Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 87.9% → 99.9%

Time: 10.6s

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: 87.9% 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 87.7%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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: 72.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\frac{y}{x}}\\ t_1 := \frac{x + -1}{y}\\ \mathbf{if}\;x \leq -1.75 \cdot 10^{+56}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.52 \cdot 10^{+32}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-127}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (/ y x))) (t_1 (/ (+ x -1.0) y)))
   (if (<= x -1.75e+56)
     (/ x y)
     (if (<= x -1.52e+32)
       1.0
       (if (<= x -1.0)
         t_1
         (if (<= x -6.2e-45)
           t_0
           (if (<= x 3.2e-127) x (if (<= x 1.3) t_0 t_1))))))))

C

double code(double x, double y) {
	double t_0 = x / (y / x);
	double t_1 = (x + -1.0) / y;
	double tmp;
	if (x <= -1.75e+56) {
		tmp = x / y;
	} else if (x <= -1.52e+32) {
		tmp = 1.0;
	} else if (x <= -1.0) {
		tmp = t_1;
	} else if (x <= -6.2e-45) {
		tmp = t_0;
	} else if (x <= 3.2e-127) {
		tmp = x;
	} else if (x <= 1.3) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x / (y / x)
    t_1 = (x + (-1.0d0)) / y
    if (x <= (-1.75d+56)) then
        tmp = x / y
    else if (x <= (-1.52d+32)) then
        tmp = 1.0d0
    else if (x <= (-1.0d0)) then
        tmp = t_1
    else if (x <= (-6.2d-45)) then
        tmp = t_0
    else if (x <= 3.2d-127) then
        tmp = x
    else if (x <= 1.3d0) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y / x);
	double t_1 = (x + -1.0) / y;
	double tmp;
	if (x <= -1.75e+56) {
		tmp = x / y;
	} else if (x <= -1.52e+32) {
		tmp = 1.0;
	} else if (x <= -1.0) {
		tmp = t_1;
	} else if (x <= -6.2e-45) {
		tmp = t_0;
	} else if (x <= 3.2e-127) {
		tmp = x;
	} else if (x <= 1.3) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y / x)
	t_1 = (x + -1.0) / y
	tmp = 0
	if x <= -1.75e+56:
		tmp = x / y
	elif x <= -1.52e+32:
		tmp = 1.0
	elif x <= -1.0:
		tmp = t_1
	elif x <= -6.2e-45:
		tmp = t_0
	elif x <= 3.2e-127:
		tmp = x
	elif x <= 1.3:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y / x))
	t_1 = Float64(Float64(x + -1.0) / y)
	tmp = 0.0
	if (x <= -1.75e+56)
		tmp = Float64(x / y);
	elseif (x <= -1.52e+32)
		tmp = 1.0;
	elseif (x <= -1.0)
		tmp = t_1;
	elseif (x <= -6.2e-45)
		tmp = t_0;
	elseif (x <= 3.2e-127)
		tmp = x;
	elseif (x <= 1.3)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y / x);
	t_1 = (x + -1.0) / y;
	tmp = 0.0;
	if (x <= -1.75e+56)
		tmp = x / y;
	elseif (x <= -1.52e+32)
		tmp = 1.0;
	elseif (x <= -1.0)
		tmp = t_1;
	elseif (x <= -6.2e-45)
		tmp = t_0;
	elseif (x <= 3.2e-127)
		tmp = x;
	elseif (x <= 1.3)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, -1.75e+56], N[(x / y), $MachinePrecision], If[LessEqual[x, -1.52e+32], 1.0, If[LessEqual[x, -1.0], t$95$1, If[LessEqual[x, -6.2e-45], t$95$0, If[LessEqual[x, 3.2e-127], x, If[LessEqual[x, 1.3], t$95$0, t$95$1]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1.75e56

   1. Initial program 74.7%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 87.9%

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

   if -1.75e56 < x < -1.5200000000000001e32

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. *-rgt-identity 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in y around inf 86.4%

