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

Percentage Accurate: 66.1% → 99.2%

Time: 10.2s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot \left(1 - x\right)}{1 + y}\\ t_1 := \frac{y}{1 + y}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-13}:\\ \;\;\;\;1 + t\_1 \cdot \left(x + -1\right)\\ \mathbf{elif}\;t\_0 \leq 1.002:\\ \;\;\;\;x - \frac{\left(x + -1\right) - \frac{\left(x + -1\right) + \frac{1 - x}{y}}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{x \cdot \left(-1 - y\right)} + \left(t\_1 + \frac{1}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* y (- 1.0 x)) (+ 1.0 y))) (t_1 (/ y (+ 1.0 y))))
   (if (<= t_0 5e-13)
     (+ 1.0 (* t_1 (+ x -1.0)))
     (if (<= t_0 1.002)
       (- x (/ (- (+ x -1.0) (/ (+ (+ x -1.0) (/ (- 1.0 x) y)) y)) y))
       (* x (+ (/ y (* x (- -1.0 y))) (+ t_1 (/ 1.0 x))))))))

C

double code(double x, double y) {
	double t_0 = (y * (1.0 - x)) / (1.0 + y);
	double t_1 = y / (1.0 + y);
	double tmp;
	if (t_0 <= 5e-13) {
		tmp = 1.0 + (t_1 * (x + -1.0));
	} else if (t_0 <= 1.002) {
		tmp = x - (((x + -1.0) - (((x + -1.0) + ((1.0 - x) / y)) / y)) / y);
	} else {
		tmp = x * ((y / (x * (-1.0 - y))) + (t_1 + (1.0 / x)));
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (y * (1.0d0 - x)) / (1.0d0 + y)
    t_1 = y / (1.0d0 + y)
    if (t_0 <= 5d-13) then
        tmp = 1.0d0 + (t_1 * (x + (-1.0d0)))
    else if (t_0 <= 1.002d0) then
        tmp = x - (((x + (-1.0d0)) - (((x + (-1.0d0)) + ((1.0d0 - x) / y)) / y)) / y)
    else
        tmp = x * ((y / (x * ((-1.0d0) - y))) + (t_1 + (1.0d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * (1.0 - x)) / (1.0 + y);
	double t_1 = y / (1.0 + y);
	double tmp;
	if (t_0 <= 5e-13) {
		tmp = 1.0 + (t_1 * (x + -1.0));
	} else if (t_0 <= 1.002) {
		tmp = x - (((x + -1.0) - (((x + -1.0) + ((1.0 - x) / y)) / y)) / y);
	} else {
		tmp = x * ((y / (x * (-1.0 - y))) + (t_1 + (1.0 / x)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * (1.0 - x)) / (1.0 + y)
	t_1 = y / (1.0 + y)
	tmp = 0
	if t_0 <= 5e-13:
		tmp = 1.0 + (t_1 * (x + -1.0))
	elif t_0 <= 1.002:
		tmp = x - (((x + -1.0) - (((x + -1.0) + ((1.0 - x) / y)) / y)) / y)
	else:
		tmp = x * ((y / (x * (-1.0 - y))) + (t_1 + (1.0 / x)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * Float64(1.0 - x)) / Float64(1.0 + y))
	t_1 = Float64(y / Float64(1.0 + y))
	tmp = 0.0
	if (t_0 <= 5e-13)
		tmp = Float64(1.0 + Float64(t_1 * Float64(x + -1.0)));
	elseif (t_0 <= 1.002)
		tmp = Float64(x - Float64(Float64(Float64(x + -1.0) - Float64(Float64(Float64(x + -1.0) + Float64(Float64(1.0 - x) / y)) / y)) / y));
	else
		tmp = Float64(x * Float64(Float64(y / Float64(x * Float64(-1.0 - y))) + Float64(t_1 + Float64(1.0 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * (1.0 - x)) / (1.0 + y);
	t_1 = y / (1.0 + y);
	tmp = 0.0;
	if (t_0 <= 5e-13)
		tmp = 1.0 + (t_1 * (x + -1.0));
	elseif (t_0 <= 1.002)
		tmp = x - (((x + -1.0) - (((x + -1.0) + ((1.0 - x) / y)) / y)) / y);
	else
		tmp = x * ((y / (x * (-1.0 - y))) + (t_1 + (1.0 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-13], N[(1.0 + N[(t$95$1 * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.002], N[(x - N[(N[(N[(x + -1.0), $MachinePrecision] - N[(N[(N[(x + -1.0), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y / N[(x * N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < 4.9999999999999999e-13

   1. Initial program 86.6%

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

      1. associate-/l* 100.0%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   if 4.9999999999999999e-13 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < 1.002

   1. Initial program 10.9%

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

      1. associate-/l* 10.9%

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

      2. +-commutative 10.9%

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

   3. Simplified 10.9%

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

   5. Taylor expanded in y around -inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. distribute-lft-out-- 99.9%

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

      3. distribute-lft-out-- 99.9%

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

      4. sub-neg 99.9%

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

      5. metadata-eval 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

      8. sub-neg 99.9%

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

      9. metadata-eval 99.9%

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

   7. Simplified 99.9%

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

   if 1.002 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))

