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

Percentage Accurate: 65.2% → 99.7%

Time: 9.8s

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: 65.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(1 - x\right) \cdot y}{1 + y}\\ \mathbf{if}\;t\_0 \leq 0.1:\\ \;\;\;\;1 + \left(1 - x\right) \cdot \frac{y}{-1 - y}\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;x - \frac{\frac{1 - x}{y} + \left(x + -1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{1 + y} + \frac{\left(-1 - y\right) + x \cdot \frac{y}{x}}{x \cdot \left(-1 - y\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- 1.0 x) y) (+ 1.0 y))))
   (if (<= t_0 0.1)
     (+ 1.0 (* (- 1.0 x) (/ y (- -1.0 y))))
     (if (<= t_0 1.0)
       (- x (/ (+ (/ (- 1.0 x) y) (+ x -1.0)) y))
       (*
        x
        (+
         (/ y (+ 1.0 y))
         (/ (+ (- -1.0 y) (* x (/ y x))) (* x (- -1.0 y)))))))))

C

double code(double x, double y) {
	double t_0 = ((1.0 - x) * y) / (1.0 + y);
	double tmp;
	if (t_0 <= 0.1) {
		tmp = 1.0 + ((1.0 - x) * (y / (-1.0 - y)));
	} else if (t_0 <= 1.0) {
		tmp = x - ((((1.0 - x) / y) + (x + -1.0)) / y);
	} else {
		tmp = x * ((y / (1.0 + y)) + (((-1.0 - y) + (x * (y / x))) / (x * (-1.0 - y))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((1.0d0 - x) * y) / (1.0d0 + y)
    if (t_0 <= 0.1d0) then
        tmp = 1.0d0 + ((1.0d0 - x) * (y / ((-1.0d0) - y)))
    else if (t_0 <= 1.0d0) then
        tmp = x - ((((1.0d0 - x) / y) + (x + (-1.0d0))) / y)
    else
        tmp = x * ((y / (1.0d0 + y)) + ((((-1.0d0) - y) + (x * (y / x))) / (x * ((-1.0d0) - y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((1.0 - x) * y) / (1.0 + y);
	double tmp;
	if (t_0 <= 0.1) {
		tmp = 1.0 + ((1.0 - x) * (y / (-1.0 - y)));
	} else if (t_0 <= 1.0) {
		tmp = x - ((((1.0 - x) / y) + (x + -1.0)) / y);
	} else {
		tmp = x * ((y / (1.0 + y)) + (((-1.0 - y) + (x * (y / x))) / (x * (-1.0 - y))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((1.0 - x) * y) / (1.0 + y)
	tmp = 0
	if t_0 <= 0.1:
		tmp = 1.0 + ((1.0 - x) * (y / (-1.0 - y)))
	elif t_0 <= 1.0:
		tmp = x - ((((1.0 - x) / y) + (x + -1.0)) / y)
	else:
		tmp = x * ((y / (1.0 + y)) + (((-1.0 - y) + (x * (y / x))) / (x * (-1.0 - y))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(1.0 - x) * y) / Float64(1.0 + y))
	tmp = 0.0
	if (t_0 <= 0.1)
		tmp = Float64(1.0 + Float64(Float64(1.0 - x) * Float64(y / Float64(-1.0 - y))));
	elseif (t_0 <= 1.0)
		tmp = Float64(x - Float64(Float64(Float64(Float64(1.0 - x) / y) + Float64(x + -1.0)) / y));
	else
		tmp = Float64(x * Float64(Float64(y / Float64(1.0 + y)) + Float64(Float64(Float64(-1.0 - y) + Float64(x * Float64(y / x))) / Float64(x * Float64(-1.0 - y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((1.0 - x) * y) / (1.0 + y);
	tmp = 0.0;
	if (t_0 <= 0.1)
		tmp = 1.0 + ((1.0 - x) * (y / (-1.0 - y)));
	elseif (t_0 <= 1.0)
		tmp = x - ((((1.0 - x) / y) + (x + -1.0)) / y);
	else
		tmp = x * ((y / (1.0 + y)) + (((-1.0 - y) + (x * (y / x))) / (x * (-1.0 - y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.1], N[(1.0 + N[(N[(1.0 - x), $MachinePrecision] * N[(y / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(x - N[(N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] + N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-1.0 - y), $MachinePrecision] + N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * N[(-1.0 - y), $MachinePrecision]), $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))) < 0.10000000000000001

   1. Initial program 89.8%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   if 0.10000000000000001 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < 1