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

      1. +-commutative 86.4%

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

   6. Simplified 86.4%

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

   7. Taylor expanded in x around inf 86.4%

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

   if -1.5200000000000001e32 < x < -1 or 1.30000000000000004 < x

   1. Initial program 78.9%

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

   2. Taylor expanded in x around inf 54.8%

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

      1. unpow2 54.8%

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

      2. associate-*r/ 59.1%

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

   4. Simplified 59.1%

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

   5. Taylor expanded in x around inf 80.1%

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

   6. Taylor expanded in y around 0 80.1%

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

   if -1 < x < -6.2000000000000002e-45 or 3.20000000000000017e-127 < x < 1.30000000000000004

   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%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around inf 68.6%

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

      1. associate--l+ 68.6%

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

      2. div-sub 68.7%

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

      3. unpow2 68.7%

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

   6. Simplified 68.7%

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

   7. Taylor expanded in y around 0 68.9%

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

   8. Taylor expanded in x around 0 61.2%

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

   if -6.2000000000000002e-45 < x < 3.20000000000000017e-127

   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%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 87.3%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+56}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.52 \cdot 10^{+32}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;\frac{x + -1}{y}\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-45}:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-127}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + -1}{y}\\ \end{array} \]

Alternative 3: 73.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+56}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{+32}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-51}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1e+56)
   (/ x y)
   (if (<= x -1.35e+32)
     1.0
     (if (<= x -1.0)
       (/ x y)
       (if (<= x 6.8e-51) x (if (<= x 1.0) (* x (/ x y)) (/ x y)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1e+56) {
		tmp = x / y;
	} else if (x <= -1.35e+32) {
		tmp = 1.0;
	} else if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 6.8e-51) {
		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 <= (-1d+56)) then
        tmp = x / y
    else if (x <= (-1.35d+32)) then
        tmp = 1.0d0
    else if (x <= (-1.0d0)) then
        tmp = x / y
    else if (x <= 6.8d-51) 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 <= -1e+56) {
		tmp = x / y;
	} else if (x <= -1.35e+32) {
		tmp = 1.0;
	} else if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 6.8e-51) {
		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 <= -1e+56:
		tmp = x / y
	elif x <= -1.35e+32:
		tmp = 1.0
	elif x <= -1.0:
		tmp = x / y
	elif x <= 6.8e-51:
		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 <= -1e+56)
		tmp = Float64(x / y);
	elseif (x <= -1.35e+32)
		tmp = 1.0;
	elseif (x <= -1.0)
		tmp = Float64(x / y);
	elseif (x <= 6.8e-51)
		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 <= -1e+56)
		tmp = x / y;
	elseif (x <= -1.35e+32)
		tmp = 1.0;
	elseif (x <= -1.0)
		tmp = x / y;
	elseif (x <= 6.8e-51)
		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, -1e+56], N[(x / y), $MachinePrecision], If[LessEqual[x, -1.35e+32], 1.0, If[LessEqual[x, -1.0], N[(x / y), $MachinePrecision], If[LessEqual[x, 6.8e-51], x, If[LessEqual[x, 1.0], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.00000000000000009e56 or -1.35000000000000006e32 < x < -1 or 1 < x

   1. Initial program 76.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 83.6%

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

   if -1.00000000000000009e56 < x < -1.35000000000000006e32

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. *-rgt-identity 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in y around inf 86.4%

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

      1. +-commutative 86.4%

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

   6. Simplified 86.4%

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

   7. Taylor expanded in x around inf 86.4%

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

   if -1 < x < 6.80000000000000005e-51

   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 78.7%

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

   if 6.80000000000000005e-51 < 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%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around inf 75.4%

