Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.3% → 99.8%

Time: 8.6s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

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(x * Float64(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[(x * N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.0%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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. Final simplification 99.8%

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

Alternative 2: 72.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{x}{y}\\ \mathbf{if}\;x \leq -3 \cdot 10^{+22}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -9.8 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 10^{-82}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (/ x y))))
   (if (<= x -3e+22)
     (/ x y)
     (if (<= x -9.8e-17)
       (/ x (+ x 1.0))
       (if (<= x -2e-52)
         t_0
         (if (<= x 1e-82) x (if (<= x 1.0) t_0 (/ x y))))))))

C

double code(double x, double y) {
	double t_0 = x * (x / y);
	double tmp;
	if (x <= -3e+22) {
		tmp = x / y;
	} else if (x <= -9.8e-17) {
		tmp = x / (x + 1.0);
	} else if (x <= -2e-52) {
		tmp = t_0;
	} else if (x <= 1e-82) {
		tmp = x;
	} else if (x <= 1.0) {
		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 / y)
    if (x <= (-3d+22)) then
        tmp = x / y
    else if (x <= (-9.8d-17)) then
        tmp = x / (x + 1.0d0)
    else if (x <= (-2d-52)) then
        tmp = t_0
    else if (x <= 1d-82) then
        tmp = x
    else if (x <= 1.0d0) 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 / y);
	double tmp;
	if (x <= -3e+22) {
		tmp = x / y;
	} else if (x <= -9.8e-17) {
		tmp = x / (x + 1.0);
	} else if (x <= -2e-52) {
		tmp = t_0;
	} else if (x <= 1e-82) {
		tmp = x;
	} else if (x <= 1.0) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (x / y)
	tmp = 0
	if x <= -3e+22:
		tmp = x / y
	elif x <= -9.8e-17:
		tmp = x / (x + 1.0)
	elif x <= -2e-52:
		tmp = t_0
	elif x <= 1e-82:
		tmp = x
	elif x <= 1.0:
		tmp = t_0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(x / y))
	tmp = 0.0
	if (x <= -3e+22)
		tmp = Float64(x / y);
	elseif (x <= -9.8e-17)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (x <= -2e-52)
		tmp = t_0;
	elseif (x <= 1e-82)
		tmp = x;
	elseif (x <= 1.0)
		tmp = t_0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (x / y);
	tmp = 0.0;
	if (x <= -3e+22)
		tmp = x / y;
	elseif (x <= -9.8e-17)
		tmp = x / (x + 1.0);
	elseif (x <= -2e-52)
		tmp = t_0;
	elseif (x <= 1e-82)
		tmp = x;
	elseif (x <= 1.0)
		tmp = t_0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3e+22], N[(x / y), $MachinePrecision], If[LessEqual[x, -9.8e-17], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2e-52], t$95$0, If[LessEqual[x, 1e-82], x, If[LessEqual[x, 1.0], t$95$0, N[(x / y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3e22 or 1 < x