   1. Initial program 4.9%

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

      1. associate-/l* 4.9%

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

      2. remove-double-neg 4.9%

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

      3. remove-double-neg 4.9%

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

      4. +-commutative 4.9%

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

   3. Simplified 4.9%

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

   5. Taylor expanded in y around -inf 100.0%

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

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

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

   1. Initial program 76.4%

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

      1. sub-neg 76.4%

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

      2. +-commutative 76.4%

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

      3. *-commutative 76.4%

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

      4. associate-/l* 96.7%

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

      5. distribute-rgt-neg-in 96.7%

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

      6. fma-define 96.7%

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

      7. distribute-frac-neg2 96.7%

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

      8. +-commutative 96.7%

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

      9. distribute-neg-in 96.7%

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

      10. metadata-eval 96.7%

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

      11. unsub-neg 96.7%

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

   3. Simplified 96.7%

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

   5. Taylor expanded in x around inf 97.5%

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

      1. +-commutative 97.5%

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

      2. +-commutative 97.5%

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

      3. associate-+l+ 97.5%

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

      4. +-commutative 97.5%

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

      5. mul-1-neg 97.5%

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

      6. distribute-neg-frac2 97.5%

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

      7. distribute-rgt-neg-out 97.5%

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

      8. distribute-neg-in 97.5%

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

      9. metadata-eval 97.5%

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

      10. sub-neg 97.5%

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

   7. Simplified 97.5%

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

      1. associate-/r* 98.8%

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

      2. frac-add 99.9%

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

      3. *-un-lft-identity 99.9%

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 0.1:\\ \;\;\;\;1 + \left(1 - x\right) \cdot \frac{y}{-1 - y}\\ \mathbf{elif}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 1:\\ \;\;\;\;x - \frac{\frac{1 - x}{y} + \left(x + -1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{1 + y} + \frac{\left(-1 - y\right) + x \cdot \frac{y}{x}}{x \cdot \left(-1 - y\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(1 - x\right) \cdot y}{1 + y}\\ \mathbf{if}\;t\_0 \leq 0.1:\\ \;\;\;\;1 + \left(1 - x\right) \cdot \frac{y}{-1 - y}\\ \mathbf{elif}\;t\_0 \leq 1.0000000002:\\ \;\;\;\;x - \frac{\frac{1 - x}{y} + \left(x + -1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(\frac{y}{1 + y} + \frac{1}{x}\right) + \frac{\frac{y}{x}}{-1 - y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- 1.0 x) y) (+ 1.0 y))))
   (if (<= t_0 0.1)
     (+ 1.0 (* (- 1.0 x) (/ y (- -1.0 y))))
     (if (<= t_0 1.0000000002)
       (- x (/ (+ (/ (- 1.0 x) y) (+ x -1.0)) y))
       (* x (+ (+ (/ y (+ 1.0 y)) (/ 1.0 x)) (/ (/ y x) (- -1.0 y))))))))

C

double code(double x, double y) {
	double t_0 = ((1.0 - x) * y) / (1.0 + y);
	double tmp;
	if (t_0 <= 0.1) {
		tmp = 1.0 + ((1.0 - x) * (y / (-1.0 - y)));
	} else if (t_0 <= 1.0000000002) {
		tmp = x - ((((1.0 - x) / y) + (x + -1.0)) / y);
	} else {
		tmp = x * (((y / (1.0 + y)) + (1.0 / x)) + ((y / 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) :: t_0
    real(8) :: tmp
    t_0 = ((1.0d0 - x) * y) / (1.0d0 + y)
    if (t_0 <= 0.1d0) then
        tmp = 1.0d0 + ((1.0d0 - x) * (y / ((-1.0d0) - y)))
    else if (t_0 <= 1.0000000002d0) then
        tmp = x - ((((1.0d0 - x) / y) + (x + (-1.0d0))) / y)
    else
        tmp = x * (((y / (1.0d0 + y)) + (1.0d0 / x)) + ((y / x) / ((-1.0d0) - y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((1.0 - x) * y) / (1.0 + y);
	double tmp;
	if (t_0 <= 0.1) {
		tmp = 1.0 + ((1.0 - x) * (y / (-1.0 - y)));
	} else if (t_0 <= 1.0000000002) {
		tmp = x - ((((1.0 - x) / y) + (x + -1.0)) / y);
	} else {
		tmp = x * (((y / (1.0 + y)) + (1.0 / x)) + ((y / x) / (-1.0 - y)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((1.0 - x) * y) / (1.0 + y)
	tmp = 0
	if t_0 <= 0.1:
		tmp = 1.0 + ((1.0 - x) * (y / (-1.0 - y)))
	elif t_0 <= 1.0000000002:
		tmp = x - ((((1.0 - x) / y) + (x + -1.0)) / y)
	else:
		tmp = x * (((y / (1.0 + y)) + (1.0 / x)) + ((y / x) / (-1.0 - y)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(1.0 - x) * y) / Float64(1.0 + y))
	tmp = 0.0
	if (t_0 <= 0.1)
		tmp = Float64(1.0 + Float64(Float64(1.0 - x) * Float64(y / Float64(-1.0 - y))));
	elseif (t_0 <= 1.0000000002)
		tmp = Float64(x - Float64(Float64(Float64(Float64(1.0 - x) / y) + Float64(x + -1.0)) / y));
	else
		tmp = Float64(x * Float64(Float64(Float64(y / Float64(1.0 + y)) + Float64(1.0 / x)) + Float64(Float64(y / x) / Float64(-1.0 - y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((1.0 - x) * y) / (1.0 + y);
	tmp = 0.0;
	if (t_0 <= 0.1)
		tmp = 1.0 + ((1.0 - x) * (y / (-1.0 - y)));
	elseif (t_0 <= 1.0000000002)
		tmp = x - ((((1.0 - x) / y) + (x + -1.0)) / y);
	else
		tmp = x * (((y / (1.0 + y)) + (1.0 / x)) + ((y / x) / (-1.0 - y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.1], N[(1.0 + N[(N[(1.0 - x), $MachinePrecision] * N[(y / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0000000002], N[(x - N[(N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] + N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(N[(y / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] + N[(N[(y / x), $MachinePrecision] / N[(-1.0 - y), $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))) < 0.10000000000000001

   1. Initial program 89.8%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   if 0.10000000000000001 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < 1.0000000002

   1. Initial program 5.8%

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

      1. associate-/l* 5.8%

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

      2. remove-double-neg 5.8%

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

      3. remove-double-neg 5.8%

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

      4. +-commutative 5.8%

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

   3. Simplified 5.8%

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

   5. Taylor expanded in y around -inf 100.0%

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

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

   if 1.0000000002 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))