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

      1. associate--l+ 75.4%

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

      2. div-sub 75.4%

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

      3. unpow2 75.4%

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

   6. Simplified 75.4%

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

   7. Taylor expanded in y around 0 75.7%

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

   8. Taylor expanded in x around 0 68.4%

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

      1. associate-/r/ 68.3%

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

   10. Applied egg-rr 68.3%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+56}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{+32}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-51}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4: 73.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+56}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.52 \cdot 10^{+32}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.26 \cdot 10^{-50}:\\ \;\;\;\;x - x \cdot x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1e+56)
   (/ x y)
   (if (<= x -1.52e+32)
     1.0
     (if (<= x -1.0)
       (/ x y)
       (if (<= x 1.26e-50)
         (- x (* x x))
         (if (<= x 1.0) (* x (/ x y)) (/ x y)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1e+56) {
		tmp = x / y;
	} else if (x <= -1.52e+32) {
		tmp = 1.0;
	} else if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 1.26e-50) {
		tmp = x - (x * 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 <= (-1d+56)) then
        tmp = x / y
    else if (x <= (-1.52d+32)) then
        tmp = 1.0d0
    else if (x <= (-1.0d0)) then
        tmp = x / y
    else if (x <= 1.26d-50) then
        tmp = x - (x * 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 <= -1e+56) {
		tmp = x / y;
	} else if (x <= -1.52e+32) {
		tmp = 1.0;
	} else if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 1.26e-50) {
		tmp = x - (x * 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 <= -1e+56:
		tmp = x / y
	elif x <= -1.52e+32:
		tmp = 1.0
	elif x <= -1.0:
		tmp = x / y
	elif x <= 1.26e-50:
		tmp = x - (x * 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 <= -1e+56)
		tmp = Float64(x / y);
	elseif (x <= -1.52e+32)
		tmp = 1.0;
	elseif (x <= -1.0)
		tmp = Float64(x / y);
	elseif (x <= 1.26e-50)
		tmp = Float64(x - Float64(x * 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 <= -1e+56)
		tmp = x / y;
	elseif (x <= -1.52e+32)
		tmp = 1.0;
	elseif (x <= -1.0)
		tmp = x / y;
	elseif (x <= 1.26e-50)
		tmp = x - (x * 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, -1e+56], N[(x / y), $MachinePrecision], If[LessEqual[x, -1.52e+32], 1.0, If[LessEqual[x, -1.0], N[(x / y), $MachinePrecision], If[LessEqual[x, 1.26e-50], N[(x - N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.00000000000000009e56 or -1.5200000000000001e32 < x < -1 or 1 < x

   1. Initial program 76.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 83.6%

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

   if -1.00000000000000009e56 < x < -1.5200000000000001e32

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. *-rgt-identity 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in y around inf 86.4%

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

      1. +-commutative 86.4%

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

   6. Simplified 86.4%

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

   7. Taylor expanded in x around inf 86.4%

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

   if -1 < x < 1.26e-50

   1. Initial program 99.8%

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

      1. distribute-lft-in 99.8%

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

      2. *-rgt-identity 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in y around inf 79.0%

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

      1. +-commutative 79.0%

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

   6. Simplified 79.0%

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

   7. Taylor expanded in x around 0 79.1%

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

      1. mul-1-neg 79.1%

         \[\leadsto x + \color{blue}{\left(-{x}^{2}\right)} \]

      2. unsub-neg 79.1%

         \[\leadsto \color{blue}{x - {x}^{2}} \]

      3. unpow2 79.1%

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

   9. Simplified 79.1%

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

   if 1.26e-50 < 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%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around inf 75.4%

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

      1. associate--l+ 75.4%

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

      2. div-sub 75.4%

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

      3. unpow2 75.4%

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

   6. Simplified 75.4%

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

   7. Taylor expanded in y around 0 75.7%

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

   8. Taylor expanded in x around 0 68.4%

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

      1. associate-/r/ 68.3%

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

   10. Applied egg-rr 68.3%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+56}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.52 \cdot 10^{+32}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.26 \cdot 10^{-50}:\\ \;\;\;\;x - x \cdot x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 5: 95.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 78.1%

