Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 87.8% → 99.9%

Time: 9.0s

Alternatives: 14

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: 87.8% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \frac{x}{y} \cdot t\_0 + t\_0 \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0)))) (+ (* (/ x y) t_0) t_0)))

C

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	return ((x / y) * t_0) + t_0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = x / (x + 1.0d0)
    code = ((x / y) * t_0) + t_0
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	return ((x / y) * t_0) + t_0;
}

Python

def code(x, y):
	t_0 = x / (x + 1.0)
	return ((x / y) * t_0) + t_0

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + 1.0))
	return Float64(Float64(Float64(x / y) * t_0) + t_0)
end

MATLAB

function tmp = code(x, y)
	t_0 = x / (x + 1.0);
	tmp = ((x / y) * t_0) + t_0;
end

Wolfram

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

Derivation

1. Initial program 89.3%

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

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

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

   1. clear-num 99.8%

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

   2. un-div-inv 99.9%

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

6. Applied egg-rr 99.9%

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

   1. associate-/r/ 99.9%

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

   2. +-commutative 99.9%

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

   3. un-div-inv 99.8%

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

   4. +-commutative 99.8%

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

   5. distribute-lft-in 99.8%

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

   6. *-commutative 99.8%

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

   7. *-un-lft-identity 99.8%

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

   8. un-div-inv 99.9%

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

   9. +-commutative 99.9%

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

   10. un-div-inv 99.9%

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

   11. +-commutative 99.9%

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

   \[\leadsto \frac{x}{y} \cdot \frac{x}{x + 1} + \frac{x}{x + 1} \]
10. Add Preprocessing

Alternative 2: 74.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -3.6 \cdot 10^{+129}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{+106}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1950000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{+14}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= x -3.6e+129)
     (/ x y)
     (if (<= x -2.05e+106)
       1.0
       (if (<= x -1950000000.0)
         (/ x y)
         (if (<= x 1.45e-52)
           t_0
           (if (<= x 9e-9)
             (* x (/ x y))
             (if (<= x 2.65e+14) t_0 (/ x y)))))))))

C

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -3.6e+129) {
		tmp = x / y;
	} else if (x <= -2.05e+106) {
		tmp = 1.0;
	} else if (x <= -1950000000.0) {
		tmp = x / y;
	} else if (x <= 1.45e-52) {
		tmp = t_0;
	} else if (x <= 9e-9) {
		tmp = x * (x / y);
	} else if (x <= 2.65e+14) {
		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 <= (-3.6d+129)) then
        tmp = x / y
    else if (x <= (-2.05d+106)) then
        tmp = 1.0d0
    else if (x <= (-1950000000.0d0)) then
        tmp = x / y
    else if (x <= 1.45d-52) then
        tmp = t_0
    else if (x <= 9d-9) then
        tmp = x * (x / y)
    else if (x <= 2.65d+14) 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 <= -3.6e+129) {
		tmp = x / y;
	} else if (x <= -2.05e+106) {
		tmp = 1.0;
	} else if (x <= -1950000000.0) {
		tmp = x / y;
	} else if (x <= 1.45e-52) {
		tmp = t_0;
	} else if (x <= 9e-9) {
		tmp = x * (x / y);
	} else if (x <= 2.65e+14) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + 1.0)
	tmp = 0
	if x <= -3.6e+129:
		tmp = x / y
	elif x <= -2.05e+106:
		tmp = 1.0
	elif x <= -1950000000.0:
		tmp = x / y
	elif x <= 1.45e-52:
		tmp = t_0
	elif x <= 9e-9:
		tmp = x * (x / y)
	elif x <= 2.65e+14:
		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 <= -3.6e+129)
		tmp = Float64(x / y);
	elseif (x <= -2.05e+106)
		tmp = 1.0;
	elseif (x <= -1950000000.0)
		tmp = Float64(x / y);
	elseif (x <= 1.45e-52)
		tmp = t_0;
	elseif (x <= 9e-9)
		tmp = Float64(x * Float64(x / y));
	elseif (x <= 2.65e+14)
		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 <= -3.6e+129)
		tmp = x / y;
	elseif (x <= -2.05e+106)
		tmp = 1.0;
	elseif (x <= -1950000000.0)
		tmp = x / y;
	elseif (x <= 1.45e-52)
		tmp = t_0;
	elseif (x <= 9e-9)
		tmp = x * (x / y);
	elseif (x <= 2.65e+14)
		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, -3.6e+129], N[(x / y), $MachinePrecision], If[LessEqual[x, -2.05e+106], 1.0, If[LessEqual[x, -1950000000.0], N[(x / y), $MachinePrecision], If[LessEqual[x, 1.45e-52], t$95$0, If[LessEqual[x, 9e-9], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.65e+14], t$95$0, N[(x / y), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.6000000000000001e129 or -2.0500000000000001e106 < x < -1.95e9 or 2.65e14 < x