   1. Initial program 77.8%

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

      1. sub-neg 77.8%

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

      2. +-commutative 77.8%

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

      3. *-commutative 77.8%

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

      4. associate-/l* 98.8%

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

      5. distribute-rgt-neg-in 98.8%

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

      6. fma-define 98.8%

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

      7. distribute-frac-neg2 98.8%

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

      8. +-commutative 98.8%

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

      9. distribute-neg-in 98.8%

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

      10. metadata-eval 98.8%

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

      11. unsub-neg 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in x around inf 99.7%

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

      1. +-commutative 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. +-commutative 99.7%

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

      5. mul-1-neg 99.7%

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

      6. distribute-neg-frac2 99.7%

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

      7. distribute-rgt-neg-out 99.7%

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

      8. distribute-neg-in 99.7%

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

      9. metadata-eval 99.7%

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

      10. sub-neg 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around inf 99.7%

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

      1. +-commutative 99.7%

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

      2. +-commutative 99.7%

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

      3. mul-1-neg 99.7%

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

      4. unsub-neg 99.7%

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

      5. associate-/r* 99.8%

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

   10. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 0.1:\\ \;\;\;\;1 + \left(1 - x\right) \cdot \frac{y}{-1 - y}\\ \mathbf{elif}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 1.0000000002:\\ \;\;\;\;x - \frac{\frac{1 - x}{y} + \left(x + -1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(\frac{y}{1 + y} + \frac{1}{x}\right) + \frac{\frac{y}{x}}{-1 - y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(1 - x\right) \cdot y}{1 + y}\\ \mathbf{if}\;t\_0 \leq 0.1:\\ \;\;\;\;1 + \left(1 - x\right) \cdot \frac{y}{-1 - y}\\ \mathbf{elif}\;t\_0 \leq 1.0000000002:\\ \;\;\;\;x - \frac{\frac{1 - x}{y} + \left(x + -1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{1 + y} + \left(\frac{1}{x} + \frac{y}{x \cdot \left(-1 - y\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- 1.0 x) y) (+ 1.0 y))))
   (if (<= t_0 0.1)
     (+ 1.0 (* (- 1.0 x) (/ y (- -1.0 y))))
     (if (<= t_0 1.0000000002)
       (- x (/ (+ (/ (- 1.0 x) y) (+ x -1.0)) y))
       (* x (+ (/ y (+ 1.0 y)) (+ (/ 1.0 x) (/ y (* x (- -1.0 y))))))))))

C

double code(double x, double y) {
	double t_0 = ((1.0 - x) * y) / (1.0 + y);
	double tmp;
	if (t_0 <= 0.1) {
		tmp = 1.0 + ((1.0 - x) * (y / (-1.0 - y)));
	} else if (t_0 <= 1.0000000002) {
		tmp = x - ((((1.0 - x) / y) + (x + -1.0)) / y);
	} else {
		tmp = x * ((y / (1.0 + y)) + ((1.0 / x) + (y / (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) :: t_0
    real(8) :: tmp
    t_0 = ((1.0d0 - x) * y) / (1.0d0 + y)
    if (t_0 <= 0.1d0) then
        tmp = 1.0d0 + ((1.0d0 - x) * (y / ((-1.0d0) - y)))
    else if (t_0 <= 1.0000000002d0) then
        tmp = x - ((((1.0d0 - x) / y) + (x + (-1.0d0))) / y)
    else
        tmp = x * ((y / (1.0d0 + y)) + ((1.0d0 / x) + (y / (x * ((-1.0d0) - y)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((1.0 - x) * y) / (1.0 + y);
	double tmp;
	if (t_0 <= 0.1) {
		tmp = 1.0 + ((1.0 - x) * (y / (-1.0 - y)));
	} else if (t_0 <= 1.0000000002) {
		tmp = x - ((((1.0 - x) / y) + (x + -1.0)) / y);
	} else {
		tmp = x * ((y / (1.0 + y)) + ((1.0 / x) + (y / (x * (-1.0 - y)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((1.0 - x) * y) / (1.0 + y)
	tmp = 0
	if t_0 <= 0.1:
		tmp = 1.0 + ((1.0 - x) * (y / (-1.0 - y)))
	elif t_0 <= 1.0000000002:
		tmp = x - ((((1.0 - x) / y) + (x + -1.0)) / y)
	else:
		tmp = x * ((y / (1.0 + y)) + ((1.0 / x) + (y / (x * (-1.0 - y)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(1.0 - x) * y) / Float64(1.0 + y))
	tmp = 0.0
	if (t_0 <= 0.1)
		tmp = Float64(1.0 + Float64(Float64(1.0 - x) * Float64(y / Float64(-1.0 - y))));
	elseif (t_0 <= 1.0000000002)
		tmp = Float64(x - Float64(Float64(Float64(Float64(1.0 - x) / y) + Float64(x + -1.0)) / y));
	else
		tmp = Float64(x * Float64(Float64(y / Float64(1.0 + y)) + Float64(Float64(1.0 / x) + Float64(y / Float64(x * Float64(-1.0 - y))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((1.0 - x) * y) / (1.0 + y);
	tmp = 0.0;
	if (t_0 <= 0.1)
		tmp = 1.0 + ((1.0 - x) * (y / (-1.0 - y)));
	elseif (t_0 <= 1.0000000002)
		tmp = x - ((((1.0 - x) / y) + (x + -1.0)) / y);
	else
		tmp = x * ((y / (1.0 + y)) + ((1.0 / x) + (y / (x * (-1.0 - y)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.1], N[(1.0 + N[(N[(1.0 - x), $MachinePrecision] * N[(y / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0000000002], N[(x - N[(N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] + N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] + N[(y / N[(x * N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $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))) < 0.10000000000000001

   1. Initial program 89.8%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   if 0.10000000000000001 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < 1.0000000002

