Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.6% → 99.9%

Time: 8.6s

Alternatives: 12

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: 88.6% 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, 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 88.0%

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

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 71.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -4.9 \cdot 10^{+101}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.32 \cdot 10^{+48}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{+26}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-66}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+121}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= x -4.9e+101)
     (/ x y)
     (if (<= x -1.32e+48)
       1.0
       (if (<= x -8.5e+26)
         (/ x y)
         (if (<= x 1.3e-66)
           t_0
           (if (<= x 3.6e-12)
             (* x (/ x y))
             (if (<= x 1.7e+121) t_0 (/ x y)))))))))

C

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -4.9e+101) {
		tmp = x / y;
	} else if (x <= -1.32e+48) {
		tmp = 1.0;
	} else if (x <= -8.5e+26) {
		tmp = x / y;
	} else if (x <= 1.3e-66) {
		tmp = t_0;
	} else if (x <= 3.6e-12) {
		tmp = x * (x / y);
	} else if (x <= 1.7e+121) {
		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 <= (-4.9d+101)) then
        tmp = x / y
    else if (x <= (-1.32d+48)) then
        tmp = 1.0d0
    else if (x <= (-8.5d+26)) then
        tmp = x / y
    else if (x <= 1.3d-66) then
        tmp = t_0
    else if (x <= 3.6d-12) then
        tmp = x * (x / y)
    else if (x <= 1.7d+121) 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 <= -4.9e+101) {
		tmp = x / y;
	} else if (x <= -1.32e+48) {
		tmp = 1.0;
	} else if (x <= -8.5e+26) {
		tmp = x / y;
	} else if (x <= 1.3e-66) {
		tmp = t_0;
	} else if (x <= 3.6e-12) {
		tmp = x * (x / y);
	} else if (x <= 1.7e+121) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + 1.0)
	tmp = 0
	if x <= -4.9e+101:
		tmp = x / y
	elif x <= -1.32e+48:
		tmp = 1.0
	elif x <= -8.5e+26:
		tmp = x / y
	elif x <= 1.3e-66:
		tmp = t_0
	elif x <= 3.6e-12:
		tmp = x * (x / y)
	elif x <= 1.7e+121:
		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 <= -4.9e+101)
		tmp = Float64(x / y);
	elseif (x <= -1.32e+48)
		tmp = 1.0;
	elseif (x <= -8.5e+26)
		tmp = Float64(x / y);
	elseif (x <= 1.3e-66)
		tmp = t_0;
	elseif (x <= 3.6e-12)
		tmp = Float64(x * Float64(x / y));
	elseif (x <= 1.7e+121)
		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 <= -4.9e+101)
		tmp = x / y;
	elseif (x <= -1.32e+48)
		tmp = 1.0;
	elseif (x <= -8.5e+26)
		tmp = x / y;
	elseif (x <= 1.3e-66)
		tmp = t_0;
	elseif (x <= 3.6e-12)
		tmp = x * (x / y);
	elseif (x <= 1.7e+121)
		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, -4.9e+101], N[(x / y), $MachinePrecision], If[LessEqual[x, -1.32e+48], 1.0, If[LessEqual[x, -8.5e+26], N[(x / y), $MachinePrecision], If[LessEqual[x, 1.3e-66], t$95$0, If[LessEqual[x, 3.6e-12], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.7e+121], t$95$0, N[(x / y), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.89999999999999983e101 or -1.32e48 < x < -8.5e26 or 1.70000000000000005e121 < x

   1. Initial program 69.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 87.2%

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

   if -4.89999999999999983e101 < x < -1.32e48

   1. Initial program 100.0%

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

      1. div-inv 99.8%

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

      2. *-commutative 99.8%

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

      3. fma-def 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in y around inf 79.1%

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

      1. +-commutative 79.1%

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

   6. Simplified 79.1%

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

   7. Taylor expanded in x around inf 79.1%

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

   if -8.5e26 < x < 1.2999999999999999e-66 or 3.6e-12 < x < 1.70000000000000005e121

   1. Initial program 98.1%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around inf 73.4%

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

   if 1.2999999999999999e-66 < x < 3.6e-12

   1. Initial program 99.4%

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

      1. associate-/l* 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 78.7%

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

   5. Taylor expanded in x around 0 78.4%

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

      1. associate-/r/ 78.2%

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

   7. Applied egg-rr 78.2%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{+101}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.32 \cdot 10^{+48}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{+26}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-66}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+121}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 3: 71.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -5 \cdot 10^{+101}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{+47}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -3 \cdot 10^{+28}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-66}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+121}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= x -5e+101)
     (/ x y)
     (if (<= x -1.2e+47)
       1.0
       (if (<= x -3e+28)
         (/ x y)
         (if (<= x 1.3e-66)
           t_0
           (if (<= x 9.5e-16)
             (/ x (/ y x))
             (if (<= x 1.65e+121) t_0 (/ x y)))))))))