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

      1. flip-+ 38.3%

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

      2. associate-*r/ 33.6%

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

      3. metadata-eval 33.6%

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

      4. sub-neg 33.6%

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

      5. pow2 33.6%

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

      6. metadata-eval 33.6%

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

      7. sub-neg 33.6%

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

      8. metadata-eval 33.6%

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

   3. Applied egg-rr 33.6%

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

      1. *-commutative 33.6%

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

      2. associate-/l* 38.2%

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

      3. +-commutative 38.2%

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

      4. +-commutative 38.2%

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

   5. Simplified 38.2%

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

   6. Taylor expanded in x around inf 99.3%

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

      1. associate--l+ 99.3%

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

      2. div-sub 99.3%

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

      3. sub-neg 99.3%

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

      4. metadata-eval 99.3%

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

      5. +-commutative 99.3%

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

   8. Simplified 99.3%

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

   if -1 < x < 1

   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 91.9%

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

      1. unpow2 91.9%

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

      2. sub-neg 91.9%

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

      3. metadata-eval 91.9%

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

   6. Simplified 91.9%

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

4. Final simplification 96.0%

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

Alternative 6: 84.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x + -1}{y}\\ \mathbf{if}\;x \leq -290000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-127}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (+ x -1.0) y))))
   (if (<= x -290000000.0)
     t_0
     (if (<= x 5.2e-127) (/ x (+ x 1.0)) (if (<= x 0.95) (/ x (/ y x)) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + ((x + -1.0) / y);
	double tmp;
	if (x <= -290000000.0) {
		tmp = t_0;
	} else if (x <= 5.2e-127) {
		tmp = x / (x + 1.0);
	} else if (x <= 0.95) {
		tmp = x / (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 = 1.0d0 + ((x + (-1.0d0)) / y)
    if (x <= (-290000000.0d0)) then
        tmp = t_0
    else if (x <= 5.2d-127) then
        tmp = x / (x + 1.0d0)
    else if (x <= 0.95d0) then
        tmp = x / (y / x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + ((x + -1.0) / y);
	double tmp;
	if (x <= -290000000.0) {
		tmp = t_0;
	} else if (x <= 5.2e-127) {
		tmp = x / (x + 1.0);
	} else if (x <= 0.95) {
		tmp = x / (y / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + ((x + -1.0) / y)
	tmp = 0
	if x <= -290000000.0:
		tmp = t_0
	elif x <= 5.2e-127:
		tmp = x / (x + 1.0)
	elif x <= 0.95:
		tmp = x / (y / x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(Float64(x + -1.0) / y))
	tmp = 0.0
	if (x <= -290000000.0)
		tmp = t_0;
	elseif (x <= 5.2e-127)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (x <= 0.95)
		tmp = Float64(x / Float64(y / x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + ((x + -1.0) / y);
	tmp = 0.0;
	if (x <= -290000000.0)
		tmp = t_0;
	elseif (x <= 5.2e-127)
		tmp = x / (x + 1.0);
	elseif (x <= 0.95)
		tmp = x / (y / x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -290000000.0], t$95$0, If[LessEqual[x, 5.2e-127], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.95], N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.9e8 or 0.94999999999999996 < x

   1. Initial program 77.8%

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

      1. flip-+ 37.4%

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

      2. associate-*r/ 32.6%

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

      3. metadata-eval 32.6%

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

      4. sub-neg 32.6%

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

      5. pow2 32.6%

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

      6. metadata-eval 32.6%

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

      7. sub-neg 32.6%

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

      8. metadata-eval 32.6%

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

   3. Applied egg-rr 32.6%

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

      1. *-commutative 32.6%

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

      2. associate-/l* 37.3%

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

      3. +-commutative 37.3%

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

      4. +-commutative 37.3%

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

   5. Simplified 37.3%

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

   6. Taylor expanded in x around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. div-sub 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

      5. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   if -2.9e8 < x < 5.19999999999999982e-127

   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%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 82.4%