   1. Initial program 5.8%

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

      1. associate-/l* 5.8%

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

      2. remove-double-neg 5.8%

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

      3. remove-double-neg 5.8%

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

      4. +-commutative 5.8%

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

   3. Simplified 5.8%

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

   5. Taylor expanded in y around -inf 100.0%

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

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

   if 1.0000000002 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))

   1. Initial program 77.8%

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

      1. sub-neg 77.8%

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

      2. +-commutative 77.8%

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

      3. *-commutative 77.8%

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

      4. associate-/l* 98.8%

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

      5. distribute-rgt-neg-in 98.8%

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

      6. fma-define 98.8%

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

      7. distribute-frac-neg2 98.8%

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

      8. +-commutative 98.8%

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

      9. distribute-neg-in 98.8%

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

      10. metadata-eval 98.8%

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

      11. unsub-neg 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in x around inf 99.7%

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

      1. +-commutative 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. +-commutative 99.7%

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

      5. mul-1-neg 99.7%

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

      6. distribute-neg-frac2 99.7%

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

      7. distribute-rgt-neg-out 99.7%

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

      8. distribute-neg-in 99.7%

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

      9. metadata-eval 99.7%

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

      10. sub-neg 99.7%

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

   7. Simplified 99.7%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 0.1:\\ \;\;\;\;1 + \left(1 - x\right) \cdot \frac{y}{-1 - y}\\ \mathbf{elif}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 1.0000000002:\\ \;\;\;\;x - \frac{\frac{1 - x}{y} + \left(x + -1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{1 + y} + \left(\frac{1}{x} + \frac{y}{x \cdot \left(-1 - y\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(1 - x\right) \cdot y}{1 + y}\\ t_1 := \frac{y}{-1 - y}\\ \mathbf{if}\;t\_0 \leq 0.1:\\ \;\;\;\;1 + \left(1 - x\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 1.0000000002:\\ \;\;\;\;x - \frac{\frac{1 - x}{y} + \left(x + -1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{1 + y} + \frac{1 + t\_1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- 1.0 x) y) (+ 1.0 y))) (t_1 (/ y (- -1.0 y))))
   (if (<= t_0 0.1)
     (+ 1.0 (* (- 1.0 x) t_1))
     (if (<= t_0 1.0000000002)
       (- x (/ (+ (/ (- 1.0 x) y) (+ x -1.0)) y))
       (* x (+ (/ y (+ 1.0 y)) (/ (+ 1.0 t_1) x)))))))

C

double code(double x, double y) {
	double t_0 = ((1.0 - x) * y) / (1.0 + y);
	double t_1 = y / (-1.0 - y);
	double tmp;
	if (t_0 <= 0.1) {
		tmp = 1.0 + ((1.0 - x) * t_1);
	} else if (t_0 <= 1.0000000002) {
		tmp = x - ((((1.0 - x) / y) + (x + -1.0)) / y);
	} else {
		tmp = x * ((y / (1.0 + y)) + ((1.0 + t_1) / 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 = ((1.0d0 - x) * y) / (1.0d0 + y)
    t_1 = y / ((-1.0d0) - y)
    if (t_0 <= 0.1d0) then
        tmp = 1.0d0 + ((1.0d0 - x) * t_1)
    else if (t_0 <= 1.0000000002d0) then
        tmp = x - ((((1.0d0 - x) / y) + (x + (-1.0d0))) / y)
    else
        tmp = x * ((y / (1.0d0 + y)) + ((1.0d0 + t_1) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((1.0 - x) * y) / (1.0 + y);
	double t_1 = y / (-1.0 - y);
	double tmp;
	if (t_0 <= 0.1) {
		tmp = 1.0 + ((1.0 - x) * t_1);
	} else if (t_0 <= 1.0000000002) {
		tmp = x - ((((1.0 - x) / y) + (x + -1.0)) / y);
	} else {
		tmp = x * ((y / (1.0 + y)) + ((1.0 + t_1) / x));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((1.0 - x) * y) / (1.0 + y)
	t_1 = y / (-1.0 - y)
	tmp = 0
	if t_0 <= 0.1:
		tmp = 1.0 + ((1.0 - x) * t_1)
	elif t_0 <= 1.0000000002:
		tmp = x - ((((1.0 - x) / y) + (x + -1.0)) / y)
	else:
		tmp = x * ((y / (1.0 + y)) + ((1.0 + t_1) / x))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(1.0 - x) * y) / Float64(1.0 + y))
	t_1 = Float64(y / Float64(-1.0 - y))
	tmp = 0.0
	if (t_0 <= 0.1)
		tmp = Float64(1.0 + Float64(Float64(1.0 - x) * t_1));
	elseif (t_0 <= 1.0000000002)
		tmp = Float64(x - Float64(Float64(Float64(Float64(1.0 - x) / y) + Float64(x + -1.0)) / y));
	else
		tmp = Float64(x * Float64(Float64(y / Float64(1.0 + y)) + Float64(Float64(1.0 + t_1) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((1.0 - x) * y) / (1.0 + y);
	t_1 = y / (-1.0 - y);
	tmp = 0.0;
	if (t_0 <= 0.1)
		tmp = 1.0 + ((1.0 - x) * t_1);
	elseif (t_0 <= 1.0000000002)
		tmp = x - ((((1.0 - x) / y) + (x + -1.0)) / y);
	else
		tmp = x * ((y / (1.0 + y)) + ((1.0 + t_1) / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.1], N[(1.0 + N[(N[(1.0 - x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0000000002], N[(x - N[(N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] + N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + t$95$1), $MachinePrecision] / x), $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))) < 0.10000000000000001

