Data.Colour.SRGB:invTransferFunction from colour-2.3.3

Percentage Accurate: 100.0% → 100.0%

Time: 5.2s

Alternatives: 9

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. Add Preprocessing

Alternative 2: 71.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{-117}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-77}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 8500:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{+68}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.1)
   1.0
   (if (<= y -2.6e-117)
     y
     (if (<= y 1.75e-77)
       x
       (if (<= y 8500.0) y (if (<= y 7.4e+68) (/ x y) 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.1) {
		tmp = 1.0;
	} else if (y <= -2.6e-117) {
		tmp = y;
	} else if (y <= 1.75e-77) {
		tmp = x;
	} else if (y <= 8500.0) {
		tmp = y;
	} else if (y <= 7.4e+68) {
		tmp = x / y;
	} 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 <= (-1.1d0)) then
        tmp = 1.0d0
    else if (y <= (-2.6d-117)) then
        tmp = y
    else if (y <= 1.75d-77) then
        tmp = x
    else if (y <= 8500.0d0) then
        tmp = y
    else if (y <= 7.4d+68) then
        tmp = x / y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.1) {
		tmp = 1.0;
	} else if (y <= -2.6e-117) {
		tmp = y;
	} else if (y <= 1.75e-77) {
		tmp = x;
	} else if (y <= 8500.0) {
		tmp = y;
	} else if (y <= 7.4e+68) {
		tmp = x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.1:
		tmp = 1.0
	elif y <= -2.6e-117:
		tmp = y
	elif y <= 1.75e-77:
		tmp = x
	elif y <= 8500.0:
		tmp = y
	elif y <= 7.4e+68:
		tmp = x / y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.1)
		tmp = 1.0;
	elseif (y <= -2.6e-117)
		tmp = y;
	elseif (y <= 1.75e-77)
		tmp = x;
	elseif (y <= 8500.0)
		tmp = y;
	elseif (y <= 7.4e+68)
		tmp = Float64(x / y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.1)
		tmp = 1.0;
	elseif (y <= -2.6e-117)
		tmp = y;
	elseif (y <= 1.75e-77)
		tmp = x;
	elseif (y <= 8500.0)
		tmp = y;
	elseif (y <= 7.4e+68)
		tmp = x / y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.1], 1.0, If[LessEqual[y, -2.6e-117], y, If[LessEqual[y, 1.75e-77], x, If[LessEqual[y, 8500.0], y, If[LessEqual[y, 7.4e+68], N[(x / y), $MachinePrecision], 1.0]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.1000000000000001 or 7.39999999999999996e68 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 80.1%

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

   if -1.1000000000000001 < y < -2.59999999999999983e-117 or 1.75000000000000006e-77 < y < 8500

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 68.2%

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

      1. +-commutative 68.2%

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

   5. Simplified 68.2%

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

   6. Taylor expanded in y around 0 63.2%

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

   if -2.59999999999999983e-117 < y < 1.75000000000000006e-77

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 83.9%

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

   if 8500 < y < 7.39999999999999996e68

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 76.0%

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

      1. +-commutative 76.0%

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

   5. Simplified 76.0%

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

   6. Taylor expanded in y around inf 64.4%

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

Alternative 3: 72.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{y + 1}\\ \mathbf{if}\;y \leq -3.2 \cdot 10^{-117}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-77}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 106000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.05 \cdot 10^{+68}:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ y 1.0))))
   (if (<= y -3.2e-117)
     t_0
     (if (<= y 8.5e-77)
       x
       (if (<= y 106000.0) t_0 (if (<= y 3.05e+68) (/ x (+ y 1.0)) 1.0))))))