C

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -5e+101) {
		tmp = x / y;
	} else if (x <= -1.2e+47) {
		tmp = 1.0;
	} else if (x <= -3e+28) {
		tmp = x / y;
	} else if (x <= 1.3e-66) {
		tmp = t_0;
	} else if (x <= 9.5e-16) {
		tmp = x / (y / x);
	} else if (x <= 1.65e+121) {
		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 <= (-5d+101)) then
        tmp = x / y
    else if (x <= (-1.2d+47)) then
        tmp = 1.0d0
    else if (x <= (-3d+28)) then
        tmp = x / y
    else if (x <= 1.3d-66) then
        tmp = t_0
    else if (x <= 9.5d-16) then
        tmp = x / (y / x)
    else if (x <= 1.65d+121) 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 <= -5e+101) {
		tmp = x / y;
	} else if (x <= -1.2e+47) {
		tmp = 1.0;
	} else if (x <= -3e+28) {
		tmp = x / y;
	} else if (x <= 1.3e-66) {
		tmp = t_0;
	} else if (x <= 9.5e-16) {
		tmp = x / (y / x);
	} else if (x <= 1.65e+121) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + 1.0)
	tmp = 0
	if x <= -5e+101:
		tmp = x / y
	elif x <= -1.2e+47:
		tmp = 1.0
	elif x <= -3e+28:
		tmp = x / y
	elif x <= 1.3e-66:
		tmp = t_0
	elif x <= 9.5e-16:
		tmp = x / (y / x)
	elif x <= 1.65e+121:
		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 <= -5e+101)
		tmp = Float64(x / y);
	elseif (x <= -1.2e+47)
		tmp = 1.0;
	elseif (x <= -3e+28)
		tmp = Float64(x / y);
	elseif (x <= 1.3e-66)
		tmp = t_0;
	elseif (x <= 9.5e-16)
		tmp = Float64(x / Float64(y / x));
	elseif (x <= 1.65e+121)
		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 <= -5e+101)
		tmp = x / y;
	elseif (x <= -1.2e+47)
		tmp = 1.0;
	elseif (x <= -3e+28)
		tmp = x / y;
	elseif (x <= 1.3e-66)
		tmp = t_0;
	elseif (x <= 9.5e-16)
		tmp = x / (y / x);
	elseif (x <= 1.65e+121)
		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, -5e+101], N[(x / y), $MachinePrecision], If[LessEqual[x, -1.2e+47], 1.0, If[LessEqual[x, -3e+28], N[(x / y), $MachinePrecision], If[LessEqual[x, 1.3e-66], t$95$0, If[LessEqual[x, 9.5e-16], N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.65e+121], t$95$0, N[(x / y), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.99999999999999989e101 or -1.20000000000000009e47 < x < -3.0000000000000001e28 or 1.6499999999999999e121 < x

   1. Initial program 69.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 87.2%

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

   if -4.99999999999999989e101 < x < -1.20000000000000009e47

   1. Initial program 100.0%

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

      1. div-inv 99.8%

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

      2. *-commutative 99.8%

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

      3. fma-def 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in y around inf 79.1%

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

      1. +-commutative 79.1%

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

   6. Simplified 79.1%

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

   7. Taylor expanded in x around inf 79.1%

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

   if -3.0000000000000001e28 < x < 1.2999999999999999e-66 or 9.5000000000000005e-16 < x < 1.6499999999999999e121