   1. Initial program 89.8%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   if 0.10000000000000001 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < 1.0000000002

   1. Initial program 5.8%

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

      1. associate-/l* 5.8%

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

      2. remove-double-neg 5.8%

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

      3. remove-double-neg 5.8%

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

      4. +-commutative 5.8%

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

   3. Simplified 5.8%

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

   5. Taylor expanded in y around -inf 100.0%

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

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

   if 1.0000000002 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))

   1. Initial program 77.8%

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

      1. sub-neg 77.8%

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

      2. +-commutative 77.8%

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

      3. *-commutative 77.8%

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

      4. associate-/l* 98.8%

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

      5. distribute-rgt-neg-in 98.8%

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

      6. fma-define 98.8%

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

      7. distribute-frac-neg2 98.8%

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

      8. +-commutative 98.8%

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

      9. distribute-neg-in 98.8%

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

      10. metadata-eval 98.8%

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

      11. unsub-neg 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in x around -inf 99.7%

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

      1. associate-*r* 99.7%

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

      2. neg-mul-1 99.7%

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

      3. distribute-lft-out 99.7%

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

      4. associate-*r* 99.7%

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

      5. distribute-lft-neg-in 99.7%

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

      6. *-commutative 99.7%

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

      7. neg-mul-1 99.7%

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

      8. remove-double-neg 99.7%

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

      9. +-commutative 99.7%

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

      10. mul-1-neg 99.7%

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

      11. sub-neg 99.7%

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

      12. sub-neg 99.7%

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

      13. distribute-frac-neg2 99.7%

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

      14. distribute-neg-in 99.7%

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

      15. metadata-eval 99.7%

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

      16. sub-neg 99.7%

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

   7. Simplified 99.7%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 0.1:\\ \;\;\;\;1 + \left(1 - x\right) \cdot \frac{y}{-1 - y}\\ \mathbf{elif}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 1.0000000002:\\ \;\;\;\;x - \frac{\frac{1 - x}{y} + \left(x + -1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{1 + y} + \frac{1 + \frac{y}{-1 - y}}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4500000 \lor \neg \left(y \leq 340000\right):\\ \;\;\;\;x - \frac{\frac{1 - x}{y} + \left(x + -1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - x\right) \cdot y\right) \cdot \frac{1}{-1 - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4500000.0) (not (<= y 340000.0)))
   (- x (/ (+ (/ (- 1.0 x) y) (+ x -1.0)) y))
   (+ 1.0 (* (* (- 1.0 x) y) (/ 1.0 (- -1.0 y))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4500000.0) || !(y <= 340000.0)) {
		tmp = x - ((((1.0 - x) / y) + (x + -1.0)) / y);
	} else {
		tmp = 1.0 + (((1.0 - x) * y) * (1.0 / (-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 <= (-4500000.0d0)) .or. (.not. (y <= 340000.0d0))) then
        tmp = x - ((((1.0d0 - x) / y) + (x + (-1.0d0))) / y)
    else
        tmp = 1.0d0 + (((1.0d0 - x) * y) * (1.0d0 / ((-1.0d0) - y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -4500000.0) || !(y <= 340000.0)) {
		tmp = x - ((((1.0 - x) / y) + (x + -1.0)) / y);
	} else {
		tmp = 1.0 + (((1.0 - x) * y) * (1.0 / (-1.0 - y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4500000.0) or not (y <= 340000.0):
		tmp = x - ((((1.0 - x) / y) + (x + -1.0)) / y)
	else:
		tmp = 1.0 + (((1.0 - x) * y) * (1.0 / (-1.0 - y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4500000.0) || !(y <= 340000.0))
		tmp = Float64(x - Float64(Float64(Float64(Float64(1.0 - x) / y) + Float64(x + -1.0)) / y));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(1.0 - x) * y) * Float64(1.0 / Float64(-1.0 - y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -4500000.0) || ~((y <= 340000.0)))
		tmp = x - ((((1.0 - x) / y) + (x + -1.0)) / y);
	else
		tmp = 1.0 + (((1.0 - x) * y) * (1.0 / (-1.0 - y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4500000.0], N[Not[LessEqual[y, 340000.0]], $MachinePrecision]], N[(x - N[(N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] + N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] * N[(1.0 / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.5e6 or 3.4e5 < y