C

double code(double x, double y) {
	double t_0 = y / (y + 1.0);
	double tmp;
	if (y <= -3.2e-117) {
		tmp = t_0;
	} else if (y <= 8.5e-77) {
		tmp = x;
	} else if (y <= 106000.0) {
		tmp = t_0;
	} else if (y <= 3.05e+68) {
		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) :: t_0
    real(8) :: tmp
    t_0 = y / (y + 1.0d0)
    if (y <= (-3.2d-117)) then
        tmp = t_0
    else if (y <= 8.5d-77) then
        tmp = x
    else if (y <= 106000.0d0) then
        tmp = t_0
    else if (y <= 3.05d+68) 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 t_0 = y / (y + 1.0);
	double tmp;
	if (y <= -3.2e-117) {
		tmp = t_0;
	} else if (y <= 8.5e-77) {
		tmp = x;
	} else if (y <= 106000.0) {
		tmp = t_0;
	} else if (y <= 3.05e+68) {
		tmp = x / (y + 1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (y + 1.0)
	tmp = 0
	if y <= -3.2e-117:
		tmp = t_0
	elif y <= 8.5e-77:
		tmp = x
	elif y <= 106000.0:
		tmp = t_0
	elif y <= 3.05e+68:
		tmp = x / (y + 1.0)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(y + 1.0))
	tmp = 0.0
	if (y <= -3.2e-117)
		tmp = t_0;
	elseif (y <= 8.5e-77)
		tmp = x;
	elseif (y <= 106000.0)
		tmp = t_0;
	elseif (y <= 3.05e+68)
		tmp = Float64(x / Float64(y + 1.0));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / (y + 1.0);
	tmp = 0.0;
	if (y <= -3.2e-117)
		tmp = t_0;
	elseif (y <= 8.5e-77)
		tmp = x;
	elseif (y <= 106000.0)
		tmp = t_0;
	elseif (y <= 3.05e+68)
		tmp = x / (y + 1.0);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.2e-117], t$95$0, If[LessEqual[y, 8.5e-77], x, If[LessEqual[y, 106000.0], t$95$0, If[LessEqual[y, 3.05e+68], N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], 1.0]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.19999999999999995e-117 or 8.4999999999999998e-77 < y < 106000

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 76.1%

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

      1. +-commutative 76.1%

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

   5. Simplified 76.1%

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

   if -3.19999999999999995e-117 < y < 8.4999999999999998e-77

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 83.9%

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

   if 106000 < y < 3.05e68

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 76.0%

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

      1. +-commutative 76.0%

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

   5. Simplified 76.0%

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

   if 3.05e68 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 82.0%

      \[\leadsto \color{blue}{1} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 73.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y + 1}\\ \mathbf{if}\;y \leq -3 \cdot 10^{+82}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-77}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-22}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+67}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y 1.0))))
   (if (<= y -3e+82)
     1.0
     (if (<= y 8.5e-77)
       t_0
       (if (<= y 2.3e-22) y (if (<= y 3.7e+67) t_0 1.0))))))

C

double code(double x, double y) {
	double t_0 = x / (y + 1.0);
	double tmp;
	if (y <= -3e+82) {
		tmp = 1.0;
	} else if (y <= 8.5e-77) {
		tmp = t_0;
	} else if (y <= 2.3e-22) {
		tmp = y;
	} else if (y <= 3.7e+67) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = x / (y + 1.0d0)
    if (y <= (-3d+82)) then
        tmp = 1.0d0
    else if (y <= 8.5d-77) then
        tmp = t_0
    else if (y <= 2.3d-22) then
        tmp = y
    else if (y <= 3.7d+67) then
        tmp = t_0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y + 1.0);
	double tmp;
	if (y <= -3e+82) {
		tmp = 1.0;
	} else if (y <= 8.5e-77) {
		tmp = t_0;
	} else if (y <= 2.3e-22) {
		tmp = y;
	} else if (y <= 3.7e+67) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y + 1.0)
	tmp = 0
	if y <= -3e+82:
		tmp = 1.0
	elif y <= 8.5e-77:
		tmp = t_0
	elif y <= 2.3e-22:
		tmp = y
	elif y <= 3.7e+67:
		tmp = t_0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y + 1.0))
	tmp = 0.0
	if (y <= -3e+82)
		tmp = 1.0;
	elseif (y <= 8.5e-77)
		tmp = t_0;
	elseif (y <= 2.3e-22)
		tmp = y;
	elseif (y <= 3.7e+67)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y + 1.0);
	tmp = 0.0;
	if (y <= -3e+82)
		tmp = 1.0;
	elseif (y <= 8.5e-77)
		tmp = t_0;
	elseif (y <= 2.3e-22)
		tmp = y;
	elseif (y <= 3.7e+67)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3e+82], 1.0, If[LessEqual[y, 8.5e-77], t$95$0, If[LessEqual[y, 2.3e-22], y, If[LessEqual[y, 3.7e+67], t$95$0, 1.0]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.99999999999999989e82 or 3.6999999999999997e67 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 85.6%