   1. Initial program 77.0%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 77.7%

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

   if -3.6000000000000001e129 < x < -2.0500000000000001e106

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 100.0%

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

      1. un-div-inv 100.0%

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

      2. clear-num 100.0%

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

      3. +-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around inf 100.0%

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

   if -1.95e9 < x < 1.4500000000000001e-52 or 8.99999999999999953e-9 < x < 2.65e14

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 82.4%

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

   if 1.4500000000000001e-52 < x < 8.99999999999999953e-9

   1. Initial program 99.6%

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

      1. associate-/l* 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 77.5%

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

   6. Taylor expanded in x around 0 69.3%

      \[\leadsto x \cdot \frac{x}{\color{blue}{y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+129}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{+106}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1950000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-52}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{+129}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{+106}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -600000000:\\ \;\;\;\;\frac{x + -1}{y}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+14}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= x -3.4e+129)
     (/ x y)
     (if (<= x -2.05e+106)
       1.0
       (if (<= x -600000000.0)
         (/ (+ x -1.0) y)
         (if (<= x 1.25e-52)
           t_0
           (if (<= x 1.05e-8)
             (* x (/ x y))
             (if (<= x 2.3e+14) t_0 (/ x y)))))))))

C

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -3.4e+129) {
		tmp = x / y;
	} else if (x <= -2.05e+106) {
		tmp = 1.0;
	} else if (x <= -600000000.0) {
		tmp = (x + -1.0) / y;
	} else if (x <= 1.25e-52) {
		tmp = t_0;
	} else if (x <= 1.05e-8) {
		tmp = x * (x / y);
	} else if (x <= 2.3e+14) {
		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 <= (-3.4d+129)) then
        tmp = x / y
    else if (x <= (-2.05d+106)) then
        tmp = 1.0d0
    else if (x <= (-600000000.0d0)) then
        tmp = (x + (-1.0d0)) / y
    else if (x <= 1.25d-52) then
        tmp = t_0
    else if (x <= 1.05d-8) then
        tmp = x * (x / y)
    else if (x <= 2.3d+14) 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 <= -3.4e+129) {
		tmp = x / y;
	} else if (x <= -2.05e+106) {
		tmp = 1.0;
	} else if (x <= -600000000.0) {
		tmp = (x + -1.0) / y;
	} else if (x <= 1.25e-52) {
		tmp = t_0;
	} else if (x <= 1.05e-8) {
		tmp = x * (x / y);
	} else if (x <= 2.3e+14) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + 1.0)
	tmp = 0
	if x <= -3.4e+129:
		tmp = x / y
	elif x <= -2.05e+106:
		tmp = 1.0
	elif x <= -600000000.0:
		tmp = (x + -1.0) / y
	elif x <= 1.25e-52:
		tmp = t_0
	elif x <= 1.05e-8:
		tmp = x * (x / y)
	elif x <= 2.3e+14:
		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 <= -3.4e+129)
		tmp = Float64(x / y);
	elseif (x <= -2.05e+106)
		tmp = 1.0;
	elseif (x <= -600000000.0)
		tmp = Float64(Float64(x + -1.0) / y);
	elseif (x <= 1.25e-52)
		tmp = t_0;
	elseif (x <= 1.05e-8)
		tmp = Float64(x * Float64(x / y));
	elseif (x <= 2.3e+14)
		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 <= -3.4e+129)
		tmp = x / y;
	elseif (x <= -2.05e+106)
		tmp = 1.0;
	elseif (x <= -600000000.0)
		tmp = (x + -1.0) / y;
	elseif (x <= 1.25e-52)
		tmp = t_0;
	elseif (x <= 1.05e-8)
		tmp = x * (x / y);
	elseif (x <= 2.3e+14)
		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, -3.4e+129], N[(x / y), $MachinePrecision], If[LessEqual[x, -2.05e+106], 1.0, If[LessEqual[x, -600000000.0], N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 1.25e-52], t$95$0, If[LessEqual[x, 1.05e-8], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.3e+14], t$95$0, N[(x / y), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -3.40000000000000018e129 or 2.3e14 < x