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

   if 5.19999999999999982e-127 < x < 0.94999999999999996

   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%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around inf 64.5%

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

      1. associate--l+ 64.5%

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

      2. div-sub 64.6%

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

      3. unpow2 64.6%

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

   6. Simplified 64.6%

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

   7. Taylor expanded in y around 0 64.8%

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

   8. Taylor expanded in x around 0 60.8%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -290000000:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-127}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \end{array} \]

Alternative 7: 84.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x + -1}{y}\\ \mathbf{if}\;x \leq -290000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-127}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 240000:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (+ x -1.0) y))))
   (if (<= x -290000000.0)
     t_0
     (if (<= x 5.2e-127)
       (/ x (+ x 1.0))
       (if (<= x 240000.0) (/ x (+ y (/ y x))) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + ((x + -1.0) / y);
	double tmp;
	if (x <= -290000000.0) {
		tmp = t_0;
	} else if (x <= 5.2e-127) {
		tmp = x / (x + 1.0);
	} else if (x <= 240000.0) {
		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 = 1.0d0 + ((x + (-1.0d0)) / y)
    if (x <= (-290000000.0d0)) then
        tmp = t_0
    else if (x <= 5.2d-127) then
        tmp = x / (x + 1.0d0)
    else if (x <= 240000.0d0) 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 = 1.0 + ((x + -1.0) / y);
	double tmp;
	if (x <= -290000000.0) {
		tmp = t_0;
	} else if (x <= 5.2e-127) {
		tmp = x / (x + 1.0);
	} else if (x <= 240000.0) {
		tmp = x / (y + (y / x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + ((x + -1.0) / y)
	tmp = 0
	if x <= -290000000.0:
		tmp = t_0
	elif x <= 5.2e-127:
		tmp = x / (x + 1.0)
	elif x <= 240000.0:
		tmp = x / (y + (y / x))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(Float64(x + -1.0) / y))
	tmp = 0.0
	if (x <= -290000000.0)
		tmp = t_0;
	elseif (x <= 5.2e-127)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (x <= 240000.0)
		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 = 1.0 + ((x + -1.0) / y);
	tmp = 0.0;
	if (x <= -290000000.0)
		tmp = t_0;
	elseif (x <= 5.2e-127)
		tmp = x / (x + 1.0);
	elseif (x <= 240000.0)
		tmp = x / (y + (y / x));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -290000000.0], t$95$0, If[LessEqual[x, 5.2e-127], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 240000.0], N[(x / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.9e8 or 2.4e5 < x

   1. Initial program 77.8%

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

      1. flip-+ 37.4%

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

      2. associate-*r/ 32.6%

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

      3. metadata-eval 32.6%

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

      4. sub-neg 32.6%

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

      5. pow2 32.6%

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

      6. metadata-eval 32.6%

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

      7. sub-neg 32.6%

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

      8. metadata-eval 32.6%

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

   3. Applied egg-rr 32.6%

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

      1. *-commutative 32.6%

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

      2. associate-/l* 37.3%

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

      3. +-commutative 37.3%

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

      4. +-commutative 37.3%

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

   5. Simplified 37.3%

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

   6. Taylor expanded in x around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. div-sub 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

      5. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   if -2.9e8 < x < 5.19999999999999982e-127

   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%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 82.4%

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

   if 5.19999999999999982e-127 < x < 2.4e5

   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%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around inf 64.5%

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

      1. associate--l+ 64.5%

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

      2. div-sub 64.6%

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

      3. unpow2 64.6%

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

   6. Simplified 64.6%

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

   7. Taylor expanded in y around 0 64.8%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -290000000:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-127}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 240000:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \end{array} \]