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

   if -2.99999999999999989e82 < y < 8.4999999999999998e-77 or 2.2999999999999998e-22 < y < 3.6999999999999997e67

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 72.6%

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

      1. +-commutative 72.6%

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

   5. Simplified 72.6%

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

   if 8.4999999999999998e-77 < y < 2.2999999999999998e-22

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 85.0%

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

      1. +-commutative 85.0%

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

   5. Simplified 85.0%

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

   6. Taylor expanded in y around 0 85.0%

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

Alternative 5: 86.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x + -1}{y}\\ \mathbf{if}\;y \leq -54000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-77}:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{elif}\;y \leq 15500:\\ \;\;\;\;\frac{1}{1 + \frac{1}{y}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (+ x -1.0) y))))
   (if (<= y -54000.0)
     t_0
     (if (<= y 3.3e-77)
       (/ x (+ y 1.0))
       (if (<= y 15500.0) (/ 1.0 (+ 1.0 (/ 1.0 y))) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + ((x + -1.0) / y);
	double tmp;
	if (y <= -54000.0) {
		tmp = t_0;
	} else if (y <= 3.3e-77) {
		tmp = x / (y + 1.0);
	} else if (y <= 15500.0) {
		tmp = 1.0 / (1.0 + (1.0 / y));
	} 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 <= (-54000.0d0)) then
        tmp = t_0
    else if (y <= 3.3d-77) then
        tmp = x / (y + 1.0d0)
    else if (y <= 15500.0d0) then
        tmp = 1.0d0 / (1.0d0 + (1.0d0 / y))
    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 <= -54000.0) {
		tmp = t_0;
	} else if (y <= 3.3e-77) {
		tmp = x / (y + 1.0);
	} else if (y <= 15500.0) {
		tmp = 1.0 / (1.0 + (1.0 / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + ((x + -1.0) / y)
	tmp = 0
	if y <= -54000.0:
		tmp = t_0
	elif y <= 3.3e-77:
		tmp = x / (y + 1.0)
	elif y <= 15500.0:
		tmp = 1.0 / (1.0 + (1.0 / y))
	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 <= -54000.0)
		tmp = t_0;
	elseif (y <= 3.3e-77)
		tmp = Float64(x / Float64(y + 1.0));
	elseif (y <= 15500.0)
		tmp = Float64(1.0 / Float64(1.0 + Float64(1.0 / y)));
	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 <= -54000.0)
		tmp = t_0;
	elseif (y <= 3.3e-77)
		tmp = x / (y + 1.0);
	elseif (y <= 15500.0)
		tmp = 1.0 / (1.0 + (1.0 / y));
	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, -54000.0], t$95$0, If[LessEqual[y, 3.3e-77], N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 15500.0], N[(1.0 / N[(1.0 + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -54000 or 15500 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 98.2%

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

      1. associate--l+ 98.2%

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

      2. div-sub 98.2%

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

      3. sub-neg 98.2%

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

      4. metadata-eval 98.2%

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

   5. Simplified 98.2%

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

   if -54000 < y < 3.29999999999999991e-77

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 76.7%

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

      1. +-commutative 76.7%

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

   5. Simplified 76.7%

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

   if 3.29999999999999991e-77 < y < 15500

   1. Initial program 99.8%

      \[\frac{x + y}{y + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in y around inf 78.0%