   1. Initial program 98.1%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around inf 73.4%

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

   if 1.2999999999999999e-66 < x < 9.5000000000000005e-16

   1. Initial program 99.4%

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

      1. associate-/l* 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 78.7%

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

   5. Taylor expanded in x around 0 78.4%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+101}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{+47}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -3 \cdot 10^{+28}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-66}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+121}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4: 71.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{+101}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{+47}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -3.6 \cdot 10^{+28}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-66}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-16}:\\ \;\;\;\;\frac{x \cdot x}{y}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+121}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= x -4.8e+101)
     (/ x y)
     (if (<= x -1.25e+47)
       1.0
       (if (<= x -3.6e+28)
         (/ x y)
         (if (<= x 1.3e-66)
           t_0
           (if (<= x 7e-16)
             (/ (* x x) y)
             (if (<= x 6.5e+121) t_0 (/ x y)))))))))

C

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -4.8e+101) {
		tmp = x / y;
	} else if (x <= -1.25e+47) {
		tmp = 1.0;
	} else if (x <= -3.6e+28) {
		tmp = x / y;
	} else if (x <= 1.3e-66) {
		tmp = t_0;
	} else if (x <= 7e-16) {
		tmp = (x * x) / y;
	} else if (x <= 6.5e+121) {
		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 <= (-4.8d+101)) then
        tmp = x / y
    else if (x <= (-1.25d+47)) then
        tmp = 1.0d0
    else if (x <= (-3.6d+28)) then
        tmp = x / y
    else if (x <= 1.3d-66) then
        tmp = t_0
    else if (x <= 7d-16) then
        tmp = (x * x) / y
    else if (x <= 6.5d+121) 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 <= -4.8e+101) {
		tmp = x / y;
	} else if (x <= -1.25e+47) {
		tmp = 1.0;
	} else if (x <= -3.6e+28) {
		tmp = x / y;
	} else if (x <= 1.3e-66) {
		tmp = t_0;
	} else if (x <= 7e-16) {
		tmp = (x * x) / y;
	} else if (x <= 6.5e+121) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + 1.0)
	tmp = 0
	if x <= -4.8e+101:
		tmp = x / y
	elif x <= -1.25e+47:
		tmp = 1.0
	elif x <= -3.6e+28:
		tmp = x / y
	elif x <= 1.3e-66:
		tmp = t_0
	elif x <= 7e-16:
		tmp = (x * x) / y
	elif x <= 6.5e+121:
		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 <= -4.8e+101)
		tmp = Float64(x / y);
	elseif (x <= -1.25e+47)
		tmp = 1.0;
	elseif (x <= -3.6e+28)
		tmp = Float64(x / y);
	elseif (x <= 1.3e-66)
		tmp = t_0;
	elseif (x <= 7e-16)
		tmp = Float64(Float64(x * x) / y);
	elseif (x <= 6.5e+121)
		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 <= -4.8e+101)
		tmp = x / y;
	elseif (x <= -1.25e+47)
		tmp = 1.0;
	elseif (x <= -3.6e+28)
		tmp = x / y;
	elseif (x <= 1.3e-66)
		tmp = t_0;
	elseif (x <= 7e-16)
		tmp = (x * x) / y;
	elseif (x <= 6.5e+121)
		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, -4.8e+101], N[(x / y), $MachinePrecision], If[LessEqual[x, -1.25e+47], 1.0, If[LessEqual[x, -3.6e+28], N[(x / y), $MachinePrecision], If[LessEqual[x, 1.3e-66], t$95$0, If[LessEqual[x, 7e-16], N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 6.5e+121], t$95$0, N[(x / y), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.79999999999999977e101 or -1.25000000000000005e47 < x < -3.5999999999999999e28 or 6.50000000000000019e121 < x

   1. Initial program 69.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 87.2%

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

   if -4.79999999999999977e101 < x < -1.25000000000000005e47

   1. Initial program 100.0%

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

      1. div-inv 99.8%

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

      2. *-commutative 99.8%

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

      3. fma-def 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in y around inf 79.1%

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

      1. +-commutative 79.1%

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

   6. Simplified 79.1%

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

   7. Taylor expanded in x around inf 79.1%

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

   if -3.5999999999999999e28 < x < 1.2999999999999999e-66 or 7.00000000000000035e-16 < x < 6.50000000000000019e121