Alternative 8: 73.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{+56}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.52 \cdot 10^{+32}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-13}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -8.2e+56)
   (/ x y)
   (if (<= x -1.52e+32)
     1.0
     (if (<= x -1.0) (/ x y) (if (<= x 6.5e-13) x (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -8.2e+56) {
		tmp = x / y;
	} else if (x <= -1.52e+32) {
		tmp = 1.0;
	} else if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 6.5e-13) {
		tmp = x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-8.2d+56)) then
        tmp = x / y
    else if (x <= (-1.52d+32)) then
        tmp = 1.0d0
    else if (x <= (-1.0d0)) then
        tmp = x / y
    else if (x <= 6.5d-13) then
        tmp = x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -8.2e+56) {
		tmp = x / y;
	} else if (x <= -1.52e+32) {
		tmp = 1.0;
	} else if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 6.5e-13) {
		tmp = x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -8.2e+56:
		tmp = x / y
	elif x <= -1.52e+32:
		tmp = 1.0
	elif x <= -1.0:
		tmp = x / y
	elif x <= 6.5e-13:
		tmp = x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -8.2e+56)
		tmp = Float64(x / y);
	elseif (x <= -1.52e+32)
		tmp = 1.0;
	elseif (x <= -1.0)
		tmp = Float64(x / y);
	elseif (x <= 6.5e-13)
		tmp = x;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -8.2e+56)
		tmp = x / y;
	elseif (x <= -1.52e+32)
		tmp = 1.0;
	elseif (x <= -1.0)
		tmp = x / y;
	elseif (x <= 6.5e-13)
		tmp = x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -8.2e+56], N[(x / y), $MachinePrecision], If[LessEqual[x, -1.52e+32], 1.0, If[LessEqual[x, -1.0], N[(x / y), $MachinePrecision], If[LessEqual[x, 6.5e-13], x, N[(x / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.2000000000000007e56 or -1.5200000000000001e32 < x < -1 or 6.49999999999999957e-13 < x

   1. Initial program 77.3%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 82.5%

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

   if -8.2000000000000007e56 < x < -1.5200000000000001e32

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. *-rgt-identity 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in y around inf 86.4%

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

      1. +-commutative 86.4%

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

   6. Simplified 86.4%

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

   7. Taylor expanded in x around inf 86.4%

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

   if -1 < x < 6.49999999999999957e-13

   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 74.6%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{+56}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.52 \cdot 10^{+32}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-13}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 9: 73.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+56}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-52}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.1e+56)
   (/ x y)
   (if (<= x 3.6e-52) (/ x (+ x 1.0)) (if (<= x 1.0) (* x (/ x y)) (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.1e+56) {
		tmp = x / y;
	} else if (x <= 3.6e-52) {
		tmp = x / (x + 1.0);
	} 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 <= (-1.1d+56)) then
        tmp = x / y
    else if (x <= 3.6d-52) then
        tmp = x / (x + 1.0d0)
    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 <= -1.1e+56) {
		tmp = x / y;
	} else if (x <= 3.6e-52) {
		tmp = x / (x + 1.0);
	} else if (x <= 1.0) {
		tmp = x * (x / y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.1e+56:
		tmp = x / y
	elif x <= 3.6e-52:
		tmp = x / (x + 1.0)
	elif x <= 1.0:
		tmp = x * (x / y)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.1e+56)
		tmp = Float64(x / y);
	elseif (x <= 3.6e-52)
		tmp = Float64(x / Float64(x + 1.0));
	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 <= -1.1e+56)
		tmp = x / y;
	elseif (x <= 3.6e-52)
		tmp = x / (x + 1.0);
	elseif (x <= 1.0)
		tmp = x * (x / y);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.1e+56], N[(x / y), $MachinePrecision], If[LessEqual[x, 3.6e-52], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.10000000000000008e56 or 1 < x

   1. Initial program 75.7%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 85.1%

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

   if -1.10000000000000008e56 < x < 3.59999999999999988e-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%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around inf 77.3%

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

   if 3.59999999999999988e-52 < 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%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around inf 75.4%