   1. Initial program 62.3%

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

      1. sub-neg 62.3%

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

      2. +-commutative 62.3%

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

      3. associate-/l* 99.7%

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

      4. distribute-rgt-neg-in 99.7%

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

      5. fma-define 99.7%

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

      6. distribute-neg-frac2 99.7%

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

      7. +-commutative 99.7%

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

      8. distribute-neg-in 99.7%

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

      9. metadata-eval 99.7%

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

      10. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. distribute-neg-frac2 99.8%

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

      3. distribute-rgt-neg-out 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. metadata-eval 99.8%

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

      6. sub-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(1 - x\right)}{1 + y} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;1 + \frac{y}{1 + y} \cdot \left(x + -1\right)\\ \mathbf{elif}\;\frac{y \cdot \left(1 - x\right)}{1 + y} \leq 1.002:\\ \;\;\;\;x - \frac{\left(x + -1\right) - \frac{\left(x + -1\right) + \frac{1 - x}{y}}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{x \cdot \left(-1 - y\right)} + \left(\frac{y}{1 + y} + \frac{1}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot \left(1 - x\right)}{1 + y}\\ t_1 := \frac{y}{1 + y}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-13}:\\ \;\;\;\;1 + t\_1 \cdot \left(x + -1\right)\\ \mathbf{elif}\;t\_0 \leq 1.002:\\ \;\;\;\;x + \frac{-1}{\frac{y}{\left(x + -1\right) + \frac{1 - x}{y}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{x \cdot \left(-1 - y\right)} + \left(t\_1 + \frac{1}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* y (- 1.0 x)) (+ 1.0 y))) (t_1 (/ y (+ 1.0 y))))
   (if (<= t_0 5e-13)
     (+ 1.0 (* t_1 (+ x -1.0)))
     (if (<= t_0 1.002)
       (+ x (/ -1.0 (/ y (+ (+ x -1.0) (/ (- 1.0 x) y)))))
       (* x (+ (/ y (* x (- -1.0 y))) (+ t_1 (/ 1.0 x))))))))

C

double code(double x, double y) {
	double t_0 = (y * (1.0 - x)) / (1.0 + y);
	double t_1 = y / (1.0 + y);
	double tmp;
	if (t_0 <= 5e-13) {
		tmp = 1.0 + (t_1 * (x + -1.0));
	} else if (t_0 <= 1.002) {
		tmp = x + (-1.0 / (y / ((x + -1.0) + ((1.0 - x) / y))));
	} else {
		tmp = x * ((y / (x * (-1.0 - y))) + (t_1 + (1.0 / x)));
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (y * (1.0d0 - x)) / (1.0d0 + y)
    t_1 = y / (1.0d0 + y)
    if (t_0 <= 5d-13) then
        tmp = 1.0d0 + (t_1 * (x + (-1.0d0)))
    else if (t_0 <= 1.002d0) then
        tmp = x + ((-1.0d0) / (y / ((x + (-1.0d0)) + ((1.0d0 - x) / y))))
    else
        tmp = x * ((y / (x * ((-1.0d0) - y))) + (t_1 + (1.0d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * (1.0 - x)) / (1.0 + y);
	double t_1 = y / (1.0 + y);
	double tmp;
	if (t_0 <= 5e-13) {
		tmp = 1.0 + (t_1 * (x + -1.0));
	} else if (t_0 <= 1.002) {
		tmp = x + (-1.0 / (y / ((x + -1.0) + ((1.0 - x) / y))));
	} else {
		tmp = x * ((y / (x * (-1.0 - y))) + (t_1 + (1.0 / x)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * (1.0 - x)) / (1.0 + y)
	t_1 = y / (1.0 + y)
	tmp = 0
	if t_0 <= 5e-13:
		tmp = 1.0 + (t_1 * (x + -1.0))
	elif t_0 <= 1.002:
		tmp = x + (-1.0 / (y / ((x + -1.0) + ((1.0 - x) / y))))
	else:
		tmp = x * ((y / (x * (-1.0 - y))) + (t_1 + (1.0 / x)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * Float64(1.0 - x)) / Float64(1.0 + y))
	t_1 = Float64(y / Float64(1.0 + y))
	tmp = 0.0
	if (t_0 <= 5e-13)
		tmp = Float64(1.0 + Float64(t_1 * Float64(x + -1.0)));
	elseif (t_0 <= 1.002)
		tmp = Float64(x + Float64(-1.0 / Float64(y / Float64(Float64(x + -1.0) + Float64(Float64(1.0 - x) / y)))));
	else
		tmp = Float64(x * Float64(Float64(y / Float64(x * Float64(-1.0 - y))) + Float64(t_1 + Float64(1.0 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * (1.0 - x)) / (1.0 + y);
	t_1 = y / (1.0 + y);
	tmp = 0.0;
	if (t_0 <= 5e-13)
		tmp = 1.0 + (t_1 * (x + -1.0));
	elseif (t_0 <= 1.002)
		tmp = x + (-1.0 / (y / ((x + -1.0) + ((1.0 - x) / y))));
	else
		tmp = x * ((y / (x * (-1.0 - y))) + (t_1 + (1.0 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-13], N[(1.0 + N[(t$95$1 * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.002], N[(x + N[(-1.0 / N[(y / N[(N[(x + -1.0), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y / N[(x * N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < 4.9999999999999999e-13