   1. Initial program 98.1%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around inf 73.4%

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

   if 1.2999999999999999e-66 < x < 7.00000000000000035e-16

   1. Initial program 99.4%

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

      1. associate-/l* 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 78.7%

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

   5. Taylor expanded in x around 0 78.7%

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

   6. Taylor expanded in x around 0 78.7%

      \[\leadsto \color{blue}{\frac{{x}^{2}}{y}} \]
   7. Step-by-step derivation

      1. unpow2 78.7%

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

   8. Simplified 78.7%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+101}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{+47}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -3.6 \cdot 10^{+28}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-66}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-16}:\\ \;\;\;\;\frac{x \cdot x}{y}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+121}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 5: 71.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -4.9 \cdot 10^{+101}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -8.4 \cdot 10^{+47}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -8 \cdot 10^{+26}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-66}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{y} \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+121}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= x -4.9e+101)
     (/ x y)
     (if (<= x -8.4e+47)
       1.0
       (if (<= x -8e+26)
         (/ x y)
         (if (<= x 1.25e-66)
           t_0
           (if (<= x 1.7e-13)
             (* (/ 1.0 y) (* x x))
             (if (<= x 1.65e+121) t_0 (/ x y)))))))))

C

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -4.9e+101) {
		tmp = x / y;
	} else if (x <= -8.4e+47) {
		tmp = 1.0;
	} else if (x <= -8e+26) {
		tmp = x / y;
	} else if (x <= 1.25e-66) {
		tmp = t_0;
	} else if (x <= 1.7e-13) {
		tmp = (1.0 / y) * (x * x);
	} else if (x <= 1.65e+121) {
		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 <= (-4.9d+101)) then
        tmp = x / y
    else if (x <= (-8.4d+47)) then
        tmp = 1.0d0
    else if (x <= (-8d+26)) then
        tmp = x / y
    else if (x <= 1.25d-66) then
        tmp = t_0
    else if (x <= 1.7d-13) then
        tmp = (1.0d0 / y) * (x * x)
    else if (x <= 1.65d+121) 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 <= -4.9e+101) {
		tmp = x / y;
	} else if (x <= -8.4e+47) {
		tmp = 1.0;
	} else if (x <= -8e+26) {
		tmp = x / y;
	} else if (x <= 1.25e-66) {
		tmp = t_0;
	} else if (x <= 1.7e-13) {
		tmp = (1.0 / y) * (x * x);
	} else if (x <= 1.65e+121) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + 1.0)
	tmp = 0
	if x <= -4.9e+101:
		tmp = x / y
	elif x <= -8.4e+47:
		tmp = 1.0
	elif x <= -8e+26:
		tmp = x / y
	elif x <= 1.25e-66:
		tmp = t_0
	elif x <= 1.7e-13:
		tmp = (1.0 / y) * (x * x)
	elif x <= 1.65e+121:
		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 <= -4.9e+101)
		tmp = Float64(x / y);
	elseif (x <= -8.4e+47)
		tmp = 1.0;
	elseif (x <= -8e+26)
		tmp = Float64(x / y);
	elseif (x <= 1.25e-66)
		tmp = t_0;
	elseif (x <= 1.7e-13)
		tmp = Float64(Float64(1.0 / y) * Float64(x * x));
	elseif (x <= 1.65e+121)
		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 <= -4.9e+101)
		tmp = x / y;
	elseif (x <= -8.4e+47)
		tmp = 1.0;
	elseif (x <= -8e+26)
		tmp = x / y;
	elseif (x <= 1.25e-66)
		tmp = t_0;
	elseif (x <= 1.7e-13)
		tmp = (1.0 / y) * (x * x);
	elseif (x <= 1.65e+121)
		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, -4.9e+101], N[(x / y), $MachinePrecision], If[LessEqual[x, -8.4e+47], 1.0, If[LessEqual[x, -8e+26], N[(x / y), $MachinePrecision], If[LessEqual[x, 1.25e-66], t$95$0, If[LessEqual[x, 1.7e-13], N[(N[(1.0 / y), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.65e+121], t$95$0, N[(x / y), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.89999999999999983e101 or -8.4e47 < x < -8.00000000000000038e26 or 1.6499999999999999e121 < x

   1. Initial program 69.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 87.2%

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

   if -4.89999999999999983e101 < x < -8.4e47

   1. Initial program 100.0%

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

      1. div-inv 99.8%

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

      2. *-commutative 99.8%

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

      3. fma-def 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in y around inf 79.1%

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

      1. +-commutative 79.1%

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

   6. Simplified 79.1%

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

   7. Taylor expanded in x around inf 79.1%

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

   if -8.00000000000000038e26 < x < 1.2499999999999999e-66 or 1.70000000000000008e-13 < x < 1.6499999999999999e121