   1. Initial program 73.1%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 81.1%

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

   if -3.40000000000000018e129 < x < -2.0500000000000001e106

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 100.0%

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

      1. un-div-inv 100.0%

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

      2. clear-num 100.0%

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

      3. +-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around inf 100.0%

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

   if -2.0500000000000001e106 < x < -6e8

   1. Initial program 95.4%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 100.0%

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

   6. Taylor expanded in y around 0 63.5%

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

   if -6e8 < x < 1.25e-52 or 1.04999999999999997e-8 < x < 2.3e14

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 82.4%

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

   if 1.25e-52 < x < 1.04999999999999997e-8

   1. Initial program 99.6%

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

      1. associate-/l* 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 77.5%

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

   6. Taylor expanded in x around 0 69.3%

      \[\leadsto x \cdot \frac{x}{\color{blue}{y}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+129}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{+106}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -600000000:\\ \;\;\;\;\frac{x + -1}{y}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-52}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+129}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{+106}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -180000000:\\ \;\;\;\;\frac{x + -1}{y}\\ \mathbf{elif}\;x \leq 10^{-52}:\\ \;\;\;\;x \cdot \frac{1}{x + 1}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -9e+129)
   (/ x y)
   (if (<= x -2.05e+106)
     1.0
     (if (<= x -180000000.0)
       (/ (+ x -1.0) y)
       (if (<= x 1e-52)
         (* x (/ 1.0 (+ x 1.0)))
         (if (<= x 6.8e-9)
           (* x (/ x y))
           (if (<= x 7.5e+18) (/ x (+ x 1.0)) (/ x y))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -9e+129) {
		tmp = x / y;
	} else if (x <= -2.05e+106) {
		tmp = 1.0;
	} else if (x <= -180000000.0) {
		tmp = (x + -1.0) / y;
	} else if (x <= 1e-52) {
		tmp = x * (1.0 / (x + 1.0));
	} else if (x <= 6.8e-9) {
		tmp = x * (x / y);
	} else if (x <= 7.5e+18) {
		tmp = x / (x + 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 <= (-9d+129)) then
        tmp = x / y
    else if (x <= (-2.05d+106)) then
        tmp = 1.0d0
    else if (x <= (-180000000.0d0)) then
        tmp = (x + (-1.0d0)) / y
    else if (x <= 1d-52) then
        tmp = x * (1.0d0 / (x + 1.0d0))
    else if (x <= 6.8d-9) then
        tmp = x * (x / y)
    else if (x <= 7.5d+18) then
        tmp = x / (x + 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 <= -9e+129) {
		tmp = x / y;
	} else if (x <= -2.05e+106) {
		tmp = 1.0;
	} else if (x <= -180000000.0) {
		tmp = (x + -1.0) / y;
	} else if (x <= 1e-52) {
		tmp = x * (1.0 / (x + 1.0));
	} else if (x <= 6.8e-9) {
		tmp = x * (x / y);
	} else if (x <= 7.5e+18) {
		tmp = x / (x + 1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -9e+129:
		tmp = x / y
	elif x <= -2.05e+106:
		tmp = 1.0
	elif x <= -180000000.0:
		tmp = (x + -1.0) / y
	elif x <= 1e-52:
		tmp = x * (1.0 / (x + 1.0))
	elif x <= 6.8e-9:
		tmp = x * (x / y)
	elif x <= 7.5e+18:
		tmp = x / (x + 1.0)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -9e+129)
		tmp = Float64(x / y);
	elseif (x <= -2.05e+106)
		tmp = 1.0;
	elseif (x <= -180000000.0)
		tmp = Float64(Float64(x + -1.0) / y);
	elseif (x <= 1e-52)
		tmp = Float64(x * Float64(1.0 / Float64(x + 1.0)));
	elseif (x <= 6.8e-9)
		tmp = Float64(x * Float64(x / y));
	elseif (x <= 7.5e+18)
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9e+129)
		tmp = x / y;
	elseif (x <= -2.05e+106)
		tmp = 1.0;
	elseif (x <= -180000000.0)
		tmp = (x + -1.0) / y;
	elseif (x <= 1e-52)
		tmp = x * (1.0 / (x + 1.0));
	elseif (x <= 6.8e-9)
		tmp = x * (x / y);
	elseif (x <= 7.5e+18)
		tmp = x / (x + 1.0);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -9e+129], N[(x / y), $MachinePrecision], If[LessEqual[x, -2.05e+106], 1.0, If[LessEqual[x, -180000000.0], N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 1e-52], N[(x * N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.8e-9], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.5e+18], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < -9.0000000000000003e129 or 7.5e18 < x

