Data.Colour.SRGB:invTransferFunction from colour-2.3.3

Percentage Accurate: 100.0% → 100.0%

Time: 3.6s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 85.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x + -1}{y}\\ \mathbf{if}\;y \leq -1.85:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{-103}:\\ \;\;\;\;\frac{y}{y + 1}\\ \mathbf{elif}\;y \leq 270000:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (+ x -1.0) y))))
   (if (<= y -1.85)
     t_0
     (if (<= y -5.5e-103)
       (/ y (+ y 1.0))
       (if (<= y 270000.0) (/ x (+ y 1.0)) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + ((x + -1.0) / y);
	double tmp;
	if (y <= -1.85) {
		tmp = t_0;
	} else if (y <= -5.5e-103) {
		tmp = y / (y + 1.0);
	} else if (y <= 270000.0) {
		tmp = x / (y + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + ((x + (-1.0d0)) / y)
    if (y <= (-1.85d0)) then
        tmp = t_0
    else if (y <= (-5.5d-103)) then
        tmp = y / (y + 1.0d0)
    else if (y <= 270000.0d0) then
        tmp = x / (y + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + ((x + -1.0) / y);
	double tmp;
	if (y <= -1.85) {
		tmp = t_0;
	} else if (y <= -5.5e-103) {
		tmp = y / (y + 1.0);
	} else if (y <= 270000.0) {
		tmp = x / (y + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + ((x + -1.0) / y)
	tmp = 0
	if y <= -1.85:
		tmp = t_0
	elif y <= -5.5e-103:
		tmp = y / (y + 1.0)
	elif y <= 270000.0:
		tmp = x / (y + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(Float64(x + -1.0) / y))
	tmp = 0.0
	if (y <= -1.85)
		tmp = t_0;
	elseif (y <= -5.5e-103)
		tmp = Float64(y / Float64(y + 1.0));
	elseif (y <= 270000.0)
		tmp = Float64(x / Float64(y + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + ((x + -1.0) / y);
	tmp = 0.0;
	if (y <= -1.85)
		tmp = t_0;
	elseif (y <= -5.5e-103)
		tmp = y / (y + 1.0);
	elseif (y <= 270000.0)
		tmp = x / (y + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.85], t$95$0, If[LessEqual[y, -5.5e-103], N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 270000.0], N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.8500000000000001 or 2.7e5 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 99.3%

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

      1. +-commutative 99.3%

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

      2. associate--l+ 99.3%

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

      3. +-commutative 99.3%

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

      4. associate--r- 99.3%

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

      5. div-sub 99.3%

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

   5. Simplified 99.3%

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

   if -1.8500000000000001 < y < -5.50000000000000032e-103

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 66.3%

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

      1. +-commutative 66.3%

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

   5. Simplified 66.3%

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

   if -5.50000000000000032e-103 < y < 2.7e5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 81.6%

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

      1. +-commutative 81.6%

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

   5. Simplified 81.6%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.85:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{-103}:\\ \;\;\;\;\frac{y}{y + 1}\\ \mathbf{elif}\;y \leq 270000:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;y \leq -7.4 \cdot 10^{+21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -4.9 \cdot 10^{-103}:\\ \;\;\;\;\frac{y}{y + 1}\\ \mathbf{elif}\;y \leq 1600000:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))))
   (if (<= y -7.4e+21)
     t_0
     (if (<= y -4.9e-103)
       (/ y (+ y 1.0))
       (if (<= y 1600000.0) (/ x (+ y 1.0)) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (y <= -7.4e+21) {
		tmp = t_0;
	} else if (y <= -4.9e-103) {
		tmp = y / (y + 1.0);
	} else if (y <= 1600000.0) {
		tmp = x / (y + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (x / y)
    if (y <= (-7.4d+21)) then
        tmp = t_0
    else if (y <= (-4.9d-103)) then
        tmp = y / (y + 1.0d0)
    else if (y <= 1600000.0d0) then
        tmp = x / (y + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (y <= -7.4e+21) {
		tmp = t_0;
	} else if (y <= -4.9e-103) {
		tmp = y / (y + 1.0);
	} else if (y <= 1600000.0) {
		tmp = x / (y + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x / y)
	tmp = 0
	if y <= -7.4e+21:
		tmp = t_0
	elif y <= -4.9e-103:
		tmp = y / (y + 1.0)
	elif y <= 1600000.0:
		tmp = x / (y + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / y))
	tmp = 0.0
	if (y <= -7.4e+21)
		tmp = t_0;
	elseif (y <= -4.9e-103)
		tmp = Float64(y / Float64(y + 1.0));
	elseif (y <= 1600000.0)
		tmp = Float64(x / Float64(y + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (x / y);
	tmp = 0.0;
	if (y <= -7.4e+21)
		tmp = t_0;
	elseif (y <= -4.9e-103)
		tmp = y / (y + 1.0);
	elseif (y <= 1600000.0)
		tmp = x / (y + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.4e+21], t$95$0, If[LessEqual[y, -4.9e-103], N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1600000.0], N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.4e21 or 1.6e6 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 100.0%

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

      1. +-commutative 100.0%

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

      2. associate--l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. associate--r- 100.0%