   1. Initial program 76.4%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 82.0%

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

   if -3e22 < x < -9.80000000000000024e-17

   1. Initial program 100.0%

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

      1. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf 73.5%

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

   if -9.80000000000000024e-17 < x < -2e-52 or 1e-82 < x < 1

   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 77.6%

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

   6. Taylor expanded in x around 0 77.5%

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

   if -2e-52 < x < 1e-82

   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 x around 0 86.9%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+22}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -9.8 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-52}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 10^{-82}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+19}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -9.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-52}:\\ \;\;\;\;x \cdot \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-89}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5e+19)
   (/ x y)
   (if (<= x -9.4e-17)
     (/ x (+ x 1.0))
     (if (<= x -3e-52)
       (* x (- (/ x y) x))
       (if (<= x 3.9e-89) x (if (<= x 1.0) (* x (/ x y)) (/ x y)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5e+19) {
		tmp = x / y;
	} else if (x <= -9.4e-17) {
		tmp = x / (x + 1.0);
	} else if (x <= -3e-52) {
		tmp = x * ((x / y) - x);
	} else if (x <= 3.9e-89) {
		tmp = x;
	} else if (x <= 1.0) {
		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 <= (-5d+19)) then
        tmp = x / y
    else if (x <= (-9.4d-17)) then
        tmp = x / (x + 1.0d0)
    else if (x <= (-3d-52)) then
        tmp = x * ((x / y) - x)
    else if (x <= 3.9d-89) then
        tmp = x
    else if (x <= 1.0d0) 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 <= -5e+19) {
		tmp = x / y;
	} else if (x <= -9.4e-17) {
		tmp = x / (x + 1.0);
	} else if (x <= -3e-52) {
		tmp = x * ((x / y) - x);
	} else if (x <= 3.9e-89) {
		tmp = x;
	} else if (x <= 1.0) {
		tmp = x * (x / y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5e+19:
		tmp = x / y
	elif x <= -9.4e-17:
		tmp = x / (x + 1.0)
	elif x <= -3e-52:
		tmp = x * ((x / y) - x)
	elif x <= 3.9e-89:
		tmp = x
	elif x <= 1.0:
		tmp = x * (x / y)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5e+19)
		tmp = Float64(x / y);
	elseif (x <= -9.4e-17)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (x <= -3e-52)
		tmp = Float64(x * Float64(Float64(x / y) - x));
	elseif (x <= 3.9e-89)
		tmp = x;
	elseif (x <= 1.0)
		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 <= -5e+19)
		tmp = x / y;
	elseif (x <= -9.4e-17)
		tmp = x / (x + 1.0);
	elseif (x <= -3e-52)
		tmp = x * ((x / y) - x);
	elseif (x <= 3.9e-89)
		tmp = x;
	elseif (x <= 1.0)
		tmp = x * (x / y);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5e+19], N[(x / y), $MachinePrecision], If[LessEqual[x, -9.4e-17], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3e-52], N[(x * N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.9e-89], x, If[LessEqual[x, 1.0], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -5e19 or 1 < x

   1. Initial program 76.4%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 82.0%

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

   if -5e19 < x < -9.3999999999999999e-17

   1. Initial program 100.0%

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

      1. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf 73.5%

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

   if -9.3999999999999999e-17 < x < -3e-52

   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%

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

   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 0 99.4%

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

   6. Taylor expanded in x around inf 74.8%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{y} - 1\right)} \]
   7. Step-by-step derivation

      1. unpow2 74.8%

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

      2. associate-*r* 74.8%

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

      3. sub-neg 74.8%

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

      4. metadata-eval 74.8%

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

      5. distribute-lft-in 74.8%

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

      6. associate-*r/ 74.9%

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

      7. *-rgt-identity 74.9%

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

      8. remove-double-neg 74.9%

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

      9. remove-double-neg 74.9%

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

      10. *-commutative 74.9%

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

      11. neg-mul-1 74.9%

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

      12. unsub-neg 74.9%

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

   8. Simplified 74.9%

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

   if -3e-52 < x < 3.89999999999999978e-89

   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 x around 0 86.9%

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

   if 3.89999999999999978e-89 < x < 1

   1. Initial program 99.5%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 79.1%

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

   6. Taylor expanded in x around 0 79.0%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+19}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -9.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-52}:\\ \;\;\;\;x \cdot \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-89}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+19}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-16}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-52} \lor \neg \left(x \leq 2.8 \cdot 10^{-82}\right):\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.2e+19)
   (/ x y)
   (if (<= x -1.25e-16)
     (/ x (+ x 1.0))
     (if (or (<= x -6e-52) (not (<= x 2.8e-82))) (/ x (+ y (/ y x))) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.2e+19) {
		tmp = x / y;
	} else if (x <= -1.25e-16) {
		tmp = x / (x + 1.0);
	} else if ((x <= -6e-52) || !(x <= 2.8e-82)) {
		tmp = x / (y + (y / x));
	} 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 <= (-5.2d+19)) then
        tmp = x / y
    else if (x <= (-1.25d-16)) then
        tmp = x / (x + 1.0d0)
    else if ((x <= (-6d-52)) .or. (.not. (x <= 2.8d-82))) then
        tmp = x / (y + (y / x))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5.2e+19) {
		tmp = x / y;
	} else if (x <= -1.25e-16) {
		tmp = x / (x + 1.0);
	} else if ((x <= -6e-52) || !(x <= 2.8e-82)) {
		tmp = x / (y + (y / x));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.2e+19:
		tmp = x / y
	elif x <= -1.25e-16:
		tmp = x / (x + 1.0)
	elif (x <= -6e-52) or not (x <= 2.8e-82):
		tmp = x / (y + (y / x))
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.2e+19)
		tmp = Float64(x / y);
	elseif (x <= -1.25e-16)
		tmp = Float64(x / Float64(x + 1.0));
	elseif ((x <= -6e-52) || !(x <= 2.8e-82))
		tmp = Float64(x / Float64(y + Float64(y / x)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.2e+19)
		tmp = x / y;
	elseif (x <= -1.25e-16)
		tmp = x / (x + 1.0);
	elseif ((x <= -6e-52) || ~((x <= 2.8e-82)))
		tmp = x / (y + (y / x));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.2e+19], N[(x / y), $MachinePrecision], If[LessEqual[x, -1.25e-16], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, -6e-52], N[Not[LessEqual[x, 2.8e-82]], $MachinePrecision]], N[(x / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.2e19