   1. Initial program 73.1%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 81.1%

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

   if -9.0000000000000003e129 < x < -2.0500000000000001e106

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 100.0%

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

      1. un-div-inv 100.0%

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

      2. clear-num 100.0%

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

      3. +-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around inf 100.0%

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

   if -2.0500000000000001e106 < x < -1.8e8

   1. Initial program 95.4%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 100.0%

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

   6. Taylor expanded in y around 0 63.5%

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

   if -1.8e8 < x < 1e-52

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 82.0%

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

   if 1e-52 < x < 6.7999999999999997e-9

   1. Initial program 99.6%

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

      1. associate-/l* 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 77.5%

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

   6. Taylor expanded in x around 0 69.3%

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

   if 6.7999999999999997e-9 < x < 7.5e18

   1. Initial program 99.6%

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

      1. associate-/l* 99.2%

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

   3. Simplified 99.2%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 95.0%

      \[\leadsto \color{blue}{\frac{x}{1 + x}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+129}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{+106}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -180000000:\\ \;\;\;\;\frac{x + -1}{y}\\ \mathbf{elif}\;x \leq 10^{-52}:\\ \;\;\;\;x \cdot \frac{1}{x + 1}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+129}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{+106}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-52}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \mathbf{elif}\;x \leq 420000000000:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.4e+129)
   (/ x y)
   (if (<= x -2.05e+106)
     1.0
     (if (<= x -1.0)
       (/ x y)
       (if (<= x 1.3e-52)
         (* x (- 1.0 x))
         (if (<= x 420000000000.0) (* x (/ x y)) (/ x y)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.4e+129) {
		tmp = x / y;
	} else if (x <= -2.05e+106) {
		tmp = 1.0;
	} else if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 1.3e-52) {
		tmp = x * (1.0 - x);
	} else if (x <= 420000000000.0) {
		tmp = x * (x / y);
	} 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 <= (-3.4d+129)) then
        tmp = x / y
    else if (x <= (-2.05d+106)) then
        tmp = 1.0d0
    else if (x <= (-1.0d0)) then
        tmp = x / y
    else if (x <= 1.3d-52) then
        tmp = x * (1.0d0 - x)
    else if (x <= 420000000000.0d0) then
        tmp = x * (x / y)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.4e+129) {
		tmp = x / y;
	} else if (x <= -2.05e+106) {
		tmp = 1.0;
	} else if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 1.3e-52) {
		tmp = x * (1.0 - x);
	} else if (x <= 420000000000.0) {
		tmp = x * (x / y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.4e+129:
		tmp = x / y
	elif x <= -2.05e+106:
		tmp = 1.0
	elif x <= -1.0:
		tmp = x / y
	elif x <= 1.3e-52:
		tmp = x * (1.0 - x)
	elif x <= 420000000000.0:
		tmp = x * (x / y)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.4e+129)
		tmp = Float64(x / y);
	elseif (x <= -2.05e+106)
		tmp = 1.0;
	elseif (x <= -1.0)
		tmp = Float64(x / y);
	elseif (x <= 1.3e-52)
		tmp = Float64(x * Float64(1.0 - x));
	elseif (x <= 420000000000.0)
		tmp = Float64(x * Float64(x / y));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.4e+129)
		tmp = x / y;
	elseif (x <= -2.05e+106)
		tmp = 1.0;
	elseif (x <= -1.0)
		tmp = x / y;
	elseif (x <= 1.3e-52)
		tmp = x * (1.0 - x);
	elseif (x <= 420000000000.0)
		tmp = x * (x / y);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.4e+129], N[(x / y), $MachinePrecision], If[LessEqual[x, -2.05e+106], 1.0, If[LessEqual[x, -1.0], N[(x / y), $MachinePrecision], If[LessEqual[x, 1.3e-52], N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 420000000000.0], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.40000000000000018e129 or -2.0500000000000001e106 < x < -1 or 4.2e11 < x