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

      1. associate--l+ 78.0%

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

      2. div-sub 78.0%

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

   7. Simplified 78.0%

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

   8. Taylor expanded in x around 0 78.0%

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

Alternative 6: 86.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x + -1}{y}\\ \mathbf{if}\;y \leq -4500:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{-77}:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{elif}\;y \leq 23000:\\ \;\;\;\;\frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (+ x -1.0) y))))
   (if (<= y -4500.0)
     t_0
     (if (<= y 5.3e-77)
       (/ x (+ y 1.0))
       (if (<= y 23000.0) (/ y (+ 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 <= -4500.0) {
		tmp = t_0;
	} else if (y <= 5.3e-77) {
		tmp = x / (y + 1.0);
	} else if (y <= 23000.0) {
		tmp = y / (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 <= (-4500.0d0)) then
        tmp = t_0
    else if (y <= 5.3d-77) then
        tmp = x / (y + 1.0d0)
    else if (y <= 23000.0d0) then
        tmp = y / (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 <= -4500.0) {
		tmp = t_0;
	} else if (y <= 5.3e-77) {
		tmp = x / (y + 1.0);
	} else if (y <= 23000.0) {
		tmp = y / (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 <= -4500.0:
		tmp = t_0
	elif y <= 5.3e-77:
		tmp = x / (y + 1.0)
	elif y <= 23000.0:
		tmp = y / (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 <= -4500.0)
		tmp = t_0;
	elseif (y <= 5.3e-77)
		tmp = Float64(x / Float64(y + 1.0));
	elseif (y <= 23000.0)
		tmp = Float64(y / 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 <= -4500.0)
		tmp = t_0;
	elseif (y <= 5.3e-77)
		tmp = x / (y + 1.0);
	elseif (y <= 23000.0)
		tmp = y / (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, -4500.0], t$95$0, If[LessEqual[y, 5.3e-77], N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 23000.0], N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4500 or 23000 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 98.2%

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

      1. associate--l+ 98.2%

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

      2. div-sub 98.2%

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

      3. sub-neg 98.2%

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

      4. metadata-eval 98.2%

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

   5. Simplified 98.2%

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

   if -4500 < y < 5.30000000000000015e-77

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 76.7%

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

      1. +-commutative 76.7%

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

   5. Simplified 76.7%

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

   if 5.30000000000000015e-77 < y < 23000

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 78.0%

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

      1. +-commutative 78.0%

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

   5. Simplified 78.0%

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

Alternative 7: 71.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{-117}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-77}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.1)
   1.0
   (if (<= y -3.2e-117) y (if (<= y 1.45e-77) x (if (<= y 1.0) y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.1) {
		tmp = 1.0;
	} else if (y <= -3.2e-117) {
		tmp = y;
	} else if (y <= 1.45e-77) {
		tmp = x;
	} else if (y <= 1.0) {
		tmp = y;
	} 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 <= (-1.1d0)) then
        tmp = 1.0d0
    else if (y <= (-3.2d-117)) then
        tmp = y
    else if (y <= 1.45d-77) then
        tmp = x
    else if (y <= 1.0d0) then
        tmp = y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.1) {
		tmp = 1.0;
	} else if (y <= -3.2e-117) {
		tmp = y;
	} else if (y <= 1.45e-77) {
		tmp = x;
	} else if (y <= 1.0) {
		tmp = y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.1:
		tmp = 1.0
	elif y <= -3.2e-117:
		tmp = y
	elif y <= 1.45e-77:
		tmp = x
	elif y <= 1.0:
		tmp = y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.1)
		tmp = 1.0;
	elseif (y <= -3.2e-117)
		tmp = y;
	elseif (y <= 1.45e-77)
		tmp = x;
	elseif (y <= 1.0)
		tmp = y;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.1)
		tmp = 1.0;
	elseif (y <= -3.2e-117)
		tmp = y;
	elseif (y <= 1.45e-77)
		tmp = x;
	elseif (y <= 1.0)
		tmp = y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.1], 1.0, If[LessEqual[y, -3.2e-117], y, If[LessEqual[y, 1.45e-77], x, If[LessEqual[y, 1.0], y, 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.1000000000000001 or 1 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 73.1%

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

   if -1.1000000000000001 < y < -3.19999999999999995e-117 or 1.4499999999999999e-77 < y < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 67.3%

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

      1. +-commutative 67.3%

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

   5. Simplified 67.3%

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

   6. Taylor expanded in y around 0 64.6%

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

   if -3.19999999999999995e-117 < y < 1.4499999999999999e-77

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 83.9%

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

Alternative 8: 73.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{+24}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) 1.0 (if (<= y 2.25e+24) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 2.25e+24) {
		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 <= 2.25d+24) 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 <= 2.25e+24) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = 1.0
	elif y <= 2.25e+24:
		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 <= 2.25e+24)
		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 <= 2.25e+24)
		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, 2.25e+24], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 2.2500000000000001e24 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 77.5%

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

   if -1 < y < 2.2500000000000001e24

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 65.3%

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

Alternative 9: 39.0% 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 38.4%

   \[\leadsto \color{blue}{1} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024091 
(FPCore (x y)
  :name "Data.Colour.SRGB:invTransferFunction from colour-2.3.3"
  :precision binary64
  (/ (+ x y) (+ y 1.0)))