   1. Initial program 98.1%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around inf 73.4%

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

   if 1.2499999999999999e-66 < x < 1.70000000000000008e-13

   1. Initial program 99.4%

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

      1. associate-/l* 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 78.7%

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

   5. Taylor expanded in x around 0 78.4%

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

      1. associate-/l* 78.7%

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

      2. div-inv 78.7%

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

      3. *-commutative 78.7%

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

   7. Applied egg-rr 78.7%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{+101}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -8.4 \cdot 10^{+47}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -8 \cdot 10^{+26}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-66}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{y} \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+121}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 6: 69.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+101}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{+47}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-67}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+15}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+121}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.8e+101)
   (/ x y)
   (if (<= x -1.05e+47)
     1.0
     (if (<= x -1.0)
       (/ x y)
       (if (<= x 4.5e-67)
         x
         (if (<= x 5.6e+15)
           (* x (/ x y))
           (if (<= x 1.65e+121) 1.0 (/ x y))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.8e+101) {
		tmp = x / y;
	} else if (x <= -1.05e+47) {
		tmp = 1.0;
	} else if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 4.5e-67) {
		tmp = x;
	} else if (x <= 5.6e+15) {
		tmp = x * (x / y);
	} else if (x <= 1.65e+121) {
		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 <= (-4.8d+101)) then
        tmp = x / y
    else if (x <= (-1.05d+47)) then
        tmp = 1.0d0
    else if (x <= (-1.0d0)) then
        tmp = x / y
    else if (x <= 4.5d-67) then
        tmp = x
    else if (x <= 5.6d+15) then
        tmp = x * (x / y)
    else if (x <= 1.65d+121) 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 <= -4.8e+101) {
		tmp = x / y;
	} else if (x <= -1.05e+47) {
		tmp = 1.0;
	} else if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 4.5e-67) {
		tmp = x;
	} else if (x <= 5.6e+15) {
		tmp = x * (x / y);
	} else if (x <= 1.65e+121) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.8e+101:
		tmp = x / y
	elif x <= -1.05e+47:
		tmp = 1.0
	elif x <= -1.0:
		tmp = x / y
	elif x <= 4.5e-67:
		tmp = x
	elif x <= 5.6e+15:
		tmp = x * (x / y)
	elif x <= 1.65e+121:
		tmp = 1.0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.8e+101)
		tmp = Float64(x / y);
	elseif (x <= -1.05e+47)
		tmp = 1.0;
	elseif (x <= -1.0)
		tmp = Float64(x / y);
	elseif (x <= 4.5e-67)
		tmp = x;
	elseif (x <= 5.6e+15)
		tmp = Float64(x * Float64(x / y));
	elseif (x <= 1.65e+121)
		tmp = 1.0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.8e+101)
		tmp = x / y;
	elseif (x <= -1.05e+47)
		tmp = 1.0;
	elseif (x <= -1.0)
		tmp = x / y;
	elseif (x <= 4.5e-67)
		tmp = x;
	elseif (x <= 5.6e+15)
		tmp = x * (x / y);
	elseif (x <= 1.65e+121)
		tmp = 1.0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.8e+101], N[(x / y), $MachinePrecision], If[LessEqual[x, -1.05e+47], 1.0, If[LessEqual[x, -1.0], N[(x / y), $MachinePrecision], If[LessEqual[x, 4.5e-67], x, If[LessEqual[x, 5.6e+15], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.65e+121], 1.0, N[(x / y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.79999999999999977e101 or -1.05e47 < x < -1 or 1.6499999999999999e121 < x

   1. Initial program 70.9%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 83.4%

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

   if -4.79999999999999977e101 < x < -1.05e47 or 5.6e15 < x < 1.6499999999999999e121