   1. Initial program 86.6%

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

      1. associate-/l* 100.0%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   if 4.9999999999999999e-13 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < 1.002

   1. Initial program 10.9%

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

      1. associate-/l* 10.9%

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

      2. +-commutative 10.9%

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

   3. Simplified 10.9%

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

   5. Taylor expanded in y around -inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. unsub-neg 99.7%

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

   7. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

   9. Applied egg-rr 99.7%

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

       1. unpow-1 99.7%

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

       2. +-commutative 99.7%

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

   11. Simplified 99.7%

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

   if 1.002 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))

   1. Initial program 62.3%

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

      1. sub-neg 62.3%

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

      2. +-commutative 62.3%

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

      3. associate-/l* 99.7%

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

      4. distribute-rgt-neg-in 99.7%

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

      5. fma-define 99.7%

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

      6. distribute-neg-frac2 99.7%

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

      7. +-commutative 99.7%

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

      8. distribute-neg-in 99.7%

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

      9. metadata-eval 99.7%

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

      10. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. distribute-neg-frac2 99.8%

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

      3. distribute-rgt-neg-out 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. metadata-eval 99.8%

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

      6. sub-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(1 - x\right)}{1 + y} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;1 + \frac{y}{1 + y} \cdot \left(x + -1\right)\\ \mathbf{elif}\;\frac{y \cdot \left(1 - x\right)}{1 + y} \leq 1.002:\\ \;\;\;\;x + \frac{-1}{\frac{y}{\left(x + -1\right) + \frac{1 - x}{y}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{x \cdot \left(-1 - y\right)} + \left(\frac{y}{1 + y} + \frac{1}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -145000000 \lor \neg \left(y \leq 33000000000\right):\\ \;\;\;\;x + \frac{\frac{-1}{y} - -1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{1 + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -145000000.0) (not (<= y 33000000000.0)))
   (+ x (/ (- (/ -1.0 y) -1.0) y))
   (- 1.0 (/ (* y (- 1.0 x)) (+ 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -145000000.0) || !(y <= 33000000000.0)) {
		tmp = x + (((-1.0 / y) - -1.0) / y);
	} else {
		tmp = 1.0 - ((y * (1.0 - x)) / (1.0 + y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-145000000.0d0)) .or. (.not. (y <= 33000000000.0d0))) then
        tmp = x + ((((-1.0d0) / y) - (-1.0d0)) / y)
    else
        tmp = 1.0d0 - ((y * (1.0d0 - x)) / (1.0d0 + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -145000000.0) || !(y <= 33000000000.0)) {
		tmp = x + (((-1.0 / y) - -1.0) / y);
	} else {
		tmp = 1.0 - ((y * (1.0 - x)) / (1.0 + y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -145000000.0) or not (y <= 33000000000.0):
		tmp = x + (((-1.0 / y) - -1.0) / y)
	else:
		tmp = 1.0 - ((y * (1.0 - x)) / (1.0 + y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -145000000.0) || !(y <= 33000000000.0))
		tmp = Float64(x + Float64(Float64(Float64(-1.0 / y) - -1.0) / y));
	else
		tmp = Float64(1.0 - Float64(Float64(y * Float64(1.0 - x)) / Float64(1.0 + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -145000000.0) || ~((y <= 33000000000.0)))
		tmp = x + (((-1.0 / y) - -1.0) / y);
	else
		tmp = 1.0 - ((y * (1.0 - x)) / (1.0 + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -145000000.0], N[Not[LessEqual[y, 33000000000.0]], $MachinePrecision]], N[(x + N[(N[(N[(-1.0 / y), $MachinePrecision] - -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.45e8 or 3.3e10 < y