   1. Initial program 77.6%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 76.1%

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

   if -3.40000000000000018e129 < x < -2.0500000000000001e106

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 100.0%

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

      1. un-div-inv 100.0%

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

      2. clear-num 100.0%

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

      3. +-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around inf 100.0%

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

   if -1 < x < 1.2999999999999999e-52

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 82.4%

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

   6. Taylor expanded in x around 0 82.0%

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

      1. neg-mul-1 82.0%

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

      2. sub-neg 82.0%

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

   8. Simplified 82.0%

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

   if 1.2999999999999999e-52 < x < 4.2e11

   1. Initial program 99.6%

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

      1. associate-/l* 99.5%

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

   3. Simplified 99.5%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 60.0%

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

   6. Taylor expanded in x around 0 53.8%

      \[\leadsto x \cdot \frac{x}{\color{blue}{y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+129}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{+106}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-52}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \mathbf{elif}\;x \leq 420000000000:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x + -1}{y}\\ \mathbf{if}\;x \leq -145000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-52}:\\ \;\;\;\;x \cdot \frac{1}{x + 1}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 420000000000:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (+ x -1.0) y))))
   (if (<= x -145000000.0)
     t_0
     (if (<= x 1.3e-52)
       (* x (/ 1.0 (+ x 1.0)))
       (if (<= x 6.8e-9)
         (* x (/ x y))
         (if (<= x 420000000000.0) (/ x (+ x 1.0)) t_0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + ((x + -1.0) / y);
	double tmp;
	if (x <= -145000000.0) {
		tmp = t_0;
	} else if (x <= 1.3e-52) {
		tmp = x * (1.0 / (x + 1.0));
	} else if (x <= 6.8e-9) {
		tmp = x * (x / y);
	} else if (x <= 420000000000.0) {
		tmp = x / (x + 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 + ((x + (-1.0d0)) / y)
    if (x <= (-145000000.0d0)) then
        tmp = t_0
    else if (x <= 1.3d-52) then
        tmp = x * (1.0d0 / (x + 1.0d0))
    else if (x <= 6.8d-9) then
        tmp = x * (x / y)
    else if (x <= 420000000000.0d0) then
        tmp = x / (x + 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 + ((x + -1.0) / y);
	double tmp;
	if (x <= -145000000.0) {
		tmp = t_0;
	} else if (x <= 1.3e-52) {
		tmp = x * (1.0 / (x + 1.0));
	} else if (x <= 6.8e-9) {
		tmp = x * (x / y);
	} else if (x <= 420000000000.0) {
		tmp = x / (x + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + ((x + -1.0) / y)
	tmp = 0
	if x <= -145000000.0:
		tmp = t_0
	elif x <= 1.3e-52:
		tmp = x * (1.0 / (x + 1.0))
	elif x <= 6.8e-9:
		tmp = x * (x / y)
	elif x <= 420000000000.0:
		tmp = x / (x + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(Float64(x + -1.0) / y))
	tmp = 0.0
	if (x <= -145000000.0)
		tmp = t_0;
	elseif (x <= 1.3e-52)
		tmp = Float64(x * Float64(1.0 / Float64(x + 1.0)));
	elseif (x <= 6.8e-9)
		tmp = Float64(x * Float64(x / y));
	elseif (x <= 420000000000.0)
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + ((x + -1.0) / y);
	tmp = 0.0;
	if (x <= -145000000.0)
		tmp = t_0;
	elseif (x <= 1.3e-52)
		tmp = x * (1.0 / (x + 1.0));
	elseif (x <= 6.8e-9)
		tmp = x * (x / y);
	elseif (x <= 420000000000.0)
		tmp = x / (x + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -145000000.0], t$95$0, If[LessEqual[x, 1.3e-52], N[(x * N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.8e-9], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 420000000000.0], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.45e8 or 4.2e11 < x