   1. Initial program 92.5%

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

      1. div-inv 92.5%

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

      2. *-commutative 92.5%

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

      3. fma-def 92.5%

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

   3. Applied egg-rr 92.5%

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

   4. Taylor expanded in y around inf 67.9%

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

      1. +-commutative 67.9%

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

   6. Simplified 67.9%

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

   7. Taylor expanded in x around inf 67.9%

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

   if -1 < x < 4.50000000000000015e-67

   1. Initial program 99.9%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 75.4%

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

   if 4.50000000000000015e-67 < x < 5.6e15

   1. Initial program 99.4%

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

      1. associate-/l* 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 76.1%

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

   5. Taylor expanded in x around 0 63.8%

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

      1. associate-/r/ 63.6%

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

   7. Applied egg-rr 63.6%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+101}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{+47}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-67}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+15}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+121}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 7: 70.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+101}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -4.15 \cdot 10^{+46}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 0.00125:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+121}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5e+101)
   (/ x y)
   (if (<= x -4.15e+46)
     1.0
     (if (<= x -1.0)
       (/ x y)
       (if (<= x 0.00125) x (if (<= x 3.3e+121) 1.0 (/ x y)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5e+101) {
		tmp = x / y;
	} else if (x <= -4.15e+46) {
		tmp = 1.0;
	} else if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 0.00125) {
		tmp = x;
	} else if (x <= 3.3e+121) {
		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 <= (-5d+101)) then
        tmp = x / y
    else if (x <= (-4.15d+46)) then
        tmp = 1.0d0
    else if (x <= (-1.0d0)) then
        tmp = x / y
    else if (x <= 0.00125d0) then
        tmp = x
    else if (x <= 3.3d+121) 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 <= -5e+101) {
		tmp = x / y;
	} else if (x <= -4.15e+46) {
		tmp = 1.0;
	} else if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 0.00125) {
		tmp = x;
	} else if (x <= 3.3e+121) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5e+101:
		tmp = x / y
	elif x <= -4.15e+46:
		tmp = 1.0
	elif x <= -1.0:
		tmp = x / y
	elif x <= 0.00125:
		tmp = x
	elif x <= 3.3e+121:
		tmp = 1.0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5e+101)
		tmp = Float64(x / y);
	elseif (x <= -4.15e+46)
		tmp = 1.0;
	elseif (x <= -1.0)
		tmp = Float64(x / y);
	elseif (x <= 0.00125)
		tmp = x;
	elseif (x <= 3.3e+121)
		tmp = 1.0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5e+101)
		tmp = x / y;
	elseif (x <= -4.15e+46)
		tmp = 1.0;
	elseif (x <= -1.0)
		tmp = x / y;
	elseif (x <= 0.00125)
		tmp = x;
	elseif (x <= 3.3e+121)
		tmp = 1.0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5e+101], N[(x / y), $MachinePrecision], If[LessEqual[x, -4.15e+46], 1.0, If[LessEqual[x, -1.0], N[(x / y), $MachinePrecision], If[LessEqual[x, 0.00125], x, If[LessEqual[x, 3.3e+121], 1.0, N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.99999999999999989e101 or -4.14999999999999976e46 < x < -1 or 3.29999999999999979e121 < x

   1. Initial program 70.9%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 83.4%

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

   if -4.99999999999999989e101 < x < -4.14999999999999976e46 or 0.00125000000000000003 < x < 3.29999999999999979e121

   1. Initial program 92.9%

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

      1. div-inv 92.8%

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

      2. *-commutative 92.8%

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

      3. fma-def 92.8%

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

   3. Applied egg-rr 92.8%

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

   4. Taylor expanded in y around inf 64.6%

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

      1. +-commutative 64.6%

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

   6. Simplified 64.6%

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

   7. Taylor expanded in x around inf 64.6%

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

   if -1 < x < 0.00125000000000000003

   1. Initial program 99.9%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 70.8%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+101}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -4.15 \cdot 10^{+46}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 0.00125:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+121}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 8: 85.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + -1}{y} + 1\\ \mathbf{if}\;x \leq -15200:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-69}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{y} \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ (+ x -1.0) y) 1.0)))
   (if (<= x -15200.0)
     t_0
     (if (<= x 4e-69)
       (/ x (+ x 1.0))
       (if (<= x 1.0) (* (/ 1.0 y) (* x x)) t_0)))))