   1. Initial program 28.7%

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

      1. associate-/l* 57.1%

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

      2. +-commutative 57.1%

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

   3. Simplified 57.1%

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

   5. Taylor expanded in y around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 100.0%

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

   if -1.45e8 < y < 3.3e10

   1. Initial program 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -145000000 \lor \neg \left(y \leq 33000000000\right):\\ \;\;\;\;x + \frac{\frac{-1}{y} - -1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{1 + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -13000000000 \lor \neg \left(y \leq 32000000000\right):\\ \;\;\;\;x + \frac{\frac{-1}{y} - -1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y}{1 + y} \cdot \left(x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -13000000000.0) (not (<= y 32000000000.0)))
   (+ x (/ (- (/ -1.0 y) -1.0) y))
   (+ 1.0 (* (/ y (+ 1.0 y)) (+ x -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -13000000000.0) || !(y <= 32000000000.0)) {
		tmp = x + (((-1.0 / y) - -1.0) / y);
	} else {
		tmp = 1.0 + ((y / (1.0 + y)) * (x + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-13000000000.0d0)) .or. (.not. (y <= 32000000000.0d0))) then
        tmp = x + ((((-1.0d0) / y) - (-1.0d0)) / y)
    else
        tmp = 1.0d0 + ((y / (1.0d0 + y)) * (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -13000000000.0) || !(y <= 32000000000.0)) {
		tmp = x + (((-1.0 / y) - -1.0) / y);
	} else {
		tmp = 1.0 + ((y / (1.0 + y)) * (x + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -13000000000.0) or not (y <= 32000000000.0):
		tmp = x + (((-1.0 / y) - -1.0) / y)
	else:
		tmp = 1.0 + ((y / (1.0 + y)) * (x + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -13000000000.0) || !(y <= 32000000000.0))
		tmp = Float64(x + Float64(Float64(Float64(-1.0 / y) - -1.0) / y));
	else
		tmp = Float64(1.0 + Float64(Float64(y / Float64(1.0 + y)) * Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -13000000000.0) || ~((y <= 32000000000.0)))
		tmp = x + (((-1.0 / y) - -1.0) / y);
	else
		tmp = 1.0 + ((y / (1.0 + y)) * (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -13000000000.0], N[Not[LessEqual[y, 32000000000.0]], $MachinePrecision]], N[(x + N[(N[(N[(-1.0 / y), $MachinePrecision] - -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3e10 or 3.2e10 < y

   1. Initial program 28.7%

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

      1. associate-/l* 57.1%

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

      2. +-commutative 57.1%

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

   3. Simplified 57.1%

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

   5. Taylor expanded in y around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 100.0%

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

   if -1.3e10 < y < 3.2e10

   1. Initial program 99.7%

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

      1. associate-/l* 99.7%

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

      2. +-commutative 99.7%

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

   3. Simplified 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -13000000000 \lor \neg \left(y \leq 32000000000\right):\\ \;\;\;\;x + \frac{\frac{-1}{y} - -1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y}{1 + y} \cdot \left(x + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -260000:\\ \;\;\;\;x + \frac{-1}{\frac{y}{\left(x + -1\right) + \frac{1 - x}{y}}}\\ \mathbf{elif}\;y \leq 32000000000:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{-1}{y} - -1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -260000.0)
   (+ x (/ -1.0 (/ y (+ (+ x -1.0) (/ (- 1.0 x) y)))))
   (if (<= y 32000000000.0)
     (- 1.0 (/ (* y (- 1.0 x)) (+ 1.0 y)))
     (+ x (/ (- (/ -1.0 y) -1.0) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -260000.0) {
		tmp = x + (-1.0 / (y / ((x + -1.0) + ((1.0 - x) / y))));
	} else if (y <= 32000000000.0) {
		tmp = 1.0 - ((y * (1.0 - x)) / (1.0 + y));
	} else {
		tmp = x + (((-1.0 / y) - -1.0) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-260000.0d0)) then
        tmp = x + ((-1.0d0) / (y / ((x + (-1.0d0)) + ((1.0d0 - x) / y))))
    else if (y <= 32000000000.0d0) then
        tmp = 1.0d0 - ((y * (1.0d0 - x)) / (1.0d0 + y))
    else
        tmp = x + ((((-1.0d0) / y) - (-1.0d0)) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -260000.0) {
		tmp = x + (-1.0 / (y / ((x + -1.0) + ((1.0 - x) / y))));
	} else if (y <= 32000000000.0) {
		tmp = 1.0 - ((y * (1.0 - x)) / (1.0 + y));
	} else {
		tmp = x + (((-1.0 / y) - -1.0) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -260000.0:
		tmp = x + (-1.0 / (y / ((x + -1.0) + ((1.0 - x) / y))))
	elif y <= 32000000000.0:
		tmp = 1.0 - ((y * (1.0 - x)) / (1.0 + y))
	else:
		tmp = x + (((-1.0 / y) - -1.0) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -260000.0)
		tmp = Float64(x + Float64(-1.0 / Float64(y / Float64(Float64(x + -1.0) + Float64(Float64(1.0 - x) / y)))));
	elseif (y <= 32000000000.0)
		tmp = Float64(1.0 - Float64(Float64(y * Float64(1.0 - x)) / Float64(1.0 + y)));
	else
		tmp = Float64(x + Float64(Float64(Float64(-1.0 / y) - -1.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -260000.0)
		tmp = x + (-1.0 / (y / ((x + -1.0) + ((1.0 - x) / y))));
	elseif (y <= 32000000000.0)
		tmp = 1.0 - ((y * (1.0 - x)) / (1.0 + y));
	else
		tmp = x + (((-1.0 / y) - -1.0) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -260000.0], N[(x + N[(-1.0 / N[(y / N[(N[(x + -1.0), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 32000000000.0], N[(1.0 - N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(-1.0 / y), $MachinePrecision] - -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.6e5