   1. Initial program 78.0%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.8%

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

      2. un-div-inv 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 100.0%

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

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

      5. +-commutative 100.0%

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

      6. metadata-eval 100.0%

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

      7. remove-double-neg 100.0%

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

      8. neg-mul-1 100.0%

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

      9. distribute-lft-in 100.0%

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

      10. neg-mul-1 100.0%

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

      11. associate-*r/ 100.0%

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

      12. mul-1-neg 100.0%

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

      13. unsub-neg 100.0%

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

      14. neg-mul-1 100.0%

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

      15. sub-neg 100.0%

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

   9. Simplified 100.0%

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

   if -1.45e8 < x < 1.2999999999999999e-52

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 82.0%

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

   if 1.2999999999999999e-52 < x < 6.7999999999999997e-9

   1. Initial program 99.6%

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

      1. associate-/l* 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 77.5%

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

   6. Taylor expanded in x around 0 69.3%

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

   if 6.7999999999999997e-9 < x < 4.2e11

   1. Initial program 99.6%

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

      1. associate-/l* 99.2%

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

   3. Simplified 99.2%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 95.0%

      \[\leadsto \color{blue}{\frac{x}{1 + x}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -145000000:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-52}:\\ \;\;\;\;x \cdot \frac{1}{x + 1}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 420000000000:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{+129}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{+106}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1 \lor \neg \left(x \leq 0.55\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.9e+129)
   (/ x y)
   (if (<= x -2.05e+106)
     1.0
     (if (or (<= x -1.0) (not (<= x 0.55))) (/ x y) (* x (- 1.0 x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.9e+129) {
		tmp = x / y;
	} else if (x <= -2.05e+106) {
		tmp = 1.0;
	} else if ((x <= -1.0) || !(x <= 0.55)) {
		tmp = x / y;
	} else {
		tmp = x * (1.0 - 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 <= (-3.9d+129)) then
        tmp = x / y
    else if (x <= (-2.05d+106)) then
        tmp = 1.0d0
    else if ((x <= (-1.0d0)) .or. (.not. (x <= 0.55d0))) then
        tmp = x / y
    else
        tmp = x * (1.0d0 - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.9e+129) {
		tmp = x / y;
	} else if (x <= -2.05e+106) {
		tmp = 1.0;
	} else if ((x <= -1.0) || !(x <= 0.55)) {
		tmp = x / y;
	} else {
		tmp = x * (1.0 - x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.9e+129:
		tmp = x / y
	elif x <= -2.05e+106:
		tmp = 1.0
	elif (x <= -1.0) or not (x <= 0.55):
		tmp = x / y
	else:
		tmp = x * (1.0 - x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.9e+129)
		tmp = Float64(x / y);
	elseif (x <= -2.05e+106)
		tmp = 1.0;
	elseif ((x <= -1.0) || !(x <= 0.55))
		tmp = Float64(x / y);
	else
		tmp = Float64(x * Float64(1.0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.9e+129)
		tmp = x / y;
	elseif (x <= -2.05e+106)
		tmp = 1.0;
	elseif ((x <= -1.0) || ~((x <= 0.55)))
		tmp = x / y;
	else
		tmp = x * (1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.9e+129], N[(x / y), $MachinePrecision], If[LessEqual[x, -2.05e+106], 1.0, If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.55]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.8999999999999997e129 or -2.0500000000000001e106 < x < -1 or 0.55000000000000004 < x

   1. Initial program 77.9%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 74.9%

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

   if -3.8999999999999997e129 < x < -2.0500000000000001e106

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 100.0%

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

      1. un-div-inv 100.0%

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

      2. clear-num 100.0%

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

      3. +-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around inf 100.0%

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

   if -1 < x < 0.55000000000000004

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 76.8%

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

   6. Taylor expanded in x around 0 76.0%

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

      1. neg-mul-1 76.0%

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

      2. sub-neg 76.0%

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

   8. Simplified 76.0%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{+129}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{+106}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1 \lor \neg \left(x \leq 0.55\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x + -1}{y}\\ \mathbf{if}\;x \leq -145000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.42 \cdot 10^{-52}:\\ \;\;\;\;x \cdot \frac{1}{x + 1}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (+ x -1.0) y))))
   (if (<= x -145000000.0)
     t_0
     (if (<= x 1.42e-52)
       (* x (/ 1.0 (+ x 1.0)))
       (if (<= x 2.4e-8) (* x (/ x (* y (+ x 1.0)))) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + ((x + -1.0) / y);
	double tmp;
	if (x <= -145000000.0) {
		tmp = t_0;
	} else if (x <= 1.42e-52) {
		tmp = x * (1.0 / (x + 1.0));
	} else if (x <= 2.4e-8) {
		tmp = x * (x / (y * (x + 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 + ((x + (-1.0d0)) / y)
    if (x <= (-145000000.0d0)) then
        tmp = t_0
    else if (x <= 1.42d-52) then
        tmp = x * (1.0d0 / (x + 1.0d0))
    else if (x <= 2.4d-8) then
        tmp = x * (x / (y * (x + 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 + ((x + -1.0) / y);
	double tmp;
	if (x <= -145000000.0) {
		tmp = t_0;
	} else if (x <= 1.42e-52) {
		tmp = x * (1.0 / (x + 1.0));
	} else if (x <= 2.4e-8) {
		tmp = x * (x / (y * (x + 1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + ((x + -1.0) / y)
	tmp = 0
	if x <= -145000000.0:
		tmp = t_0
	elif x <= 1.42e-52:
		tmp = x * (1.0 / (x + 1.0))
	elif x <= 2.4e-8:
		tmp = x * (x / (y * (x + 1.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(Float64(x + -1.0) / y))
	tmp = 0.0
	if (x <= -145000000.0)
		tmp = t_0;
	elseif (x <= 1.42e-52)
		tmp = Float64(x * Float64(1.0 / Float64(x + 1.0)));
	elseif (x <= 2.4e-8)
		tmp = Float64(x * Float64(x / Float64(y * Float64(x + 1.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + ((x + -1.0) / y);
	tmp = 0.0;
	if (x <= -145000000.0)
		tmp = t_0;
	elseif (x <= 1.42e-52)
		tmp = x * (1.0 / (x + 1.0));
	elseif (x <= 2.4e-8)
		tmp = x * (x / (y * (x + 1.0)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -145000000.0], t$95$0, If[LessEqual[x, 1.42e-52], N[(x * N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.4e-8], N[(x * N[(x / N[(y * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.45e8 or 2.39999999999999998e-8 < x