   1. Initial program 27.3%

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

      1. associate-/l* 47.6%

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

      2. remove-double-neg 47.6%

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

      3. remove-double-neg 47.6%

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

      4. +-commutative 47.6%

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

   3. Simplified 47.6%

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

   5. Taylor expanded in y around -inf 100.0%

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

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

   if -4.5e6 < y < 3.4e5

   1. Initial program 99.8%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.9%

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

      3. +-commutative 99.9%

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4500000 \lor \neg \left(y \leq 340000\right):\\ \;\;\;\;x - \frac{\frac{1 - x}{y} + \left(x + -1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - x\right) \cdot y\right) \cdot \frac{1}{-1 - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -225000000:\\ \;\;\;\;x - \frac{x + -1}{y}\\ \mathbf{elif}\;y \leq 92000000000:\\ \;\;\;\;1 + \left(\left(1 - x\right) \cdot y\right) \cdot \frac{1}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -225000000.0)
   (- x (/ (+ x -1.0) y))
   (if (<= y 92000000000.0)
     (+ 1.0 (* (* (- 1.0 x) y) (/ 1.0 (- -1.0 y))))
     (+ x (/ 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -225000000.0) {
		tmp = x - ((x + -1.0) / y);
	} else if (y <= 92000000000.0) {
		tmp = 1.0 + (((1.0 - x) * y) * (1.0 / (-1.0 - y)));
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-225000000.0d0)) then
        tmp = x - ((x + (-1.0d0)) / y)
    else if (y <= 92000000000.0d0) then
        tmp = 1.0d0 + (((1.0d0 - x) * y) * (1.0d0 / ((-1.0d0) - y)))
    else
        tmp = x + (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -225000000.0) {
		tmp = x - ((x + -1.0) / y);
	} else if (y <= 92000000000.0) {
		tmp = 1.0 + (((1.0 - x) * y) * (1.0 / (-1.0 - y)));
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -225000000.0:
		tmp = x - ((x + -1.0) / y)
	elif y <= 92000000000.0:
		tmp = 1.0 + (((1.0 - x) * y) * (1.0 / (-1.0 - y)))
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -225000000.0)
		tmp = Float64(x - Float64(Float64(x + -1.0) / y));
	elseif (y <= 92000000000.0)
		tmp = Float64(1.0 + Float64(Float64(Float64(1.0 - x) * y) * Float64(1.0 / Float64(-1.0 - y))));
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -225000000.0)
		tmp = x - ((x + -1.0) / y);
	elseif (y <= 92000000000.0)
		tmp = 1.0 + (((1.0 - x) * y) * (1.0 / (-1.0 - y)));
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -225000000.0], N[(x - N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 92000000000.0], N[(1.0 + N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] * N[(1.0 / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.25e8

   1. Initial program 25.8%

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

      1. associate-/l* 47.4%

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

      2. remove-double-neg 47.4%

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

      3. remove-double-neg 47.4%

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

      4. +-commutative 47.4%

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

   3. Simplified 47.4%

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

   5. Taylor expanded in y around inf 99.6%

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

      1. associate--l+ 99.6%

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

      2. div-sub 99.6%

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

   7. Simplified 99.6%

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

   if -2.25e8 < y < 9.2e10

   1. Initial program 99.8%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.9%

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

      3. +-commutative 99.9%

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

   4. Applied egg-rr 99.9%

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

   if 9.2e10 < y

   1. Initial program 27.7%

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

      1. associate-/l* 47.0%

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

      2. remove-double-neg 47.0%

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

      3. remove-double-neg 47.0%

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

      4. +-commutative 47.0%

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

   3. Simplified 47.0%

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

   5. Taylor expanded in y around inf 99.8%

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

      1. associate--l+ 99.8%

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

      2. div-sub 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around 0 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -225000000:\\ \;\;\;\;x - \frac{x + -1}{y}\\ \mathbf{elif}\;y \leq 92000000000:\\ \;\;\;\;1 + \left(\left(1 - x\right) \cdot y\right) \cdot \frac{1}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -310000000:\\ \;\;\;\;x - \frac{x + -1}{y}\\ \mathbf{elif}\;y \leq 150000000000:\\ \;\;\;\;1 + \left(1 - x\right) \cdot \frac{y}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -310000000.0)
   (- x (/ (+ x -1.0) y))
   (if (<= y 150000000000.0)
     (+ 1.0 (* (- 1.0 x) (/ y (- -1.0 y))))
     (+ x (/ 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -310000000.0) {
		tmp = x - ((x + -1.0) / y);
	} else if (y <= 150000000000.0) {
		tmp = 1.0 + ((1.0 - x) * (y / (-1.0 - y)));
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-310000000.0d0)) then
        tmp = x - ((x + (-1.0d0)) / y)
    else if (y <= 150000000000.0d0) then
        tmp = 1.0d0 + ((1.0d0 - x) * (y / ((-1.0d0) - y)))
    else
        tmp = x + (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -310000000.0) {
		tmp = x - ((x + -1.0) / y);
	} else if (y <= 150000000000.0) {
		tmp = 1.0 + ((1.0 - x) * (y / (-1.0 - y)));
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -310000000.0:
		tmp = x - ((x + -1.0) / y)
	elif y <= 150000000000.0:
		tmp = 1.0 + ((1.0 - x) * (y / (-1.0 - y)))
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -310000000.0)
		tmp = Float64(x - Float64(Float64(x + -1.0) / y));
	elseif (y <= 150000000000.0)
		tmp = Float64(1.0 + Float64(Float64(1.0 - x) * Float64(y / Float64(-1.0 - y))));
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -310000000.0)
		tmp = x - ((x + -1.0) / y);
	elseif (y <= 150000000000.0)
		tmp = 1.0 + ((1.0 - x) * (y / (-1.0 - y)));
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -310000000.0], N[(x - N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 150000000000.0], N[(1.0 + N[(N[(1.0 - x), $MachinePrecision] * N[(y / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.1e8