   1. Initial program 28.7%

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

      1. associate-/l* 64.0%

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

      2. +-commutative 64.0%

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

   3. Simplified 64.0%

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

   5. Taylor expanded in y around -inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. unsub-neg 99.7%

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

   7. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

   9. Applied egg-rr 99.7%

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

       1. unpow-1 99.7%

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

       2. +-commutative 99.7%

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

   11. Simplified 99.7%

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

   if -2.6e5 < y < 3.2e10

   1. Initial program 99.9%

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

   if 3.2e10 < y

   1. Initial program 29.3%

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

      1. associate-/l* 50.3%

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

      2. +-commutative 50.3%

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

   3. Simplified 50.3%

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

   5. Taylor expanded in y around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -260000:\\ \;\;\;\;x + \frac{-1}{\frac{y}{\left(x + -1\right) + \frac{1 - x}{y}}}\\ \mathbf{elif}\;y \leq 32000000000:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{-1}{y} - -1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -235000:\\ \;\;\;\;x - \frac{\left(x + -1\right) + \frac{1 - x}{y}}{y}\\ \mathbf{elif}\;y \leq 34000000000:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{-1}{y} - -1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -235000.0)
   (- x (/ (+ (+ x -1.0) (/ (- 1.0 x) y)) y))
   (if (<= y 34000000000.0)
     (- 1.0 (/ (* y (- 1.0 x)) (+ 1.0 y)))
     (+ x (/ (- (/ -1.0 y) -1.0) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -235000.0) {
		tmp = x - (((x + -1.0) + ((1.0 - x) / y)) / y);
	} else if (y <= 34000000000.0) {
		tmp = 1.0 - ((y * (1.0 - x)) / (1.0 + y));
	} else {
		tmp = x + (((-1.0 / y) - -1.0) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-235000.0d0)) then
        tmp = x - (((x + (-1.0d0)) + ((1.0d0 - x) / y)) / y)
    else if (y <= 34000000000.0d0) then
        tmp = 1.0d0 - ((y * (1.0d0 - x)) / (1.0d0 + y))
    else
        tmp = x + ((((-1.0d0) / y) - (-1.0d0)) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -235000.0) {
		tmp = x - (((x + -1.0) + ((1.0 - x) / y)) / y);
	} else if (y <= 34000000000.0) {
		tmp = 1.0 - ((y * (1.0 - x)) / (1.0 + y));
	} else {
		tmp = x + (((-1.0 / y) - -1.0) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -235000.0:
		tmp = x - (((x + -1.0) + ((1.0 - x) / y)) / y)
	elif y <= 34000000000.0:
		tmp = 1.0 - ((y * (1.0 - x)) / (1.0 + y))
	else:
		tmp = x + (((-1.0 / y) - -1.0) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -235000.0)
		tmp = Float64(x - Float64(Float64(Float64(x + -1.0) + Float64(Float64(1.0 - x) / y)) / y));
	elseif (y <= 34000000000.0)
		tmp = Float64(1.0 - Float64(Float64(y * Float64(1.0 - x)) / Float64(1.0 + y)));
	else
		tmp = Float64(x + Float64(Float64(Float64(-1.0 / y) - -1.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -235000.0)
		tmp = x - (((x + -1.0) + ((1.0 - x) / y)) / y);
	elseif (y <= 34000000000.0)
		tmp = 1.0 - ((y * (1.0 - x)) / (1.0 + y));
	else
		tmp = x + (((-1.0 / y) - -1.0) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -235000.0], N[(x - N[(N[(N[(x + -1.0), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 34000000000.0], N[(1.0 - N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(-1.0 / y), $MachinePrecision] - -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -235000