   1. Initial program 78.6%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

      1. clear-num 99.8%

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

      2. un-div-inv 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 98.2%

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

      1. associate--l+ 98.2%

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

      2. div-sub 98.2%

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

      3. sub-neg 98.2%

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

      4. metadata-eval 98.2%

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

      5. +-commutative 98.2%

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

      6. metadata-eval 98.2%

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

      7. remove-double-neg 98.2%

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

      8. neg-mul-1 98.2%

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

      9. distribute-lft-in 98.2%

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

      10. neg-mul-1 98.2%

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

      11. associate-*r/ 98.2%

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

      12. mul-1-neg 98.2%

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

      13. unsub-neg 98.2%

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

      14. neg-mul-1 98.2%

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

      15. sub-neg 98.2%

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

   9. Simplified 98.2%

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

   if -1.45e8 < x < 1.4200000000000001e-52

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 82.0%

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

   if 1.4200000000000001e-52 < x < 2.39999999999999998e-8

   1. Initial program 99.6%

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

      1. associate-/l* 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 77.5%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -145000000:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{elif}\;x \leq 1.42 \cdot 10^{-52}:\\ \;\;\;\;x \cdot \frac{1}{x + 1}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -145000000:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-52}:\\ \;\;\;\;x \cdot \frac{1}{x + 1}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -145000000.0)
   (+ 1.0 (/ (+ x -1.0) y))
   (if (<= x 1.25e-52)
     (* x (/ 1.0 (+ x 1.0)))
     (if (<= x 8.5e-9)
       (* x (/ x (* y (+ x 1.0))))
       (+ (/ x y) (/ x (+ x 1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -145000000.0) {
		tmp = 1.0 + ((x + -1.0) / y);
	} else if (x <= 1.25e-52) {
		tmp = x * (1.0 / (x + 1.0));
	} else if (x <= 8.5e-9) {
		tmp = x * (x / (y * (x + 1.0)));
	} else {
		tmp = (x / y) + (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 <= (-145000000.0d0)) then
        tmp = 1.0d0 + ((x + (-1.0d0)) / y)
    else if (x <= 1.25d-52) then
        tmp = x * (1.0d0 / (x + 1.0d0))
    else if (x <= 8.5d-9) then
        tmp = x * (x / (y * (x + 1.0d0)))
    else
        tmp = (x / y) + (x / (x + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -145000000.0) {
		tmp = 1.0 + ((x + -1.0) / y);
	} else if (x <= 1.25e-52) {
		tmp = x * (1.0 / (x + 1.0));
	} else if (x <= 8.5e-9) {
		tmp = x * (x / (y * (x + 1.0)));
	} else {
		tmp = (x / y) + (x / (x + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -145000000.0:
		tmp = 1.0 + ((x + -1.0) / y)
	elif x <= 1.25e-52:
		tmp = x * (1.0 / (x + 1.0))
	elif x <= 8.5e-9:
		tmp = x * (x / (y * (x + 1.0)))
	else:
		tmp = (x / y) + (x / (x + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -145000000.0)
		tmp = Float64(1.0 + Float64(Float64(x + -1.0) / y));
	elseif (x <= 1.25e-52)
		tmp = Float64(x * Float64(1.0 / Float64(x + 1.0)));
	elseif (x <= 8.5e-9)
		tmp = Float64(x * Float64(x / Float64(y * Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x / y) + Float64(x / Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -145000000.0)
		tmp = 1.0 + ((x + -1.0) / y);
	elseif (x <= 1.25e-52)
		tmp = x * (1.0 / (x + 1.0));
	elseif (x <= 8.5e-9)
		tmp = x * (x / (y * (x + 1.0)));
	else
		tmp = (x / y) + (x / (x + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -145000000.0], N[(1.0 + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.25e-52], N[(x * N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.5e-9], N[(x * N[(x / N[(y * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] + N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.45e8

