	elif y <= 0.0145:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= -8.5e-58)
		tmp = Float64(y * x);
	elseif (y <= 0.0145)
		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 <= -8.5e-58)
		tmp = y * x;
	elseif (y <= 0.0145)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, -8.5e-58], N[(y * x), $MachinePrecision], If[LessEqual[y, 0.0145], 1.0, x]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 0.0145000000000000007 < y

   1. Initial program 31.9%

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

      1. sub-neg 31.9%

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

      2. +-commutative 31.9%

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

      3. associate-/l* 58.5%

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

      4. distribute-neg-frac 58.5%

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

      5. *-lft-identity 58.5%

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

      6. associate-*l/ 58.4%

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

      7. fma-def 58.4%

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

      8. associate-/l* 58.5%

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

      9. *-lft-identity 58.5%

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

      10. +-commutative 58.5%

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

      11. neg-sub0 58.5%

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

      12. associate--r- 58.5%

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

      13. metadata-eval 58.5%

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

      14. +-commutative 58.5%

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

   3. Simplified 58.5%

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

   4. Taylor expanded in y around inf 70.2%

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

   if -1 < y < -8.5000000000000004e-58

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-/l* 99.3%

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

      4. distribute-neg-frac 99.3%

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

      5. *-lft-identity 99.3%

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

      6. associate-*l/ 99.7%

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

      7. fma-def 99.7%

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

      8. associate-/l* 99.9%

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

      9. *-lft-identity 99.9%

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

      10. +-commutative 99.9%

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

      11. neg-sub0 99.9%

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

      12. associate--r- 99.9%

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

      13. metadata-eval 99.9%

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

      14. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 73.5%

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

      1. associate-/l* 73.0%

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

   6. Simplified 73.0%

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

   7. Taylor expanded in y around 0 66.3%

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

   if -8.5000000000000004e-58 < y < 0.0145000000000000007

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-/l* 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. *-lft-identity 99.9%

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

      6. associate-*l/ 99.9%

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

      7. fma-def 99.9%

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

      8. associate-/l* 100.0%

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

      9. *-lft-identity 100.0%

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

      10. +-commutative 100.0%

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

      11. neg-sub0 100.0%

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

      12. associate--r- 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 72.2%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-58}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.0145:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13: 74.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.026:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1.0) x (if (<= y 0.026) 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.026) {
		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 <= 0.026d0) 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 <= 0.026) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 0.026:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 0.026)
		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 <= 0.026)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 0.026], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.0259999999999999988 < y

   1. Initial program 31.9%

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

      1. sub-neg 31.9%

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

      2. +-commutative 31.9%

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

      3. associate-/l* 58.5%

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

      4. distribute-neg-frac 58.5%

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

      5. *-lft-identity 58.5%

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

      6. associate-*l/ 58.4%

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

      7. fma-def 58.4%

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

      8. associate-/l* 58.5%

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

      9. *-lft-identity 58.5%

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

      10. +-commutative 58.5%

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

      11. neg-sub0 58.5%

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

      12. associate--r- 58.5%

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

      13. metadata-eval 58.5%

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

      14. +-commutative 58.5%

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

   3. Simplified 58.5%

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

   4. Taylor expanded in y around inf 70.2%

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

   if -1 < y < 0.0259999999999999988

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-/l* 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. *-lft-identity 99.8%

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

      6. associate-*l/ 99.9%

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

      7. fma-def 99.9%

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

      8. associate-/l* 100.0%

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

      9. *-lft-identity 100.0%

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

      10. +-commutative 100.0%

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

      11. neg-sub0 100.0%

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

      12. associate--r- 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 67.6%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.026:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 14: 38.9% 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 65.4%

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

   1. sub-neg 65.4%

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

   2. +-commutative 65.4%

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

   3. associate-/l* 78.8%

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

   4. distribute-neg-frac 78.8%

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

   5. *-lft-identity 78.8%

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

   6. associate-*l/ 78.9%

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

   7. fma-def 78.9%

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

   8. associate-/l* 78.9%

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

   9. *-lft-identity 78.9%

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

   10. +-commutative 78.9%

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

   11. neg-sub0 78.9%

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

   12. associate--r- 78.9%

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

   13. metadata-eval 78.9%

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

   14. +-commutative 78.9%

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

3. Simplified 78.9%

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

4. Taylor expanded in y around 0 35.3%

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

5. Final simplification 35.3%

   \[\leadsto 1 \]

Developer target: 99.7% 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 2023221 
(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))))