   1. Initial program 28.7%

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

      1. associate-/l* 64.0%

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

      2. +-commutative 64.0%

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

   3. Simplified 64.0%

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

   5. Taylor expanded in y around -inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. unsub-neg 99.7%

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

   7. Simplified 99.7%

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

   if -235000 < y < 3.4e10

   1. Initial program 99.9%

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

   if 3.4e10 < y

   1. Initial program 29.3%

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

      1. associate-/l* 50.3%

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

      2. +-commutative 50.3%

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

   3. Simplified 50.3%

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

   5. Taylor expanded in y around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -235000:\\ \;\;\;\;x - \frac{\left(x + -1\right) + \frac{1 - x}{y}}{y}\\ \mathbf{elif}\;y \leq 34000000000:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{-1}{y} - -1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -950000:\\ \;\;\;\;x + \frac{\left(1 - x\right) + \frac{-1}{y}}{y}\\ \mathbf{elif}\;y \leq 32000000000:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{-1}{y} - -1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -950000.0)
   (+ x (/ (+ (- 1.0 x) (/ -1.0 y)) y))
   (if (<= y 32000000000.0)
     (- 1.0 (/ (* y (- 1.0 x)) (+ 1.0 y)))
     (+ x (/ (- (/ -1.0 y) -1.0) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -950000.0) {
		tmp = x + (((1.0 - x) + (-1.0 / y)) / y);
	} else if (y <= 32000000000.0) {
		tmp = 1.0 - ((y * (1.0 - x)) / (1.0 + y));
	} else {
		tmp = x + (((-1.0 / y) - -1.0) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-950000.0d0)) then
        tmp = x + (((1.0d0 - x) + ((-1.0d0) / y)) / y)
    else if (y <= 32000000000.0d0) then
        tmp = 1.0d0 - ((y * (1.0d0 - x)) / (1.0d0 + y))
    else
        tmp = x + ((((-1.0d0) / y) - (-1.0d0)) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -950000.0) {
		tmp = x + (((1.0 - x) + (-1.0 / y)) / y);
	} else if (y <= 32000000000.0) {
		tmp = 1.0 - ((y * (1.0 - x)) / (1.0 + y));
	} else {
		tmp = x + (((-1.0 / y) - -1.0) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -950000.0:
		tmp = x + (((1.0 - x) + (-1.0 / y)) / y)
	elif y <= 32000000000.0:
		tmp = 1.0 - ((y * (1.0 - x)) / (1.0 + y))
	else:
		tmp = x + (((-1.0 / y) - -1.0) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -950000.0)
		tmp = Float64(x + Float64(Float64(Float64(1.0 - x) + Float64(-1.0 / y)) / y));
	elseif (y <= 32000000000.0)
		tmp = Float64(1.0 - Float64(Float64(y * Float64(1.0 - x)) / Float64(1.0 + y)));
	else
		tmp = Float64(x + Float64(Float64(Float64(-1.0 / y) - -1.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -950000.0)
		tmp = x + (((1.0 - x) + (-1.0 / y)) / y);
	elseif (y <= 32000000000.0)
		tmp = 1.0 - ((y * (1.0 - x)) / (1.0 + y));
	else
		tmp = x + (((-1.0 / y) - -1.0) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -950000.0], N[(x + N[(N[(N[(1.0 - x), $MachinePrecision] + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 32000000000.0], N[(1.0 - N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(-1.0 / y), $MachinePrecision] - -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.5e5

   1. Initial program 28.7%

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

      1. associate-/l* 64.0%

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

      2. +-commutative 64.0%

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

   3. Simplified 64.0%

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

   5. Taylor expanded in y around -inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. unsub-neg 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0 99.7%

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

   if -9.5e5 < y < 3.2e10

   1. Initial program 99.9%

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

   if 3.2e10 < y

   1. Initial program 29.3%

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

      1. associate-/l* 50.3%

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

      2. +-commutative 50.3%

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

   3. Simplified 50.3%

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

   5. Taylor expanded in y around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -950000:\\ \;\;\;\;x + \frac{\left(1 - x\right) + \frac{-1}{y}}{y}\\ \mathbf{elif}\;y \leq 32000000000:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{-1}{y} - -1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-11}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+83}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 5e-11) (- 1.0 y) (if (<= y 4e+83) (/ 1.0 y) x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 5e-11) {
		tmp = 1.0 - y;
	} else if (y <= 4e+83) {
		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 <= 5d-11) then
        tmp = 1.0d0 - y
    else if (y <= 4d+83) 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 <= 5e-11) {
		tmp = 1.0 - y;
	} else if (y <= 4e+83) {
		tmp = 1.0 / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 5e-11:
		tmp = 1.0 - y
	elif y <= 4e+83:
		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 <= 5e-11)
		tmp = Float64(1.0 - y);
	elseif (y <= 4e+83)
		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 <= 5e-11)
		tmp = 1.0 - y;
	elseif (y <= 4e+83)
		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, 5e-11], N[(1.0 - y), $MachinePrecision], If[LessEqual[y, 4e+83], N[(1.0 / y), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 4.00000000000000012e83 < y