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

      1. associate--l+ 75.4%

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

      2. div-sub 75.4%

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

      3. unpow2 75.4%

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

   6. Simplified 75.4%

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

   7. Taylor expanded in y around 0 75.7%

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

   8. Taylor expanded in x around 0 68.4%

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

      1. associate-/r/ 68.3%

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

   10. Applied egg-rr 68.3%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+56}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-52}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 10: 72.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+57}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-127}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.15e+57)
   (/ x y)
   (if (<= x 5.2e-127) (/ x (+ x 1.0)) (if (<= x 1.0) (/ x (/ y x)) (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.15e+57) {
		tmp = x / y;
	} else if (x <= 5.2e-127) {
		tmp = x / (x + 1.0);
	} else if (x <= 1.0) {
		tmp = x / (y / x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.15d+57)) then
        tmp = x / y
    else if (x <= 5.2d-127) then
        tmp = x / (x + 1.0d0)
    else if (x <= 1.0d0) then
        tmp = x / (y / x)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.15e+57) {
		tmp = x / y;
	} else if (x <= 5.2e-127) {
		tmp = x / (x + 1.0);
	} else if (x <= 1.0) {
		tmp = x / (y / x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.15e+57:
		tmp = x / y
	elif x <= 5.2e-127:
		tmp = x / (x + 1.0)
	elif x <= 1.0:
		tmp = x / (y / x)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.15e+57)
		tmp = Float64(x / y);
	elseif (x <= 5.2e-127)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (x <= 1.0)
		tmp = Float64(x / Float64(y / x));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.15e+57)
		tmp = x / y;
	elseif (x <= 5.2e-127)
		tmp = x / (x + 1.0);
	elseif (x <= 1.0)
		tmp = x / (y / x);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.15e+57], N[(x / y), $MachinePrecision], If[LessEqual[x, 5.2e-127], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1499999999999999e57 or 1 < x

   1. Initial program 75.7%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 85.1%

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

   if -1.1499999999999999e57 < x < 5.19999999999999982e-127

   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%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 79.8%

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

   if 5.19999999999999982e-127 < 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* 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around inf 64.5%

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

      1. associate--l+ 64.5%

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

      2. div-sub 64.6%

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

      3. unpow2 64.6%

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

   6. Simplified 64.6%

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

   7. Taylor expanded in y around 0 64.8%

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

   8. Taylor expanded in x around 0 60.8%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+57}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-127}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 11: 48.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.000112:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.00345:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.000112) 1.0 (if (<= x 0.00345) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.000112) {
		tmp = 1.0;
	} else if (x <= 0.00345) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.000112d0)) then
        tmp = 1.0d0
    else if (x <= 0.00345d0) then
        tmp = x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.000112) {
		tmp = 1.0;
	} else if (x <= 0.00345) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.000112:
		tmp = 1.0
	elif x <= 0.00345:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.000112)
		tmp = 1.0;
	elseif (x <= 0.00345)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.000112)
		tmp = 1.0;
	elseif (x <= 0.00345)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.000112], 1.0, If[LessEqual[x, 0.00345], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1.11999999999999998e-4 or 0.0034499999999999999 < x

   1. Initial program 78.5%

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

      1. distribute-lft-in 78.5%

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

      2. *-rgt-identity 78.5%

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

   3. Applied egg-rr 78.5%

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

   4. Taylor expanded in y around inf 20.2%

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

      1. +-commutative 20.2%

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

   6. Simplified 20.2%

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

   7. Taylor expanded in x around inf 19.7%

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

   if -1.11999999999999998e-4 < x < 0.0034499999999999999

   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 75.3%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.000112:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.00345:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12: 14.2% accurate, 11.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 87.7%

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

   1. distribute-lft-in 87.7%

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

   2. *-rgt-identity 87.7%

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

3. Applied egg-rr 87.7%

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

4. Taylor expanded in y around inf 44.2%

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

   1. +-commutative 44.2%

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

6. Simplified 44.2%

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

7. Taylor expanded in x around inf 12.7%

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

8. Final simplification 12.7%

   \[\leadsto 1 \]

Developer target: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023279 
(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)))