C

double code(double x, double y) {
	double t_0 = ((x + -1.0) / y) + 1.0;
	double tmp;
	if (x <= -15200.0) {
		tmp = t_0;
	} else if (x <= 4e-69) {
		tmp = x / (x + 1.0);
	} else if (x <= 1.0) {
		tmp = (1.0 / y) * (x * x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + (-1.0d0)) / y) + 1.0d0
    if (x <= (-15200.0d0)) then
        tmp = t_0
    else if (x <= 4d-69) then
        tmp = x / (x + 1.0d0)
    else if (x <= 1.0d0) then
        tmp = (1.0d0 / y) * (x * x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((x + -1.0) / y) + 1.0;
	double tmp;
	if (x <= -15200.0) {
		tmp = t_0;
	} else if (x <= 4e-69) {
		tmp = x / (x + 1.0);
	} else if (x <= 1.0) {
		tmp = (1.0 / y) * (x * x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((x + -1.0) / y) + 1.0
	tmp = 0
	if x <= -15200.0:
		tmp = t_0
	elif x <= 4e-69:
		tmp = x / (x + 1.0)
	elif x <= 1.0:
		tmp = (1.0 / y) * (x * x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x + -1.0) / y) + 1.0)
	tmp = 0.0
	if (x <= -15200.0)
		tmp = t_0;
	elseif (x <= 4e-69)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (x <= 1.0)
		tmp = Float64(Float64(1.0 / y) * Float64(x * x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((x + -1.0) / y) + 1.0;
	tmp = 0.0;
	if (x <= -15200.0)
		tmp = t_0;
	elseif (x <= 4e-69)
		tmp = x / (x + 1.0);
	elseif (x <= 1.0)
		tmp = (1.0 / y) * (x * x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -15200.0], t$95$0, If[LessEqual[x, 4e-69], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(N[(1.0 / y), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -15200 or 1 < x

   1. Initial program 77.0%

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

      1. div-inv 77.0%

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

      2. *-commutative 77.0%

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

      3. fma-def 76.9%

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

   3. Applied egg-rr 76.9%

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

   4. Taylor expanded in x around inf 99.9%

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

      1. associate--l+ 99.9%

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

      2. div-sub 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

   6. Simplified 99.9%

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

   if -15200 < x < 3.9999999999999999e-69

   1. Initial program 99.9%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around inf 77.0%

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

   if 3.9999999999999999e-69 < x < 1

   1. Initial program 99.3%

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

      1. associate-/l* 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 74.1%

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

   5. Taylor expanded in x around 0 68.3%

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

      1. associate-/l* 68.5%

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

      2. div-inv 68.5%

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

      3. *-commutative 68.5%

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

   7. Applied egg-rr 68.5%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -15200:\\ \;\;\;\;\frac{x + -1}{y} + 1\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-69}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{y} \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + -1}{y} + 1\\ \end{array} \]

Alternative 9: 85.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + -1}{y} + 1\\ \mathbf{if}\;x \leq -1650:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{-66}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 1100:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ (+ x -1.0) y) 1.0)))
   (if (<= x -1650.0)
     t_0
     (if (<= x 1.02e-66)
       (/ x (+ x 1.0))
       (if (<= x 1100.0) (/ x (+ y (/ y x))) t_0)))))

C

double code(double x, double y) {
	double t_0 = ((x + -1.0) / y) + 1.0;
	double tmp;
	if (x <= -1650.0) {
		tmp = t_0;
	} else if (x <= 1.02e-66) {
		tmp = x / (x + 1.0);
	} else if (x <= 1100.0) {
		tmp = x / (y + (y / x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + (-1.0d0)) / y) + 1.0d0
    if (x <= (-1650.0d0)) then
        tmp = t_0
    else if (x <= 1.02d-66) then
        tmp = x / (x + 1.0d0)
    else if (x <= 1100.0d0) then
        tmp = x / (y + (y / x))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((x + -1.0) / y) + 1.0;
	double tmp;
	if (x <= -1650.0) {
		tmp = t_0;
	} else if (x <= 1.02e-66) {
		tmp = x / (x + 1.0);
	} else if (x <= 1100.0) {
		tmp = x / (y + (y / x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((x + -1.0) / y) + 1.0
	tmp = 0
	if x <= -1650.0:
		tmp = t_0
	elif x <= 1.02e-66:
		tmp = x / (x + 1.0)
	elif x <= 1100.0:
		tmp = x / (y + (y / x))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x + -1.0) / y) + 1.0)
	tmp = 0.0
	if (x <= -1650.0)
		tmp = t_0;
	elseif (x <= 1.02e-66)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (x <= 1100.0)
		tmp = Float64(x / Float64(y + Float64(y / x)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((x + -1.0) / y) + 1.0;
	tmp = 0.0;
	if (x <= -1650.0)
		tmp = t_0;
	elseif (x <= 1.02e-66)
		tmp = x / (x + 1.0);
	elseif (x <= 1100.0)
		tmp = x / (y + (y / x));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -1650.0], t$95$0, If[LessEqual[x, 1.02e-66], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1100.0], N[(x / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1650 or 1100 < x

