Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.3% → 99.9%

Time: 7.8s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 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 83.8%

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

   1. clear-num 99.8%

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

   2. un-div-inv 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 73.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -4000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-125}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 3.65 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= x -4000.0)
     (/ x y)
     (if (<= x 2.6e-125)
       t_0
       (if (<= x 4.5e-66) (/ x (/ y x)) (if (<= x 3.65e+15) t_0 (/ x y)))))))

C

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -4000.0) {
		tmp = x / y;
	} else if (x <= 2.6e-125) {
		tmp = t_0;
	} else if (x <= 4.5e-66) {
		tmp = x / (y / x);
	} else if (x <= 3.65e+15) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (x + 1.0d0)
    if (x <= (-4000.0d0)) then
        tmp = x / y
    else if (x <= 2.6d-125) then
        tmp = t_0
    else if (x <= 4.5d-66) then
        tmp = x / (y / x)
    else if (x <= 3.65d+15) then
        tmp = t_0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -4000.0) {
		tmp = x / y;
	} else if (x <= 2.6e-125) {
		tmp = t_0;
	} else if (x <= 4.5e-66) {
		tmp = x / (y / x);
	} else if (x <= 3.65e+15) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + 1.0)
	tmp = 0
	if x <= -4000.0:
		tmp = x / y
	elif x <= 2.6e-125:
		tmp = t_0
	elif x <= 4.5e-66:
		tmp = x / (y / x)
	elif x <= 3.65e+15:
		tmp = t_0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (x <= -4000.0)
		tmp = Float64(x / y);
	elseif (x <= 2.6e-125)
		tmp = t_0;
	elseif (x <= 4.5e-66)
		tmp = Float64(x / Float64(y / x));
	elseif (x <= 3.65e+15)
		tmp = t_0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + 1.0);
	tmp = 0.0;
	if (x <= -4000.0)
		tmp = x / y;
	elseif (x <= 2.6e-125)
		tmp = t_0;
	elseif (x <= 4.5e-66)
		tmp = x / (y / x);
	elseif (x <= 3.65e+15)
		tmp = t_0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4000.0], N[(x / y), $MachinePrecision], If[LessEqual[x, 2.6e-125], t$95$0, If[LessEqual[x, 4.5e-66], N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.65e+15], t$95$0, N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4e3 or 3.65e15 < x

   1. Initial program 67.2%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 79.9%

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

   if -4e3 < x < 2.60000000000000006e-125 or 4.4999999999999998e-66 < x < 3.65e15

   1. Initial program 99.9%

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

      1. associate-/l* 99.8%

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

   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 74.3%

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

   if 2.60000000000000006e-125 < x < 4.4999999999999998e-66

   1. Initial program 99.7%

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

      1. associate-/l* 99.7%

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

   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 0 65.7%

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

      1. clear-num 65.4%

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

      2. un-div-inv 65.5%

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

      3. +-commutative 65.5%

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

      4. *-commutative 65.5%

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

      5. associate-/l* 65.5%

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

   7. Applied egg-rr 65.5%

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

   8. Taylor expanded in x around 0 65.5%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-125}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 3.65 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y + \frac{y}{x}}\\ t_1 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{-44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-125}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-65}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+17}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y (/ y x)))) (t_1 (/ x (+ x 1.0))))
   (if (<= x -7.2e-44)
     t_0
     (if (<= x 2.6e-125)
       t_1
       (if (<= x 1.1e-65) t_0 (if (<= x 3.2e+17) t_1 (/ x y)))))))

