Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.3% → 99.9%

Time: 8.3s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.16 \cdot 10^{+16} \lor \neg \left(x \leq 2 \cdot 10^{+15}\right):\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.16e+16) (not (<= x 2e+15)))
   (+ 1.0 (/ (+ x -1.0) y))
   (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.16e+16) || !(x <= 2e+15)) {
		tmp = 1.0 + ((x + -1.0) / y);
	} else {
		tmp = (x * ((x / y) + 1.0)) / (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 ((x <= (-1.16d+16)) .or. (.not. (x <= 2d+15))) then
        tmp = 1.0d0 + ((x + (-1.0d0)) / y)
    else
        tmp = (x * ((x / y) + 1.0d0)) / (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.16e+16) || !(x <= 2e+15)) {
		tmp = 1.0 + ((x + -1.0) / y);
	} else {
		tmp = (x * ((x / y) + 1.0)) / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.16e+16) or not (x <= 2e+15):
		tmp = 1.0 + ((x + -1.0) / y)
	else:
		tmp = (x * ((x / y) + 1.0)) / (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.16e+16) || !(x <= 2e+15))
		tmp = Float64(1.0 + Float64(Float64(x + -1.0) / y));
	else
		tmp = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.16e+16) || ~((x <= 2e+15)))
		tmp = 1.0 + ((x + -1.0) / y);
	else
		tmp = (x * ((x / y) + 1.0)) / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.16e+16], N[Not[LessEqual[x, 2e+15]], $MachinePrecision]], N[(1.0 + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.16e16 or 2e15 < x

   1. Initial program 74.1%

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

      1. *-commutative 74.1%

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

      2. associate-/l* 100.0%

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

      3. +-commutative 100.0%

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

      4. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 100.0%

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

      7. distribute-frac-neg 100.0%

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

      8. *-inverses 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. div-sub 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   if -1.16e16 < x < 2e15

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.16 \cdot 10^{+16} \lor \neg \left(x \leq 2 \cdot 10^{+15}\right):\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \end{array} \]

Alternative 2: 85.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -310000000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.0095:\\ \;\;\;\;x \cdot \left(\frac{x}{y} + 1\right)\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+119}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= x -310000000000.0)
     (/ x y)
     (if (<= x -5.5e-12)
       t_0
       (if (<= x 0.0095)
         (* x (+ (/ x y) 1.0))
         (if (<= x 4e+119) t_0 (/ x y)))))))

C

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -310000000000.0) {
		tmp = x / y;
	} else if (x <= -5.5e-12) {
		tmp = t_0;
	} else if (x <= 0.0095) {
		tmp = x * ((x / y) + 1.0);
	} else if (x <= 4e+119) {
		tmp = t_0;
	} else {
		tmp = x / 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 = x / (x + 1.0d0)
    if (x <= (-310000000000.0d0)) then
        tmp = x / y
    else if (x <= (-5.5d-12)) then
        tmp = t_0
    else if (x <= 0.0095d0) then
        tmp = x * ((x / y) + 1.0d0)
    else if (x <= 4d+119) then
        tmp = t_0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -310000000000.0) {
		tmp = x / y;
	} else if (x <= -5.5e-12) {
		tmp = t_0;
	} else if (x <= 0.0095) {
		tmp = x * ((x / y) + 1.0);
	} else if (x <= 4e+119) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + 1.0)
	tmp = 0
	if x <= -310000000000.0:
		tmp = x / y
	elif x <= -5.5e-12:
		tmp = t_0
	elif x <= 0.0095:
		tmp = x * ((x / y) + 1.0)
	elif x <= 4e+119:
		tmp = t_0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (x <= -310000000000.0)
		tmp = Float64(x / y);
	elseif (x <= -5.5e-12)
		tmp = t_0;
	elseif (x <= 0.0095)
		tmp = Float64(x * Float64(Float64(x / y) + 1.0));
	elseif (x <= 4e+119)
		tmp = t_0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + 1.0);
	tmp = 0.0;
	if (x <= -310000000000.0)
		tmp = x / y;
	elseif (x <= -5.5e-12)
		tmp = t_0;
	elseif (x <= 0.0095)
		tmp = x * ((x / y) + 1.0);
	elseif (x <= 4e+119)
		tmp = t_0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -310000000000.0], N[(x / y), $MachinePrecision], If[LessEqual[x, -5.5e-12], t$95$0, If[LessEqual[x, 0.0095], N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4e+119], t$95$0, N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.1e11 or 3.99999999999999978e119 < x

   1. Initial program 70.7%

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

   2. Taylor expanded in x around inf 83.0%

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

   if -3.1e11 < x < -5.5000000000000004e-12 or 0.00949999999999999976 < x < 3.99999999999999978e119