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

      5. div-sub 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 99.9%

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

      1. neg-mul-1 99.9%

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

      2. distribute-neg-frac 99.9%

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

   8. Simplified 99.9%

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

   if -7.4e21 < y < -4.89999999999999953e-103

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 69.1%

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

      1. +-commutative 69.1%

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

   5. Simplified 69.1%

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

   if -4.89999999999999953e-103 < y < 1.6e6

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 81.6%

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

      1. +-commutative 81.6%

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

   5. Simplified 81.6%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.4 \cdot 10^{+21}:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;y \leq -4.9 \cdot 10^{-103}:\\ \;\;\;\;\frac{y}{y + 1}\\ \mathbf{elif}\;y \leq 1600000:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.9:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{-103}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 270000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.9) 1.0 (if (<= y -5.5e-103) y (if (<= y 270000.0) x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.9) {
		tmp = 1.0;
	} else if (y <= -5.5e-103) {
		tmp = y;
	} else if (y <= 270000.0) {
		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 (y <= (-1.9d0)) then
        tmp = 1.0d0
    else if (y <= (-5.5d-103)) then
        tmp = y
    else if (y <= 270000.0d0) 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 (y <= -1.9) {
		tmp = 1.0;
	} else if (y <= -5.5e-103) {
		tmp = y;
	} else if (y <= 270000.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.9:
		tmp = 1.0
	elif y <= -5.5e-103:
		tmp = y
	elif y <= 270000.0:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.9)
		tmp = 1.0;
	elseif (y <= -5.5e-103)
		tmp = y;
	elseif (y <= 270000.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.9)
		tmp = 1.0;
	elseif (y <= -5.5e-103)
		tmp = y;
	elseif (y <= 270000.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.9], 1.0, If[LessEqual[y, -5.5e-103], y, If[LessEqual[y, 270000.0], x, 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.8999999999999999 or 2.7e5 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 73.2%

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

   if -1.8999999999999999 < y < -5.50000000000000032e-103

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 63.2%

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

      1. +-commutative 63.2%

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

   5. Simplified 63.2%

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

   6. Taylor expanded in y around 0 54.1%

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

   if -5.50000000000000032e-103 < y < 2.7e5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 78.7%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.9:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{-103}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 270000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -19500000 \lor \neg \left(x \leq 3.1 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -19500000.0) (not (<= x 3.1e-20)))
   (/ x (+ y 1.0))
   (/ y (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -19500000.0) || !(x <= 3.1e-20)) {
		tmp = x / (y + 1.0);
	} else {
		tmp = y / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-19500000.0d0)) .or. (.not. (x <= 3.1d-20))) then
        tmp = x / (y + 1.0d0)
    else
        tmp = y / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -19500000.0) || !(x <= 3.1e-20)) {
		tmp = x / (y + 1.0);
	} else {
		tmp = y / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -19500000.0) or not (x <= 3.1e-20):
		tmp = x / (y + 1.0)
	else:
		tmp = y / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -19500000.0) || !(x <= 3.1e-20))
		tmp = Float64(x / Float64(y + 1.0));
	else
		tmp = Float64(y / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -19500000.0) || ~((x <= 3.1e-20)))
		tmp = x / (y + 1.0);
	else
		tmp = y / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -19500000.0], N[Not[LessEqual[x, 3.1e-20]], $MachinePrecision]], N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.95e7 or 3.1e-20 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 78.2%

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

      1. +-commutative 78.2%

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

   5. Simplified 78.2%

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

   if -1.95e7 < x < 3.1e-20

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 82.3%

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

      1. +-commutative 82.3%

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

   5. Simplified 82.3%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -19500000 \lor \neg \left(x \leq 3.1 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2900000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+38}:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2900000.0) 1.0 (if (<= y 5.2e+38) (/ x (+ y 1.0)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2900000.0) {
		tmp = 1.0;
	} else if (y <= 5.2e+38) {
		tmp = x / (y + 1.0);
	} 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 (y <= (-2900000.0d0)) then
        tmp = 1.0d0
    else if (y <= 5.2d+38) then
        tmp = x / (y + 1.0d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2900000.0) {
		tmp = 1.0;
	} else if (y <= 5.2e+38) {
		tmp = x / (y + 1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2900000.0:
		tmp = 1.0
	elif y <= 5.2e+38:
		tmp = x / (y + 1.0)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2900000.0)
		tmp = 1.0;
	elseif (y <= 5.2e+38)
		tmp = Float64(x / Float64(y + 1.0));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2900000.0)
		tmp = 1.0;
	elseif (y <= 5.2e+38)
		tmp = x / (y + 1.0);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2900000.0], 1.0, If[LessEqual[y, 5.2e+38], N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -2.9e6 or 5.1999999999999998e38 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 74.9%

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

   if -2.9e6 < y < 5.1999999999999998e38

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 73.7%

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

      1. +-commutative 73.7%

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

   5. Simplified 73.7%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2900000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+38}:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 270000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) 1.0 (if (<= y 270000.0) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 270000.0) {
		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 (y <= (-1.0d0)) then
        tmp = 1.0d0
    else if (y <= 270000.0d0) 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 (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 270000.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = 1.0
	elif y <= 270000.0:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = 1.0;
	elseif (y <= 270000.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = 1.0;
	elseif (y <= 270000.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], 1.0, If[LessEqual[y, 270000.0], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 2.7e5 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 72.7%

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

   if -1 < y < 2.7e5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 71.1%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 270000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 39.5% accurate, 7.0× 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 100.0%

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

3. Taylor expanded in y around inf 40.5%

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

4. Final simplification 40.5%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024027 
(FPCore (x y)
  :name "Data.Colour.SRGB:invTransferFunction from colour-2.3.3"
  :precision binary64
  (/ (+ x y) (+ y 1.0)))