   1. Initial program 77.0%

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

      1. div-inv 77.0%

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

      2. *-commutative 77.0%

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

      3. fma-def 76.9%

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

   3. Applied egg-rr 76.9%

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

   4. Taylor expanded in x around inf 99.9%

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

      1. associate--l+ 99.9%

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

      2. div-sub 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

   6. Simplified 99.9%

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

   if -1650 < x < 1.01999999999999996e-66

   1. Initial program 99.9%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around inf 77.0%

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

   if 1.01999999999999996e-66 < x < 1100

   1. Initial program 99.3%

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

      1. associate-/l* 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 74.1%

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

   5. Taylor expanded in x around 0 74.1%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1650:\\ \;\;\;\;\frac{x + -1}{y} + 1\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{-66}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 1100:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + -1}{y} + 1\\ \end{array} \]

Alternative 10: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.2)) {
		tmp = ((x + -1.0) / y) + 1.0;
	} 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 = ((x + (-1.0d0)) / y) + 1.0d0
    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 = ((x + -1.0) / y) + 1.0;
	} 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 = ((x + -1.0) / y) + 1.0
	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(Float64(Float64(x + -1.0) / y) + 1.0);
	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 = ((x + -1.0) / y) + 1.0;
	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[(N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision] + 1.0), $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 77.0%

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

      1. div-inv 77.0%

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

      2. *-commutative 77.0%

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

      3. fma-def 76.9%

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

   3. Applied egg-rr 76.9%

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

   4. Taylor expanded in x around inf 99.9%

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

      1. associate--l+ 99.9%

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

      2. div-sub 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

   6. Simplified 99.9%

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

   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. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

      1. associate-/l* 99.9%

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

      2. frac-2neg 99.9%

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

      3. div-inv 99.8%

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

      4. *-commutative 99.8%

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

      5. neg-sub0 99.8%

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

      6. +-commutative 99.8%

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

      7. associate--r+ 99.8%

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

      8. metadata-eval 99.8%

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

      9. distribute-rgt-neg-in 99.8%

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

      10. neg-sub0 99.8%

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

      11. +-commutative 99.8%

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

      12. associate--r+ 99.8%

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

      13. metadata-eval 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around 0 97.4%

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

4. Final simplification 98.7%

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

Alternative 11: 49.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -100000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.00125:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -100000000000.0) 1.0 (if (<= x 0.00125) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -100000000000.0) {
		tmp = 1.0;
	} else if (x <= 0.00125) {
		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 <= (-100000000000.0d0)) then
        tmp = 1.0d0
    else if (x <= 0.00125d0) 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 <= -100000000000.0) {
		tmp = 1.0;
	} else if (x <= 0.00125) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -100000000000.0:
		tmp = 1.0
	elif x <= 0.00125:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -100000000000.0)
		tmp = 1.0;
	elseif (x <= 0.00125)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -100000000000.0)
		tmp = 1.0;
	elseif (x <= 0.00125)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -100000000000.0], 1.0, If[LessEqual[x, 0.00125], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1e11 or 0.00125000000000000003 < x

   1. Initial program 76.8%

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

      1. div-inv 76.8%

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

      2. *-commutative 76.8%

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

      3. fma-def 76.8%

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

   3. Applied egg-rr 76.8%

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

   4. Taylor expanded in y around inf 30.2%

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

      1. +-commutative 30.2%

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

   6. Simplified 30.2%

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

   7. Taylor expanded in x around inf 30.1%

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

   if -1e11 < x < 0.00125000000000000003

   1. Initial program 99.8%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 69.8%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -100000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.00125:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12: 14.5% 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 88.0%

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

   1. div-inv 87.9%

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

   2. *-commutative 87.9%

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

   3. fma-def 87.9%

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

3. Applied egg-rr 87.9%

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

4. Taylor expanded in y around inf 50.2%

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

   1. +-commutative 50.2%

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

6. Simplified 50.2%

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

7. Taylor expanded in x around inf 17.2%

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

8. Final simplification 17.2%

   \[\leadsto 1 \]

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 2023297 
(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)))