   1. Initial program 30.6%

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

      1. associate-/l* 60.9%

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

      2. +-commutative 60.9%

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

   3. Simplified 60.9%

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

   5. Taylor expanded in y around inf 82.8%

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

   if -1 < y < 5.00000000000000018e-11

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 78.3%

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

      1. +-commutative 78.3%

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

   7. Simplified 78.3%

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

   8. Taylor expanded in y around 0 77.6%

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

      1. neg-mul-1 77.6%

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

      2. unsub-neg 77.6%

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

   10. Simplified 77.6%

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

   if 5.00000000000000018e-11 < y < 4.00000000000000012e83

   1. Initial program 44.5%

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

      1. associate-/l* 44.6%

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

      2. +-commutative 44.6%

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

   3. Simplified 44.6%

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

   5. Taylor expanded in y around -inf 85.5%

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

      1. associate-*r/ 85.5%

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

      2. mul-1-neg 85.5%

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

      3. neg-sub0 85.5%

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

      4. associate-+l- 85.5%

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

      5. neg-sub0 85.5%

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

      6. +-commutative 85.5%

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

      7. sub-neg 85.5%

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

   7. Simplified 85.5%

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

   8. Taylor expanded in x around 0 60.4%

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

Alternative 9: 98.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 31.1%

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

      1. associate-/l* 58.4%

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

      2. +-commutative 58.4%

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

   3. Simplified 58.4%

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

   5. Taylor expanded in y around -inf 98.3%

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

      1. associate-*r/ 98.3%

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

      2. mul-1-neg 98.3%

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

      3. neg-sub0 98.3%

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

      4. associate-+l- 98.3%

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

      5. neg-sub0 98.3%

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

      6. +-commutative 98.3%

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

      7. sub-neg 98.3%

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

   7. Simplified 98.3%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 98.7%

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

4. Final simplification 98.5%

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

Alternative 10: 86.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.9\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(2 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 0.9)))
   (+ x (/ (- 1.0 x) y))
   (+ -1.0 (- 2.0 y))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.9)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = -1.0 + (2.0 - y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.0d0)) .or. (.not. (y <= 0.9d0))) then
        tmp = x + ((1.0d0 - x) / y)
    else
        tmp = (-1.0d0) + (2.0d0 - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.9)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = -1.0 + (2.0 - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 0.9):
		tmp = x + ((1.0 - x) / y)
	else:
		tmp = -1.0 + (2.0 - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 0.9))
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	else
		tmp = Float64(-1.0 + Float64(2.0 - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 0.9)))
		tmp = x + ((1.0 - x) / y);
	else
		tmp = -1.0 + (2.0 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 0.9]], $MachinePrecision]], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(2.0 - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.900000000000000022 < y

   1. Initial program 31.1%

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

      1. associate-/l* 58.4%

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

      2. +-commutative 58.4%

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

   3. Simplified 58.4%

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

   5. Taylor expanded in y around -inf 98.3%

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

      1. associate-*r/ 98.3%

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

      2. mul-1-neg 98.3%

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

      3. neg-sub0 98.3%

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

      4. associate-+l- 98.3%

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

      5. neg-sub0 98.3%

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

      6. +-commutative 98.3%

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

      7. sub-neg 98.3%

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

   7. Simplified 98.3%

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

   if -1 < y < 0.900000000000000022

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 77.1%

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

      1. +-commutative 77.1%

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

   7. Simplified 77.1%

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

   8. Taylor expanded in y around 0 76.4%

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

      1. neg-mul-1 76.4%

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

      2. unsub-neg 76.4%

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

   10. Simplified 76.4%

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

       1. expm1-log1p-u 76.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - y\right)\right)} \]

   12. Applied egg-rr 76.4%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - y\right)\right)} \]
   13. Step-by-step derivation

       1. expm1-undefine 76.4%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(1 - y\right)} - 1} \]

       2. sub-neg 76.4%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(1 - y\right)} + \left(-1\right)} \]

       3. log1p-undefine 76.4%

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

       4. rem-exp-log 76.4%

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

       5. associate-+r- 76.4%

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

       6. metadata-eval 76.4%

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

       7. metadata-eval 76.4%

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

   14. Simplified 76.4%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.9\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(2 - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 5 \cdot 10^{-11}\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(2 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 5e-11))) (+ x (/ 1.0 y)) (+ -1.0 (- 2.0 y))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 5e-11)) {
		tmp = x + (1.0 / y);
	} else {
		tmp = -1.0 + (2.0 - y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.0d0)) .or. (.not. (y <= 5d-11))) then
        tmp = x + (1.0d0 / y)
    else
        tmp = (-1.0d0) + (2.0d0 - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 5e-11)) {
		tmp = x + (1.0 / y);
	} else {
		tmp = -1.0 + (2.0 - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 5e-11):
		tmp = x + (1.0 / y)
	else:
		tmp = -1.0 + (2.0 - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 5e-11))
		tmp = Float64(x + Float64(1.0 / y));
	else
		tmp = Float64(-1.0 + Float64(2.0 - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 5e-11)))
		tmp = x + (1.0 / y);
	else
		tmp = -1.0 + (2.0 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 5e-11]], $MachinePrecision]], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(2.0 - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 5.00000000000000018e-11 < y

   1. Initial program 32.1%

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

      1. associate-/l* 59.0%

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

      2. +-commutative 59.0%

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

   3. Simplified 59.0%

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

   5. Taylor expanded in y around -inf 96.8%

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

      1. associate-*r/ 96.8%

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

      2. mul-1-neg 96.8%

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

      3. neg-sub0 96.8%

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

      4. associate-+l- 96.8%

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

      5. neg-sub0 96.8%

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

      6. +-commutative 96.8%

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

      7. sub-neg 96.8%

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

   7. Simplified 96.8%

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

   8. Taylor expanded in x around 0 96.4%

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

   if -1 < y < 5.00000000000000018e-11

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 78.3%

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

      1. +-commutative 78.3%

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

   7. Simplified 78.3%

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

   8. Taylor expanded in y around 0 77.6%

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

      1. neg-mul-1 77.6%

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

      2. unsub-neg 77.6%

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

   10. Simplified 77.6%

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

       1. expm1-log1p-u 77.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - y\right)\right)} \]

   12. Applied egg-rr 77.6%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - y\right)\right)} \]
   13. Step-by-step derivation

       1. expm1-undefine 77.6%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(1 - y\right)} - 1} \]

       2. sub-neg 77.6%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(1 - y\right)} + \left(-1\right)} \]