   1. Initial program 25.8%

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

      1. associate-/l* 47.4%

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

      2. remove-double-neg 47.4%

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

      3. remove-double-neg 47.4%

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

      4. +-commutative 47.4%

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

   3. Simplified 47.4%

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

   5. Taylor expanded in y around inf 99.6%

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

      1. associate--l+ 99.6%

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

      2. div-sub 99.6%

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

   7. Simplified 99.6%

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

   if -3.1e8 < y < 1.5e11

   1. Initial program 99.8%

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

      1. associate-/l* 99.9%

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

      2. remove-double-neg 99.9%

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

      3. remove-double-neg 99.9%

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

      4. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   if 1.5e11 < y

   1. Initial program 27.7%

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

      1. associate-/l* 47.0%

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

      2. remove-double-neg 47.0%

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

      3. remove-double-neg 47.0%

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

      4. +-commutative 47.0%

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

   3. Simplified 47.0%

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

   5. Taylor expanded in y around inf 99.8%

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

      1. associate--l+ 99.8%

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

      2. div-sub 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around 0 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -310000000:\\ \;\;\;\;x - \frac{x + -1}{y}\\ \mathbf{elif}\;y \leq 150000000000:\\ \;\;\;\;1 + \left(1 - x\right) \cdot \frac{y}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{-42}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 0.385:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y -5.2e-42) (* x y) (if (<= y 0.385) (- 1.0 y) x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= -5.2e-42) {
		tmp = x * y;
	} else if (y <= 0.385) {
		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 <= (-5.2d-42)) then
        tmp = x * y
    else if (y <= 0.385d0) 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 <= -5.2e-42) {
		tmp = x * y;
	} else if (y <= 0.385) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= -5.2e-42:
		tmp = x * y
	elif y <= 0.385:
		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 <= -5.2e-42)
		tmp = Float64(x * y);
	elseif (y <= 0.385)
		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 <= -5.2e-42)
		tmp = x * y;
	elseif (y <= 0.385)
		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, -5.2e-42], N[(x * y), $MachinePrecision], If[LessEqual[y, 0.385], N[(1.0 - y), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 0.38500000000000001 < y

   1. Initial program 30.5%

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

      1. associate-/l* 49.9%

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

      2. remove-double-neg 49.9%

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

      3. remove-double-neg 49.9%

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

      4. +-commutative 49.9%

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

   3. Simplified 49.9%

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

   5. Taylor expanded in y around inf 70.0%

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

   if -1 < y < -5.2e-42

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. +-commutative 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-/l* 99.9%

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

      5. distribute-rgt-neg-in 99.9%

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

      6. fma-define 99.9%

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

      7. distribute-frac-neg2 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. metadata-eval 99.9%

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

      11. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 73.5%

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

      1. *-commutative 73.5%

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

      2. *-lft-identity 73.5%

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

      3. times-frac 73.5%

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

      4. /-rgt-identity 73.5%

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

      5. +-commutative 73.5%

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

   7. Simplified 73.5%

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

   8. Taylor expanded in y around 0 66.5%

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

      1. *-commutative 66.5%

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

   10. Simplified 66.5%

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

   if -5.2e-42 < y < 0.38500000000000001

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 73.2%

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

      1. +-commutative 73.2%

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

   7. Simplified 73.2%

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

   8. Taylor expanded in y around 0 72.7%

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

      1. neg-mul-1 72.7%

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

      2. unsub-neg 72.7%

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

   10. Simplified 72.7%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{-42}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 0.385:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.5% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.1)))
   (- x (/ (+ x -1.0) y))
   (+ 1.0 (* y (+ x -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.1)) {
		tmp = x - ((x + -1.0) / y);
	} else {
		tmp = 1.0 + (y * (x + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.0d0)) .or. (.not. (y <= 1.1d0))) then
        tmp = x - ((x + (-1.0d0)) / y)
    else
        tmp = 1.0d0 + (y * (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.1)) {
		tmp = x - ((x + -1.0) / y);
	} else {
		tmp = 1.0 + (y * (x + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 1.1):
		tmp = x - ((x + -1.0) / y)
	else:
		tmp = 1.0 + (y * (x + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 1.1))
		tmp = Float64(x - Float64(Float64(x + -1.0) / y));
	else
		tmp = Float64(1.0 + Float64(y * Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 1.1)))
		tmp = x - ((x + -1.0) / y);
	else
		tmp = 1.0 + (y * (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.1]], $MachinePrecision]], N[(x - N[(N[(x + -1.0), $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 1.1000000000000001 < y

   1. Initial program 30.0%

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

      1. associate-/l* 49.5%

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

      2. remove-double-neg 49.5%

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

      3. remove-double-neg 49.5%

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

      4. +-commutative 49.5%

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

   3. Simplified 49.5%

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

   5. Taylor expanded in y around inf 97.6%

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

      1. associate--l+ 97.6%

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

      2. div-sub 97.6%

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

   7. Simplified 97.6%

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

   if -1 < y < 1.1000000000000001

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 97.2%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.1\right):\\ \;\;\;\;x - \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(x + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.25\right):\\ \;\;\;\;x - \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.25)))
   (- x (/ (+ x -1.0) y))
   (+ 1.0 (* x y))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.25)) {
		tmp = x - ((x + -1.0) / y);
	} else {
		tmp = 1.0 + (x * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.0d0)) .or. (.not. (y <= 1.25d0))) then
        tmp = x - ((x + (-1.0d0)) / y)
    else
        tmp = 1.0d0 + (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.25)) {
		tmp = x - ((x + -1.0) / y);
	} else {
		tmp = 1.0 + (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 1.25):
		tmp = x - ((x + -1.0) / y)
	else:
		tmp = 1.0 + (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 1.25))
		tmp = Float64(x - Float64(Float64(x + -1.0) / y));
	else
		tmp = Float64(1.0 + Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 1.25)))
		tmp = x - ((x + -1.0) / y);
	else
		tmp = 1.0 + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.25]], $MachinePrecision]], N[(x - N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.25 < y