   1. Initial program 82.4%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

      1. clear-num 99.8%

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

      2. un-div-inv 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 100.0%

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

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

      5. +-commutative 100.0%

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

      6. metadata-eval 100.0%

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

      7. remove-double-neg 100.0%

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

      8. neg-mul-1 100.0%

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

      9. distribute-lft-in 100.0%

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

      10. neg-mul-1 100.0%

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

      11. associate-*r/ 100.0%

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

      12. mul-1-neg 100.0%

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

      13. unsub-neg 100.0%

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

      14. neg-mul-1 100.0%

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

      15. sub-neg 100.0%

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

   9. Simplified 100.0%

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

   if -1.45e8 < x < 1.25e-52

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 82.0%

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

   if 1.25e-52 < x < 8.5e-9

   1. Initial program 99.6%

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

      1. associate-/l* 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 77.5%

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

   if 8.5e-9 < x

   1. Initial program 75.4%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

      1. clear-num 99.8%

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

      2. un-div-inv 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-/r/ 100.0%

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

      2. +-commutative 100.0%

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

      3. un-div-inv 99.7%

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

      4. +-commutative 99.7%

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

      5. distribute-lft-in 99.7%

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

      6. *-commutative 99.7%

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

      7. *-un-lft-identity 99.7%

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

      8. un-div-inv 99.8%

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

      9. +-commutative 99.8%

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

      10. un-div-inv 100.0%

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

      11. +-commutative 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around inf 100.0%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -145000000:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-52}:\\ \;\;\;\;x \cdot \frac{1}{x + 1}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \frac{x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+129}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{+106}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1 \lor \neg \left(x \leq 420000000000\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.4e+129)
   (/ x y)
   (if (<= x -2.05e+106)
     1.0
     (if (or (<= x -1.0) (not (<= x 420000000000.0))) (/ x y) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.4e+129) {
		tmp = x / y;
	} else if (x <= -2.05e+106) {
		tmp = 1.0;
	} else if ((x <= -1.0) || !(x <= 420000000000.0)) {
		tmp = 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 (x <= (-3.4d+129)) then
        tmp = x / y
    else if (x <= (-2.05d+106)) then
        tmp = 1.0d0
    else if ((x <= (-1.0d0)) .or. (.not. (x <= 420000000000.0d0))) then
        tmp = x / y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.4e+129) {
		tmp = x / y;
	} else if (x <= -2.05e+106) {
		tmp = 1.0;
	} else if ((x <= -1.0) || !(x <= 420000000000.0)) {
		tmp = x / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.4e+129:
		tmp = x / y
	elif x <= -2.05e+106:
		tmp = 1.0
	elif (x <= -1.0) or not (x <= 420000000000.0):
		tmp = x / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.4e+129)
		tmp = Float64(x / y);
	elseif (x <= -2.05e+106)
		tmp = 1.0;
	elseif ((x <= -1.0) || !(x <= 420000000000.0))
		tmp = Float64(x / y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.4e+129)
		tmp = x / y;
	elseif (x <= -2.05e+106)
		tmp = 1.0;
	elseif ((x <= -1.0) || ~((x <= 420000000000.0)))
		tmp = x / y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.4e+129], N[(x / y), $MachinePrecision], If[LessEqual[x, -2.05e+106], 1.0, If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 420000000000.0]], $MachinePrecision]], N[(x / y), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.40000000000000018e129 or -2.0500000000000001e106 < x < -1 or 4.2e11 < x

   1. Initial program 77.6%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 76.1%

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

   if -3.40000000000000018e129 < x < -2.0500000000000001e106

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 100.0%

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

      1. un-div-inv 100.0%

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

      2. clear-num 100.0%

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

      3. +-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around inf 100.0%

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

   if -1 < x < 4.2e11

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 73.4%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+129}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{+106}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1 \lor \neg \left(x \leq 420000000000\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 49.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.053:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -0.053) 1.0 (if (<= x 1.0) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.053) {
		tmp = 1.0;
	} else if (x <= 1.0) {
		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 <= (-0.053d0)) then
        tmp = 1.0d0
    else if (x <= 1.0d0) 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 <= -0.053) {
		tmp = 1.0;
	} else if (x <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.053:
		tmp = 1.0
	elif x <= 1.0:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.053)
		tmp = 1.0;
	elseif (x <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.053)
		tmp = 1.0;
	elseif (x <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.053], 1.0, If[LessEqual[x, 1.0], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0529999999999999985 or 1 < x

   1. Initial program 79.0%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 28.9%

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

      1. un-div-inv 29.0%

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

      2. clear-num 29.0%

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

      3. +-commutative 29.0%

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

   7. Applied egg-rr 29.0%

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

   8. Taylor expanded in x around inf 28.1%

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

   if -0.0529999999999999985 < x < 1

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 75.0%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.053:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return x * ((1.0 + (x / y)) / (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)) / (x + 1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.3%

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

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
4. Add Preprocessing

5. Final simplification 99.9%

   \[\leadsto x \cdot \frac{1 + \frac{x}{y}}{x + 1} \]
6. Add Preprocessing

Alternative 13: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.3%

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

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

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

   1. clear-num 99.8%

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

   2. un-div-inv 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

   \[\leadsto \frac{x}{\frac{x + 1}{1 + \frac{x}{y}}} \]
8. Add Preprocessing

Alternative 14: 14.2% 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 89.3%

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

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
4. Add Preprocessing

5. Taylor expanded in y around inf 52.8%

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

   1. un-div-inv 52.8%

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

   2. clear-num 52.7%

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

   3. +-commutative 52.7%

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

7. Applied egg-rr 52.7%

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

8. Taylor expanded in x around inf 16.0%

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

9. Final simplification 16.0%

   \[\leadsto 1 \]
10. Add Preprocessing

Developer target: 99.9% 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 2024046 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"
  :precision binary64

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

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