   1. Initial program 96.1%

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

   2. Taylor expanded in y around inf 83.4%

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

      1. +-commutative 83.4%

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

   4. Simplified 83.4%

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

   if -5.5000000000000004e-12 < x < 0.00949999999999999976

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/l* 99.6%

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

      3. +-commutative 99.6%

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

      4. remove-double-neg 99.6%

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

      5. unsub-neg 99.6%

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

      6. div-sub 99.6%

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

      7. distribute-frac-neg 99.6%

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

      8. *-inverses 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 98.3%

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

      1. associate-/r/ 98.6%

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

      2. /-rgt-identity 98.6%

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

      3. +-commutative 98.6%

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

   6. Applied egg-rr 98.6%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -310000000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 0.0095:\\ \;\;\;\;x \cdot \left(\frac{x}{y} + 1\right)\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+119}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 3: 98.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -38 \lor \neg \left(x \leq 80000\right):\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{x}{y + \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -38.0) (not (<= x 80000.0)))
   (+ 1.0 (/ (+ x -1.0) y))
   (+ x (/ x (+ y (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -38.0) || !(x <= 80000.0)) {
		tmp = 1.0 + ((x + -1.0) / y);
	} else {
		tmp = x + (x / (y + (y / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-38.0d0)) .or. (.not. (x <= 80000.0d0))) then
        tmp = 1.0d0 + ((x + (-1.0d0)) / y)
    else
        tmp = x + (x / (y + (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -38.0) || !(x <= 80000.0)) {
		tmp = 1.0 + ((x + -1.0) / y);
	} else {
		tmp = x + (x / (y + (y / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -38.0) or not (x <= 80000.0):
		tmp = 1.0 + ((x + -1.0) / y)
	else:
		tmp = x + (x / (y + (y / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -38.0) || !(x <= 80000.0))
		tmp = Float64(1.0 + Float64(Float64(x + -1.0) / y));
	else
		tmp = Float64(x + Float64(x / Float64(y + Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -38.0) || ~((x <= 80000.0)))
		tmp = 1.0 + ((x + -1.0) / y);
	else
		tmp = x + (x / (y + (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -38.0], N[Not[LessEqual[x, 80000.0]], $MachinePrecision]], N[(1.0 + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(x / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -38 or 8e4 < x

   1. Initial program 75.3%

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

      1. *-commutative 75.3%

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

      2. associate-/l* 100.0%

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

      3. +-commutative 100.0%

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

      4. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 100.0%

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

      7. distribute-frac-neg 100.0%

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

      8. *-inverses 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 100.0%

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

   5. Taylor expanded in x around inf 98.4%

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

      1. associate--l+ 98.4%

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

      2. div-sub 98.4%

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

      3. sub-neg 98.4%

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

      4. metadata-eval 98.4%

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

   7. Simplified 98.4%

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

   if -38 < x < 8e4

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/l* 99.6%

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

      3. +-commutative 99.6%

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

      4. remove-double-neg 99.6%

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

      5. unsub-neg 99.6%

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

      6. div-sub 99.6%

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

      7. distribute-frac-neg 99.6%

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

      8. *-inverses 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 99.6%

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

      1. frac-2neg 99.6%

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

      2. add-log-exp 72.3%

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

      3. neg-log 72.3%

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

      4. distribute-lft-in 72.3%

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

      5. div-inv 72.3%

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

      6. exp-sum 46.2%

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

      7. *-rgt-identity 46.2%

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

      8. exp-sum 72.3%

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

      9. neg-log 72.3%

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

      10. add-log-exp 99.7%

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

      11. clear-num 99.6%

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

      12. frac-2neg 99.6%

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

      13. inv-pow 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in x around 0 98.7%

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

   10. Taylor expanded in y around 0 98.6%

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

       1. distribute-lft-in 98.7%

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

       2. *-rgt-identity 98.7%

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

       3. associate-*r/ 98.8%

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

       4. *-rgt-identity 98.8%

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

   12. Simplified 98.8%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -38 \lor \neg \left(x \leq 80000\right):\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{x}{y + \frac{y}{x}}\\ \end{array} \]