       3. log1p-undefine 77.6%

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

       4. rem-exp-log 77.6%

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

       5. associate-+r- 77.6%

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

       6. metadata-eval 77.6%

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

       7. metadata-eval 77.6%

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

   14. Simplified 77.6%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 5 \cdot 10^{-11}\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(2 - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 85.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 5 \cdot 10^{-11}\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 5e-11))) (+ x (/ 1.0 y)) (- 1.0 y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 5e-11)) {
		tmp = x + (1.0 / y);
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.0d0)) .or. (.not. (y <= 5d-11))) then
        tmp = x + (1.0d0 / y)
    else
        tmp = 1.0d0 - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 5e-11)) {
		tmp = x + (1.0 / y);
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 5e-11):
		tmp = x + (1.0 / y)
	else:
		tmp = 1.0 - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 5e-11))
		tmp = Float64(x + Float64(1.0 / y));
	else
		tmp = Float64(1.0 - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 5e-11)))
		tmp = x + (1.0 / y);
	else
		tmp = 1.0 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 5e-11]], $MachinePrecision]], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(1.0 - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 5.00000000000000018e-11 < y

   1. Initial program 32.1%

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

      1. associate-/l* 59.0%

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

      2. +-commutative 59.0%

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

   3. Simplified 59.0%

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

   5. Taylor expanded in y around -inf 96.8%

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

      1. associate-*r/ 96.8%

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

      2. mul-1-neg 96.8%

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

      3. neg-sub0 96.8%

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

      4. associate-+l- 96.8%

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

      5. neg-sub0 96.8%

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

      6. +-commutative 96.8%

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

      7. sub-neg 96.8%

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

   7. Simplified 96.8%

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

   8. Taylor expanded in x around 0 96.4%

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

   if -1 < y < 5.00000000000000018e-11

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 78.3%

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

      1. +-commutative 78.3%

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

   7. Simplified 78.3%

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

   8. Taylor expanded in y around 0 77.6%

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

      1. neg-mul-1 77.6%

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

      2. unsub-neg 77.6%

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

   10. Simplified 77.6%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 5 \cdot 10^{-11}\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-11}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 5e-11) (- 1.0 y) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 5e-11) {
		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 <= 5d-11) 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 <= 5e-11) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 5e-11:
		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 <= 5e-11)
		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 <= 5e-11)
		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, 5e-11], N[(1.0 - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 5.00000000000000018e-11 < y

   1. Initial program 32.1%

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

      1. associate-/l* 59.0%

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

      2. +-commutative 59.0%

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

   3. Simplified 59.0%

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

   5. Taylor expanded in y around inf 76.8%

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

   if -1 < y < 5.00000000000000018e-11

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 78.3%

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

      1. +-commutative 78.3%

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

   7. Simplified 78.3%

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

   8. Taylor expanded in y around 0 77.6%

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

      1. neg-mul-1 77.6%

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

      2. unsub-neg 77.6%

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

   10. Simplified 77.6%

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

Alternative 14: 73.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-11}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1.0) x (if (<= y 5e-11) 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 5e-11) {
		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 <= 5d-11) 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 <= 5e-11) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 5e-11:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 5e-11)
		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 <= 5e-11)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 5e-11], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 5.00000000000000018e-11 < y

   1. Initial program 32.1%

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

      1. associate-/l* 59.0%

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

      2. +-commutative 59.0%

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

   3. Simplified 59.0%

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

   5. Taylor expanded in y around inf 76.8%

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

   if -1 < y < 5.00000000000000018e-11

   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* 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. fma-define 100.0%

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

      6. distribute-neg-frac2 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-neg-in 100.0%

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

      9. metadata-eval 100.0%

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

      10. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. distribute-neg-frac2 99.8%

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

      3. distribute-rgt-neg-out 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. metadata-eval 99.8%

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

      6. sub-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in y around 0 77.5%

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

Alternative 15: 38.4% 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 64.7%

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

   1. sub-neg 64.7%

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

   2. +-commutative 64.7%

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

   3. associate-/l* 78.7%

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

   4. distribute-rgt-neg-in 78.7%

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

   5. fma-define 78.7%

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

   6. distribute-neg-frac2 78.7%

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

   7. +-commutative 78.7%

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

   8. distribute-neg-in 78.7%

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

   9. metadata-eval 78.7%

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

   10. unsub-neg 78.7%

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

3. Simplified 78.7%

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

5. Taylor expanded in x around inf 78.9%

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

   1. mul-1-neg 78.9%

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

   2. distribute-neg-frac2 78.9%

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

   3. distribute-rgt-neg-out 78.9%

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

   4. distribute-neg-in 78.9%

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

   5. metadata-eval 78.9%

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

   6. sub-neg 78.9%

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

   7. +-commutative 78.9%

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

   8. +-commutative 78.9%

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

7. Simplified 78.9%

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

8. Taylor expanded in y around 0 39.3%

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

Developer Target 1: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024182 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -36938482788297247/10000000000000) (- (/ 1 y) (- (/ x y) x)) (if (< y 679931050341891/100000) (- 1 (/ (* (- 1 x) y) (+ y 1))) (- (/ 1 y) (- (/ x y) x)))))

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