   1. Initial program 74.4%

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

      1. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 86.3%

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

   if -5.2e19 < x < -1.2500000000000001e-16

   1. Initial program 100.0%

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

      1. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf 73.5%

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

   if -1.2500000000000001e-16 < x < -6e-52 or 2.80000000000000024e-82 < x

   1. Initial program 84.8%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 68.8%

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

      1. clear-num 68.8%

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

      2. un-div-inv 69.0%

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

      3. *-commutative 69.0%

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

      4. associate-/l* 78.4%

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

      5. +-commutative 78.4%

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

   7. Applied egg-rr 78.4%

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

   8. Taylor expanded in x around inf 78.4%

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

   if -6e-52 < x < 2.80000000000000024e-82

   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 x around 0 86.9%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+19}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-16}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-52} \lor \neg \left(x \leq 2.8 \cdot 10^{-82}\right):\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{x}{y}\\ \mathbf{if}\;x \leq -75000000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.36 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-89}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (/ x y))))
   (if (<= x -75000000000.0)
     (/ x y)
     (if (<= x -1.36e-52)
       t_0
       (if (<= x 7.5e-89) x (if (<= x 1.0) t_0 (/ x y)))))))

C

double code(double x, double y) {
	double t_0 = x * (x / y);
	double tmp;
	if (x <= -75000000000.0) {
		tmp = x / y;
	} else if (x <= -1.36e-52) {
		tmp = t_0;
	} else if (x <= 7.5e-89) {
		tmp = x;
	} else if (x <= 1.0) {
		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 / y)
    if (x <= (-75000000000.0d0)) then
        tmp = x / y
    else if (x <= (-1.36d-52)) then
        tmp = t_0
    else if (x <= 7.5d-89) then
        tmp = x
    else if (x <= 1.0d0) 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 / y);
	double tmp;
	if (x <= -75000000000.0) {
		tmp = x / y;
	} else if (x <= -1.36e-52) {
		tmp = t_0;
	} else if (x <= 7.5e-89) {
		tmp = x;
	} else if (x <= 1.0) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (x / y)
	tmp = 0
	if x <= -75000000000.0:
		tmp = x / y
	elif x <= -1.36e-52:
		tmp = t_0
	elif x <= 7.5e-89:
		tmp = x
	elif x <= 1.0:
		tmp = t_0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(x / y))
	tmp = 0.0
	if (x <= -75000000000.0)
		tmp = Float64(x / y);
	elseif (x <= -1.36e-52)
		tmp = t_0;
	elseif (x <= 7.5e-89)
		tmp = x;
	elseif (x <= 1.0)
		tmp = t_0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (x / y);
	tmp = 0.0;
	if (x <= -75000000000.0)
		tmp = x / y;
	elseif (x <= -1.36e-52)
		tmp = t_0;
	elseif (x <= 7.5e-89)
		tmp = x;
	elseif (x <= 1.0)
		tmp = t_0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -75000000000.0], N[(x / y), $MachinePrecision], If[LessEqual[x, -1.36e-52], t$95$0, If[LessEqual[x, 7.5e-89], x, If[LessEqual[x, 1.0], t$95$0, N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.5e10 or 1 < x