C

double code(double x, double y) {
	double t_0 = x / (y + (y / x));
	double t_1 = x / (x + 1.0);
	double tmp;
	if (x <= -7.2e-44) {
		tmp = t_0;
	} else if (x <= 2.6e-125) {
		tmp = t_1;
	} else if (x <= 1.1e-65) {
		tmp = t_0;
	} else if (x <= 3.2e+17) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = x / (y + (y / x))
    t_1 = x / (x + 1.0d0)
    if (x <= (-7.2d-44)) then
        tmp = t_0
    else if (x <= 2.6d-125) then
        tmp = t_1
    else if (x <= 1.1d-65) then
        tmp = t_0
    else if (x <= 3.2d+17) then
        tmp = t_1
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y + (y / x));
	double t_1 = x / (x + 1.0);
	double tmp;
	if (x <= -7.2e-44) {
		tmp = t_0;
	} else if (x <= 2.6e-125) {
		tmp = t_1;
	} else if (x <= 1.1e-65) {
		tmp = t_0;
	} else if (x <= 3.2e+17) {
		tmp = t_1;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y + (y / x))
	t_1 = x / (x + 1.0)
	tmp = 0
	if x <= -7.2e-44:
		tmp = t_0
	elif x <= 2.6e-125:
		tmp = t_1
	elif x <= 1.1e-65:
		tmp = t_0
	elif x <= 3.2e+17:
		tmp = t_1
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y + Float64(y / x)))
	t_1 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (x <= -7.2e-44)
		tmp = t_0;
	elseif (x <= 2.6e-125)
		tmp = t_1;
	elseif (x <= 1.1e-65)
		tmp = t_0;
	elseif (x <= 3.2e+17)
		tmp = t_1;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y + (y / x));
	t_1 = x / (x + 1.0);
	tmp = 0.0;
	if (x <= -7.2e-44)
		tmp = t_0;
	elseif (x <= 2.6e-125)
		tmp = t_1;
	elseif (x <= 1.1e-65)
		tmp = t_0;
	elseif (x <= 3.2e+17)
		tmp = t_1;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.2e-44], t$95$0, If[LessEqual[x, 2.6e-125], t$95$1, If[LessEqual[x, 1.1e-65], t$95$0, If[LessEqual[x, 3.2e+17], t$95$1, N[(x / y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.1999999999999998e-44 or 2.60000000000000006e-125 < x < 1.10000000000000011e-65

   1. Initial program 82.6%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 y around 0 68.4%

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

      1. clear-num 68.3%

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

      2. un-div-inv 68.4%

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

      3. +-commutative 68.4%

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

      4. *-commutative 68.4%

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

      5. associate-/l* 74.5%

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

   7. Applied egg-rr 74.5%

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

   8. Taylor expanded in x around inf 74.6%

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

   if -7.1999999999999998e-44 < x < 2.60000000000000006e-125 or 1.10000000000000011e-65 < x < 3.2e17

   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 78.4%

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

   if 3.2e17 < x

   1. Initial program 60.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 82.4%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-44}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-125}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-65}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+17}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+14}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.5e-8)
   (/ x (+ y (/ y x)))
   (if (<= x 1.8e+14) (* x (+ 1.0 (* x (/ 1.0 y)))) (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.5e-8) {
		tmp = x / (y + (y / x));
	} else if (x <= 1.8e+14) {
		tmp = x * (1.0 + (x * (1.0 / 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 <= (-5.5d-8)) then
        tmp = x / (y + (y / x))
    else if (x <= 1.8d+14) then
        tmp = x * (1.0d0 + (x * (1.0d0 / y)))
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5.5e-8) {
		tmp = x / (y + (y / x));
	} else if (x <= 1.8e+14) {
		tmp = x * (1.0 + (x * (1.0 / y)));
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.5e-8:
		tmp = x / (y + (y / x))
	elif x <= 1.8e+14:
		tmp = x * (1.0 + (x * (1.0 / y)))
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.5e-8)
		tmp = Float64(x / Float64(y + Float64(y / x)));
	elseif (x <= 1.8e+14)
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(1.0 / y))));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.5e-8)
		tmp = x / (y + (y / x));
	elseif (x <= 1.8e+14)
		tmp = x * (1.0 + (x * (1.0 / y)));
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.5e-8], N[(x / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.8e+14], N[(x * N[(1.0 + N[(x * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.5000000000000003e-8