Alternative 4: 72.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-16}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+119}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ x y)
   (if (<= x 9.2e-16) x (if (<= x 4e+119) (+ 1.0 (/ -1.0 x)) (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 9.2e-16) {
		tmp = x;
	} else if (x <= 4e+119) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = 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 (x <= (-1.0d0)) then
        tmp = x / y
    else if (x <= 9.2d-16) then
        tmp = x
    else if (x <= 4d+119) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 9.2e-16) {
		tmp = x;
	} else if (x <= 4e+119) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = x / y
	elif x <= 9.2e-16:
		tmp = x
	elif x <= 4e+119:
		tmp = 1.0 + (-1.0 / x)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(x / y);
	elseif (x <= 9.2e-16)
		tmp = x;
	elseif (x <= 4e+119)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = x / y;
	elseif (x <= 9.2e-16)
		tmp = x;
	elseif (x <= 4e+119)
		tmp = 1.0 + (-1.0 / x);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(x / y), $MachinePrecision], If[LessEqual[x, 9.2e-16], x, If[LessEqual[x, 4e+119], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1 or 3.99999999999999978e119 < x

   1. Initial program 71.8%

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

   2. Taylor expanded in x around inf 80.0%

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

   if -1 < x < 9.1999999999999996e-16

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 72.1%

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

   if 9.1999999999999996e-16 < x < 3.99999999999999978e119

   1. Initial program 95.7%

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

   2. Taylor expanded in x around inf 86.4%

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

      1. associate--r+ 86.4%

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

   4. Simplified 86.4%

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

   5. Taylor expanded in x around 0 68.7%

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

   6. Taylor expanded in y around inf 68.6%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-16}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+119}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 5: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.2)) {
		tmp = 1.0 + ((x + -1.0) / y);
	} else {
		tmp = x * ((x / y) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.2d0))) then
        tmp = 1.0d0 + ((x + (-1.0d0)) / y)
    else
        tmp = x * ((x / y) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.2)) {
		tmp = 1.0 + ((x + -1.0) / y);
	} else {
		tmp = x * ((x / y) + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.2):
		tmp = 1.0 + ((x + -1.0) / y)
	else:
		tmp = x * ((x / y) + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.2))
		tmp = Float64(1.0 + Float64(Float64(x + -1.0) / y));
	else
		tmp = Float64(x * Float64(Float64(x / y) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.2)))
		tmp = 1.0 + ((x + -1.0) / y);
	else
		tmp = x * ((x / y) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.2]], $MachinePrecision]], N[(1.0 + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.19999999999999996 < x

   1. Initial program 75.3%

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

      1. *-commutative 75.3%

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

      2. associate-/l* 100.0%

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

      3. +-commutative 100.0%

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

      4. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 100.0%

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

      7. distribute-frac-neg 100.0%

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

      8. *-inverses 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 100.0%

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

   5. Taylor expanded in x around inf 98.4%

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

      1. associate--l+ 98.4%

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

      2. div-sub 98.4%

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

      3. sub-neg 98.4%

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

      4. metadata-eval 98.4%

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

   7. Simplified 98.4%

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

   if -1 < x < 1.19999999999999996

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/l* 99.6%

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

      3. +-commutative 99.6%

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

      4. remove-double-neg 99.6%

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

      5. unsub-neg 99.6%

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

      6. div-sub 99.6%

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

      7. distribute-frac-neg 99.6%

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

      8. *-inverses 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 97.6%

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

      1. associate-/r/ 97.9%

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

      2. /-rgt-identity 97.9%

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

      3. +-commutative 97.9%

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

   6. Applied egg-rr 97.9%

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

4. Final simplification 98.1%

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

Alternative 6: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.25 < x

   1. Initial program 75.3%

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

      1. *-commutative 75.3%

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

      2. associate-/l* 100.0%

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

      3. +-commutative 100.0%

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

      4. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 100.0%

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

      7. distribute-frac-neg 100.0%

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

      8. *-inverses 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 100.0%

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

   5. Taylor expanded in x around inf 98.4%

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

      1. associate--l+ 98.4%

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

      2. div-sub 98.4%

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

      3. sub-neg 98.4%

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

      4. metadata-eval 98.4%

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

   7. Simplified 98.4%

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

   if -1 < x < 1.25

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/l* 99.6%

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

      3. +-commutative 99.6%

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

      4. remove-double-neg 99.6%

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

      5. unsub-neg 99.6%

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

      6. div-sub 99.6%

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

      7. distribute-frac-neg 99.6%

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

      8. *-inverses 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 97.6%

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

      1. associate-/r/ 97.9%

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

      2. /-rgt-identity 97.9%

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

      3. +-commutative 97.9%

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

   6. Applied egg-rr 97.9%

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

      1. *-commutative 97.9%

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

      2. +-commutative 97.9%

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

      3. distribute-rgt-in 97.9%

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

      4. *-un-lft-identity 97.9%

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

   8. Applied egg-rr 97.9%

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

      1. associate-*l/ 89.5%

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

      2. associate-/l* 98.0%

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

   10. Applied egg-rr 98.0%

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

4. Final simplification 98.2%

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

Alternative 7: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{x}{y} + 1}{\frac{1}{x} - -1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