   1. Initial program 76.6%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. Taylor expanded in x around inf 81.3%

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

   if -7.5e10 < x < -1.36e-52 or 7.4999999999999999e-89 < x < 1

   1. Initial program 99.7%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 65.9%

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

   6. Taylor expanded in x around 0 61.2%

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

   if -1.36e-52 < x < 7.4999999999999999e-89

   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 x around 0 86.9%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -75000000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.36 \cdot 10^{-52}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-89}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{y}\\ \mathbf{if}\;x \leq -4.4 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-82}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 66000000000000:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x y) y)))
   (if (<= x -4.4e+19)
     t_0
     (if (<= x 3.4e-82)
       (/ x (+ x 1.0))
       (if (<= x 66000000000000.0) (/ x (+ y (/ y x))) t_0)))))

C

double code(double x, double y) {
	double t_0 = (x + y) / y;
	double tmp;
	if (x <= -4.4e+19) {
		tmp = t_0;
	} else if (x <= 3.4e-82) {
		tmp = x / (x + 1.0);
	} else if (x <= 66000000000000.0) {
		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 + y) / y
    if (x <= (-4.4d+19)) then
        tmp = t_0
    else if (x <= 3.4d-82) then
        tmp = x / (x + 1.0d0)
    else if (x <= 66000000000000.0d0) 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 + y) / y;
	double tmp;
	if (x <= -4.4e+19) {
		tmp = t_0;
	} else if (x <= 3.4e-82) {
		tmp = x / (x + 1.0);
	} else if (x <= 66000000000000.0) {
		tmp = x / (y + (y / x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x + y) / y
	tmp = 0
	if x <= -4.4e+19:
		tmp = t_0
	elif x <= 3.4e-82:
		tmp = x / (x + 1.0)
	elif x <= 66000000000000.0:
		tmp = x / (y + (y / x))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x + y) / y)
	tmp = 0.0
	if (x <= -4.4e+19)
		tmp = t_0;
	elseif (x <= 3.4e-82)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (x <= 66000000000000.0)
		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 + y) / y;
	tmp = 0.0;
	if (x <= -4.4e+19)
		tmp = t_0;
	elseif (x <= 3.4e-82)
		tmp = x / (x + 1.0);
	elseif (x <= 66000000000000.0)
		tmp = x / (y + (y / x));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, -4.4e+19], t$95$0, If[LessEqual[x, 3.4e-82], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 66000000000000.0], N[(x / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.4e19 or 6.6e13 < x

   1. Initial program 75.8%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 68.5%

      \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{1 + x} + \frac{{x}^{2}}{1 + x}}{y}} \]
   6. Step-by-step derivation

      1. *-commutative 68.5%

         \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{1 + x} + \frac{{x}^{2}}{1 + x}}{y} \]

      2. +-commutative 68.5%

         \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{x + 1}} + \frac{{x}^{2}}{1 + x}}{y} \]

      3. associate-/l* 73.8%

         \[\leadsto \frac{\color{blue}{y \cdot \frac{x}{x + 1}} + \frac{{x}^{2}}{1 + x}}{y} \]

      4. *-lft-identity 73.8%

         \[\leadsto \frac{y \cdot \frac{\color{blue}{1 \cdot x}}{x + 1} + \frac{{x}^{2}}{1 + x}}{y} \]

      5. associate-*l/ 73.7%

         \[\leadsto \frac{y \cdot \color{blue}{\left(\frac{1}{x + 1} \cdot x\right)} + \frac{{x}^{2}}{1 + x}}{y} \]