   1. Initial program 75.8%

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

      1. associate-/l* 99.9%

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

   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 0 69.3%

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

      1. clear-num 69.2%

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

      2. un-div-inv 69.3%

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

      3. +-commutative 69.3%

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

      4. *-commutative 69.3%

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

      5. associate-/l* 77.9%

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

   7. Applied egg-rr 77.9%

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

   8. Taylor expanded in x around inf 78.0%

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

   if -5.5000000000000003e-8 < x < 1.8e14

   1. Initial program 99.9%

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

      1. associate-/l* 99.8%

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

   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 98.7%

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

   6. Taylor expanded in y around 0 98.3%

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

   if 1.8e14 < x

   1. Initial program 60.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 82.4%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+14}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.82 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot \left(1 + \left(\frac{x}{y} - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.82e-6)
   (/ x (+ y (/ y x)))
   (if (<= x 1.0) (* x (+ 1.0 (- (/ x y) x))) (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.82e-6) {
		tmp = x / (y + (y / x));
	} else if (x <= 1.0) {
		tmp = x * (1.0 + ((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 <= (-1.82d-6)) then
        tmp = x / (y + (y / x))
    else if (x <= 1.0d0) then
        tmp = x * (1.0d0 + ((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 <= -1.82e-6) {
		tmp = x / (y + (y / x));
	} else if (x <= 1.0) {
		tmp = x * (1.0 + ((x / y) - x));
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.82e-6:
		tmp = x / (y + (y / x))
	elif x <= 1.0:
		tmp = x * (1.0 + ((x / y) - x))
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.82e-6)
		tmp = Float64(x / Float64(y + Float64(y / x)));
	elseif (x <= 1.0)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x / y) - x)));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.82e-6)
		tmp = x / (y + (y / x));
	elseif (x <= 1.0)
		tmp = x * (1.0 + ((x / y) - x));
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.82e-6], N[(x / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(x * N[(1.0 + N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.8199999999999999e-6

   1. Initial program 75.8%

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

      1. associate-/l* 99.9%

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

   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 0 69.3%

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

      1. clear-num 69.2%

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

      2. un-div-inv 69.3%

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

      3. +-commutative 69.3%

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

      4. *-commutative 69.3%

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

      5. associate-/l* 77.9%

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

   7. Applied egg-rr 77.9%

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

   8. Taylor expanded in x around inf 78.0%

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

   if -1.8199999999999999e-6 < x < 1

   1. Initial program 99.9%

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

      1. associate-/l* 99.8%

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

   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 99.5%

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

   6. Taylor expanded in y around inf 99.5%

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

      1. neg-mul-1 99.5%

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

      2. +-commutative 99.5%

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

      3. unsub-neg 99.5%

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

   8. Simplified 99.5%

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

   if 1 < x

   1. Initial program 61.5%

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

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.82 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot \left(1 + \left(\frac{x}{y} - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.95 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x + x \cdot \left(\frac{x}{y} - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.95e-9)
   (/ x (+ y (/ y x)))
   (if (<= x 1.0) (+ x (* x (- (/ x y) x))) (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.95e-9) {
		tmp = x / (y + (y / x));
	} else if (x <= 1.0) {
		tmp = x + (x * ((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 <= (-2.95d-9)) then
        tmp = x / (y + (y / x))
    else if (x <= 1.0d0) then
        tmp = x + (x * ((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 <= -2.95e-9) {
		tmp = x / (y + (y / x));
	} else if (x <= 1.0) {
		tmp = x + (x * ((x / y) - x));
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.95e-9:
		tmp = x / (y + (y / x))
	elif x <= 1.0:
		tmp = x + (x * ((x / y) - x))
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.95e-9)
		tmp = Float64(x / Float64(y + Float64(y / x)));
	elseif (x <= 1.0)
		tmp = Float64(x + Float64(x * Float64(Float64(x / y) - x)));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.95e-9)
		tmp = x / (y + (y / x));
	elseif (x <= 1.0)
		tmp = x + (x * ((x / y) - x));
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.95e-9], N[(x / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(x + N[(x * N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.9499999999999999e-9