   1. Initial program 30.0%

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

      1. associate-/l* 49.5%

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

      2. remove-double-neg 49.5%

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

      3. remove-double-neg 49.5%

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

      4. +-commutative 49.5%

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

   3. Simplified 49.5%

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

   5. Taylor expanded in y around inf 97.6%

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

      1. associate--l+ 97.6%

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

      2. div-sub 97.6%

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

   7. Simplified 97.6%

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

   if -1 < y < 1.25

   1. Initial program 100.0%

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

      1. clear-num 99.9%

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

      2. associate-/r/ 100.0%

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

      3. +-commutative 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 98.2%

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

      1. neg-mul-1 98.2%

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

   7. Simplified 98.2%

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

   8. Taylor expanded in y around 0 96.7%

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

      1. *-commutative 96.7%

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

   10. Simplified 96.7%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.25\right):\\ \;\;\;\;x - \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{-42}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 0.65:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y -5.2e-42) (* x y) (if (<= y 0.65) 1.0 x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= -5.2e-42) {
		tmp = x * y;
	} else if (y <= 0.65) {
		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 <= (-5.2d-42)) then
        tmp = x * y
    else if (y <= 0.65d0) 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 <= -5.2e-42) {
		tmp = x * y;
	} else if (y <= 0.65) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= -5.2e-42:
		tmp = x * y
	elif y <= 0.65:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= -5.2e-42)
		tmp = Float64(x * y);
	elseif (y <= 0.65)
		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 <= -5.2e-42)
		tmp = x * y;
	elseif (y <= 0.65)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, -5.2e-42], N[(x * y), $MachinePrecision], If[LessEqual[y, 0.65], 1.0, x]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 0.650000000000000022 < y

   1. Initial program 30.5%

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

      1. associate-/l* 49.9%

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

      2. remove-double-neg 49.9%

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

      3. remove-double-neg 49.9%

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

      4. +-commutative 49.9%

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

   3. Simplified 49.9%

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

   5. Taylor expanded in y around inf 70.0%

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

   if -1 < y < -5.2e-42

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. +-commutative 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-/l* 99.9%

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

      5. distribute-rgt-neg-in 99.9%

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

      6. fma-define 99.9%

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

      7. distribute-frac-neg2 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. metadata-eval 99.9%

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

      11. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 73.5%

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

      1. *-commutative 73.5%

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

      2. *-lft-identity 73.5%

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

      3. times-frac 73.5%

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

      4. /-rgt-identity 73.5%

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

      5. +-commutative 73.5%

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

   7. Simplified 73.5%

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

   8. Taylor expanded in y around 0 66.5%

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

      1. *-commutative 66.5%

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

   10. Simplified 66.5%

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

   if -5.2e-42 < y < 0.650000000000000022

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 73.2%

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

      1. +-commutative 73.2%

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

   7. Simplified 73.2%

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

   8. Taylor expanded in y around 0 72.5%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{-42}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 0.65:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 30.5%

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

      1. associate-/l* 49.9%

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

      2. remove-double-neg 49.9%

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

      3. remove-double-neg 49.9%

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

      4. +-commutative 49.9%

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

   3. Simplified 49.9%

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

   5. Taylor expanded in y around inf 97.0%

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

      1. associate--l+ 97.0%

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

      2. div-sub 97.0%

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

   7. Simplified 97.0%

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

   8. Taylor expanded in x around 0 96.1%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

      1. clear-num 99.9%

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

      2. associate-/r/ 100.0%

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

      3. +-commutative 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 98.2%

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

      1. neg-mul-1 98.2%

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

   7. Simplified 98.2%

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

   8. Taylor expanded in y around 0 97.3%

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

      1. *-commutative 97.3%

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

   10. Simplified 97.3%

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

4. Final simplification 96.7%

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

Alternative 13: 85.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.06:\\ \;\;\;\;1 + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 1.06) (+ 1.0 (* x y)) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 1.06) {
		tmp = 1.0 + (x * y);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x
    else if (y <= 1.06d0) then
        tmp = 1.0d0 + (x * y)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 1.06) {
		tmp = 1.0 + (x * y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 1.06:
		tmp = 1.0 + (x * y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 1.06)
		tmp = Float64(1.0 + Float64(x * y));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 1.06)
		tmp = 1.0 + (x * y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 1.06], N[(1.0 + N[(x * y), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.0600000000000001 < y

   1. Initial program 30.5%

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

      1. associate-/l* 49.9%

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

      2. remove-double-neg 49.9%

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

      3. remove-double-neg 49.9%

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

      4. +-commutative 49.9%

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

   3. Simplified 49.9%

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

   5. Taylor expanded in y around inf 70.0%

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

   if -1 < y < 1.0600000000000001

   1. Initial program 100.0%

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

      1. clear-num 99.9%

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

      2. associate-/r/ 100.0%

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

      3. +-commutative 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 98.2%

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

      1. neg-mul-1 98.2%

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

   7. Simplified 98.2%

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

   8. Taylor expanded in y around 0 97.3%

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

      1. *-commutative 97.3%

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

   10. Simplified 97.3%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.06:\\ \;\;\;\;1 + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 73.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.94:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1.0) x (if (<= y 0.94) 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.94) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x
    else if (y <= 0.94d0) then
        tmp = 1.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.94) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 0.94:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 0.94)
		tmp = 1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 0.94)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 0.94], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.93999999999999995 < y

   1. Initial program 30.5%

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

      1. associate-/l* 49.9%

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

      2. remove-double-neg 49.9%

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

      3. remove-double-neg 49.9%

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

      4. +-commutative 49.9%

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

   3. Simplified 49.9%

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

   5. Taylor expanded in y around inf 70.0%

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

   if -1 < y < 0.93999999999999995

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 69.3%

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

      1. +-commutative 69.3%

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

   7. Simplified 69.3%

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

   8. Taylor expanded in y around 0 68.0%

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

Alternative 15: 38.6% 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 64.4%

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

   1. associate-/l* 74.3%

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

   2. remove-double-neg 74.3%

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

   3. remove-double-neg 74.3%

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

   4. +-commutative 74.3%

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

3. Simplified 74.3%

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

5. Taylor expanded in x around 0 36.6%

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

   1. +-commutative 36.6%

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

7. Simplified 36.6%

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

8. Taylor expanded in y around 0 35.2%

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

Developer Target 1: 99.7% 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 2024130 
(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))))