      6. unpow2 73.7%

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

      7. +-commutative 73.7%

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

      8. associate-/l* 99.9%

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

      9. *-lft-identity 99.9%

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

      10. associate-*l/ 99.8%

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

      11. distribute-rgt-out 99.8%

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

      12. associate-*l/ 100.0%

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

      13. *-lft-identity 100.0%

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

      14. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 100.0%

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

   if -4.4e19 < x < 3.39999999999999975e-82

   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 80.1%

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

   if 3.39999999999999975e-82 < x < 6.6e13

   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 77.4%

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

      1. clear-num 77.4%

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

      2. un-div-inv 77.5%

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

      3. *-commutative 77.5%

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

      4. associate-/l* 77.5%

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

      5. +-commutative 77.5%

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

   7. Applied egg-rr 77.5%

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

   8. Taylor expanded in x around inf 77.5%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+19}:\\ \;\;\;\;\frac{x + y}{y}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-82}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 66000000000000:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.8\right):\\ \;\;\;\;\frac{x + y}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \left(\frac{x}{y} - x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.8)))
   (/ (+ x y) y)
   (* x (+ 1.0 (- (/ x y) x)))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.8)) {
		tmp = (x + y) / y;
	} else {
		tmp = x * (1.0 + ((x / y) - x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 0.8d0))) then
        tmp = (x + y) / y
    else
        tmp = x * (1.0d0 + ((x / y) - x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.8)) {
		tmp = (x + y) / y;
	} else {
		tmp = x * (1.0 + ((x / y) - x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 0.8):
		tmp = (x + y) / y
	else:
		tmp = x * (1.0 + ((x / y) - x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.8))
		tmp = Float64(Float64(x + y) / y);
	else
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x / y) - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 0.8)))
		tmp = (x + y) / y;
	else
		tmp = x * (1.0 + ((x / y) - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.8]], $MachinePrecision]], N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision], N[(x * N[(1.0 + N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.80000000000000004 < x

   1. Initial program 77.2%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 y around 0 70.3%

      \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{1 + x} + \frac{{x}^{2}}{1 + x}}{y}} \]
   6. Step-by-step derivation

      1. *-commutative 70.3%

         \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{1 + x} + \frac{{x}^{2}}{1 + x}}{y} \]

      2. +-commutative 70.3%

         \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{x + 1}} + \frac{{x}^{2}}{1 + x}}{y} \]

      3. associate-/l* 75.3%

         \[\leadsto \frac{\color{blue}{y \cdot \frac{x}{x + 1}} + \frac{{x}^{2}}{1 + x}}{y} \]

      4. *-lft-identity 75.3%

         \[\leadsto \frac{y \cdot \frac{\color{blue}{1 \cdot x}}{x + 1} + \frac{{x}^{2}}{1 + x}}{y} \]

      5. associate-*l/ 75.2%

         \[\leadsto \frac{y \cdot \color{blue}{\left(\frac{1}{x + 1} \cdot x\right)} + \frac{{x}^{2}}{1 + x}}{y} \]

      6. unpow2 75.2%

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

      7. +-commutative 75.2%

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

      8. associate-/l* 99.9%

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

      9. *-lft-identity 99.9%

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

      10. associate-*l/ 99.8%

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

      11. distribute-rgt-out 99.8%

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

      12. associate-*l/ 100.0%

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

      13. *-lft-identity 100.0%

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

      14. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 98.0%

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

   if -1 < x < 0.80000000000000004

   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 97.3%

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

   6. Taylor expanded in y around inf 97.4%

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

      1. neg-mul-1 97.4%

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

      2. +-commutative 97.4%

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

      3. unsub-neg 97.4%

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

   8. Simplified 97.4%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.8\right):\\ \;\;\;\;\frac{x + y}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \left(\frac{x}{y} - x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 4 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 4e-18))) (/ x y) x))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 4e-18)) {
		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 <= (-1.0d0)) .or. (.not. (x <= 4d-18))) 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 <= -1.0) || !(x <= 4e-18)) {
		tmp = x / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 4e-18):
		tmp = x / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 4e-18))
		tmp = Float64(x / y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 4e-18)))
		tmp = x / y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 4e-18]], $MachinePrecision]], N[(x / y), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 4.0000000000000003e-18 < 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 77.2%

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

   if -1 < x < 4.0000000000000003e-18

   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 72.1%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 4 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-18}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -1.0) 1.0 (if (<= x 4e-18) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 4e-18) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = 1.0d0
    else if (x <= 4d-18) then
        tmp = x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 4e-18) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = 1.0
	elif x <= 4e-18:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 4e-18)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 4e-18)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], 1.0, If[LessEqual[x, 4e-18], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 4.0000000000000003e-18 < 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 y around inf 21.4%

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

      1. un-div-inv 21.5%

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

      2. +-commutative 21.5%

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

   7. Applied egg-rr 21.5%

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

   8. Taylor expanded in x around inf 19.9%

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

   if -1 < x < 4.0000000000000003e-18

   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 72.1%

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

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-18}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 15.5% 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 89.0%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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 y around inf 47.7%

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

   1. un-div-inv 47.7%

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

   2. +-commutative 47.7%

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

7. Applied egg-rr 47.7%

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

8. Taylor expanded in x around inf 11.4%

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

9. Final simplification 11.4%

   \[\leadsto 1 \]
10. Add Preprocessing

Developer target: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024055 
(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)))