   1. *-commutative 87.8%

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

   2. associate-/l* 99.8%

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

   3. +-commutative 99.8%

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

   4. remove-double-neg 99.8%

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

   5. unsub-neg 99.8%

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

   6. div-sub 99.8%

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

   7. distribute-frac-neg 99.8%

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

   8. *-inverses 99.8%

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

   9. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 8: 73.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 0.0078:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+119}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ x y)
   (if (<= x 0.0078) x (if (<= x 4.4e+119) 1.0 (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 0.0078) {
		tmp = x;
	} else if (x <= 4.4e+119) {
		tmp = 1.0;
	} else {
		tmp = 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 (x <= (-1.0d0)) then
        tmp = x / y
    else if (x <= 0.0078d0) then
        tmp = x
    else if (x <= 4.4d+119) then
        tmp = 1.0d0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 0.0078) {
		tmp = x;
	} else if (x <= 4.4e+119) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = x / y
	elif x <= 0.0078:
		tmp = x
	elif x <= 4.4e+119:
		tmp = 1.0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(x / y);
	elseif (x <= 0.0078)
		tmp = x;
	elseif (x <= 4.4e+119)
		tmp = 1.0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = x / y;
	elseif (x <= 0.0078)
		tmp = x;
	elseif (x <= 4.4e+119)
		tmp = 1.0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(x / y), $MachinePrecision], If[LessEqual[x, 0.0078], x, If[LessEqual[x, 4.4e+119], 1.0, N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1 or 4.4000000000000003e119 < x

   1. Initial program 71.8%

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

   2. Taylor expanded in x around inf 80.0%

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

   if -1 < x < 0.0077999999999999996

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 71.0%

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

   if 0.0077999999999999996 < x < 4.4000000000000003e119

   1. Initial program 95.2%

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

   2. Taylor expanded in y around inf 76.2%

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

      1. +-commutative 76.2%

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

   4. Simplified 76.2%

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

   5. Taylor expanded in x around inf 73.4%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 0.0078:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+119}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 9: 74.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -460000000000 \lor \neg \left(x \leq 4 \cdot 10^{+119}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -460000000000.0) (not (<= x 4e+119))) (/ x y) (/ x (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -460000000000.0) || !(x <= 4e+119)) {
		tmp = x / y;
	} else {
		tmp = x / (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 ((x <= (-460000000000.0d0)) .or. (.not. (x <= 4d+119))) then
        tmp = x / y
    else
        tmp = x / (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -460000000000.0) || !(x <= 4e+119)) {
		tmp = x / y;
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -460000000000.0) or not (x <= 4e+119):
		tmp = x / y
	else:
		tmp = x / (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -460000000000.0) || !(x <= 4e+119))
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -460000000000.0) || ~((x <= 4e+119)))
		tmp = x / y;
	else
		tmp = x / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -460000000000.0], N[Not[LessEqual[x, 4e+119]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.6e11 or 3.99999999999999978e119 < x

   1. Initial program 70.7%

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

   2. Taylor expanded in x around inf 83.0%

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

   if -4.6e11 < x < 3.99999999999999978e119

   1. Initial program 99.3%

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

   2. Taylor expanded in y around inf 72.8%

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

      1. +-commutative 72.8%

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

   4. Simplified 72.8%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -460000000000 \lor \neg \left(x \leq 4 \cdot 10^{+119}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]

Alternative 10: 48.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.0078:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) 1.0 (if (<= x 0.0078) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 0.0078) {
		tmp = x;
	} else {
		tmp = 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 (x <= (-1.0d0)) then
        tmp = 1.0d0
    else if (x <= 0.0078d0) then
        tmp = x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 0.0078) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = 1.0
	elif x <= 0.0078:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 0.0078)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 0.0078)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], 1.0, If[LessEqual[x, 0.0078], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.0077999999999999996 < x

   1. Initial program 75.5%

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

   2. Taylor expanded in y around inf 28.0%

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

      1. +-commutative 28.0%

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

   4. Simplified 28.0%

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

   5. Taylor expanded in x around inf 26.6%

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

   if -1 < x < 0.0077999999999999996

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 71.0%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.0078:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11: 13.9% accurate, 11.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 87.8%

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

2. Taylor expanded in y around inf 50.0%

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

   1. +-commutative 50.0%

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

4. Simplified 50.0%

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

5. Taylor expanded in x around inf 15.0%

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

6. Final simplification 15.0%

   \[\leadsto 1 \]

Developer target: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023310 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"
  :precision binary64

  :herbie-target
  (* (/ x 1.0) (/ (+ (/ x y) 1.0) (+ x 1.0)))

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