   1. Initial program 75.8%

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

      1. associate-/l* 99.9%

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

   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 0 69.3%

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

      1. clear-num 69.2%

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

      2. un-div-inv 69.3%

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

      3. +-commutative 69.3%

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

      4. *-commutative 69.3%

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

      5. associate-/l* 77.9%

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

   7. Applied egg-rr 77.9%

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

   8. Taylor expanded in x around inf 78.0%

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

   if -2.9499999999999999e-9 < x < 1

   1. Initial program 99.9%

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

      1. associate-/l* 99.8%

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

   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 99.5%

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

   6. Taylor expanded in y around inf 99.5%

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

      1. neg-mul-1 99.5%

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

      2. +-commutative 99.5%

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

      3. unsub-neg 99.5%

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

   8. Simplified 99.5%

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

      1. +-commutative 99.5%

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

      2. distribute-lft-in 99.5%

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

      3. *-rgt-identity 99.5%

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

   10. Applied egg-rr 99.5%

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

   if 1 < x

   1. Initial program 61.5%

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

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.95 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x + x \cdot \left(\frac{x}{y} - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.052\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.052))) (/ x y) (* x (- 1.0 x))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.052)) {
		tmp = x / y;
	} else {
		tmp = x * (1.0 - x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 0.052d0))) then
        tmp = x / y
    else
        tmp = x * (1.0d0 - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.052)) {
		tmp = x / y;
	} else {
		tmp = x * (1.0 - x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 0.052):
		tmp = x / y
	else:
		tmp = x * (1.0 - x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.052))
		tmp = Float64(x / y);
	else
		tmp = Float64(x * Float64(1.0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 0.052)))
		tmp = x / y;
	else
		tmp = x * (1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.052]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.0519999999999999976 < x

   1. Initial program 67.4%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 79.3%

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

   if -1 < x < 0.0519999999999999976

   1. Initial program 99.9%

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

      1. associate-/l* 99.8%

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

   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 69.1%

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

   6. Taylor expanded in x around 0 69.1%

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

      1. neg-mul-1 69.1%

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

      2. sub-neg 69.1%

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

   8. Simplified 69.1%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.052\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3600 \lor \neg \left(x \leq 2.55 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3600.0) (not (<= x 2.55e+14))) (/ x y) (/ x (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3600.0) || !(x <= 2.55e+14)) {
		tmp = x / y;
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-3600.0d0)) .or. (.not. (x <= 2.55d+14))) then
        tmp = x / y
    else
        tmp = x / (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -3600.0) || !(x <= 2.55e+14)) {
		tmp = x / y;
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -3600.0) or not (x <= 2.55e+14):
		tmp = x / y
	else:
		tmp = x / (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -3600.0) || !(x <= 2.55e+14))
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -3600.0) || ~((x <= 2.55e+14)))
		tmp = x / y;
	else
		tmp = x / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -3600.0], N[Not[LessEqual[x, 2.55e+14]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3600 or 2.55e14 < x

   1. Initial program 67.2%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 79.9%

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

   if -3600 < x < 2.55e14

   1. Initial program 99.9%

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

      1. associate-/l* 99.8%

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

   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 69.4%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3600 \lor \neg \left(x \leq 2.55 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.8 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.8e+14))) (/ x y) x))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.8e+14)) {
		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 <= 1.8d+14))) 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 <= 1.8e+14)) {
		tmp = x / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.8e+14):
		tmp = x / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.8e+14))
		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 <= 1.8e+14)))
		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, 1.8e+14]], $MachinePrecision]], N[(x / y), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.8e14 < x

   1. Initial program 67.2%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 79.9%

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

   if -1 < x < 1.8e14

   1. Initial program 99.9%

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

      1. associate-/l* 99.8%

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

   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 68.3%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.8 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 48.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+20}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.1e+20) 1.0 (if (<= x 1.0) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.1e+20) {
		tmp = 1.0;
	} else if (x <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.1d+20)) then
        tmp = 1.0d0
    else if (x <= 1.0d0) then
        tmp = x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.1e+20) {
		tmp = 1.0;
	} else if (x <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.1e+20:
		tmp = 1.0
	elif x <= 1.0:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.1e+20)
		tmp = 1.0;
	elseif (x <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.1e+20)
		tmp = 1.0;
	elseif (x <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.1e+20], 1.0, If[LessEqual[x, 1.0], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1e20 or 1 < x

   1. Initial program 66.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 21.5%

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

   6. Taylor expanded in x around inf 21.4%

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

   if -1.1e20 < x < 1

   1. Initial program 99.2%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 x around 0 66.4%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+20}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

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

Alternative 12: 14.6% 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 83.8%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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 inf 45.1%

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

6. Taylor expanded in x around inf 12.1%

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

7. Final simplification 12.1%

   \[\leadsto 1 \]
8. 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 2024079 
(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)))