Data.Colour.SRGB:invTransferFunction from colour-2.3.3

Percentage Accurate: 100.0% → 100.0%

Time: 3.3s

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. Final simplification 100.0%

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

Alternative 2: 72.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y + 1}\\ \mathbf{if}\;y \leq -1.1 \cdot 10^{+77}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.26 \cdot 10^{-33}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.95 \cdot 10^{-94}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{-10}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+79}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y 1.0))))
   (if (<= y -1.1e+77)
     1.0
     (if (<= y -1.26e-33)
       t_0
       (if (<= y -1.95e-94)
         y
         (if (<= y 1.3e-57)
           t_0
           (if (<= y 1.16e-10) y (if (<= y 1.12e+79) t_0 1.0))))))))

C

double code(double x, double y) {
	double t_0 = x / (y + 1.0);
	double tmp;
	if (y <= -1.1e+77) {
		tmp = 1.0;
	} else if (y <= -1.26e-33) {
		tmp = t_0;
	} else if (y <= -1.95e-94) {
		tmp = y;
	} else if (y <= 1.3e-57) {
		tmp = t_0;
	} else if (y <= 1.16e-10) {
		tmp = y;
	} else if (y <= 1.12e+79) {
		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 <= (-1.1d+77)) then
        tmp = 1.0d0
    else if (y <= (-1.26d-33)) then
        tmp = t_0
    else if (y <= (-1.95d-94)) then
        tmp = y
    else if (y <= 1.3d-57) then
        tmp = t_0
    else if (y <= 1.16d-10) then
        tmp = y
    else if (y <= 1.12d+79) 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 <= -1.1e+77) {
		tmp = 1.0;
	} else if (y <= -1.26e-33) {
		tmp = t_0;
	} else if (y <= -1.95e-94) {
		tmp = y;
	} else if (y <= 1.3e-57) {
		tmp = t_0;
	} else if (y <= 1.16e-10) {
		tmp = y;
	} else if (y <= 1.12e+79) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y + 1.0)
	tmp = 0
	if y <= -1.1e+77:
		tmp = 1.0
	elif y <= -1.26e-33:
		tmp = t_0
	elif y <= -1.95e-94:
		tmp = y
	elif y <= 1.3e-57:
		tmp = t_0
	elif y <= 1.16e-10:
		tmp = y
	elif y <= 1.12e+79:
		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 <= -1.1e+77)
		tmp = 1.0;
	elseif (y <= -1.26e-33)
		tmp = t_0;
	elseif (y <= -1.95e-94)
		tmp = y;
	elseif (y <= 1.3e-57)
		tmp = t_0;
	elseif (y <= 1.16e-10)
		tmp = y;
	elseif (y <= 1.12e+79)
		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 <= -1.1e+77)
		tmp = 1.0;
	elseif (y <= -1.26e-33)
		tmp = t_0;
	elseif (y <= -1.95e-94)
		tmp = y;
	elseif (y <= 1.3e-57)
		tmp = t_0;
	elseif (y <= 1.16e-10)
		tmp = y;
	elseif (y <= 1.12e+79)
		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, -1.1e+77], 1.0, If[LessEqual[y, -1.26e-33], t$95$0, If[LessEqual[y, -1.95e-94], y, If[LessEqual[y, 1.3e-57], t$95$0, If[LessEqual[y, 1.16e-10], y, If[LessEqual[y, 1.12e+79], t$95$0, 1.0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.1e77 or 1.12e79 < y

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]

   2. Taylor expanded in y around inf 85.0%

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

   if -1.1e77 < y < -1.26000000000000005e-33 or -1.9500000000000001e-94 < y < 1.29999999999999993e-57 or 1.16e-10 < y < 1.12e79

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]

   2. Taylor expanded in x around inf 82.6%

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

      1. +-commutative 82.6%

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

   4. Simplified 82.6%

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

   if -1.26000000000000005e-33 < y < -1.9500000000000001e-94 or 1.29999999999999993e-57 < y < 1.16e-10

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]

   2. Taylor expanded in x around 0 70.0%

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

      1. +-commutative 70.0%

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

   4. Simplified 70.0%

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

   5. Taylor expanded in y around 0 68.6%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+77}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.26 \cdot 10^{-33}:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{elif}\;y \leq -1.95 \cdot 10^{-94}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-57}:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{-10}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+79}:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3: 72.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y + 1}\\ \mathbf{if}\;y \leq -5.4 \cdot 10^{+76}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.42 \cdot 10^{-33}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.15 \cdot 10^{-94}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-9}:\\ \;\;\;\;\frac{y}{y + 1}\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{+79}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y 1.0))))
   (if (<= y -5.4e+76)
     1.0
     (if (<= y -1.42e-33)
       t_0
       (if (<= y -2.15e-94)
         y
         (if (<= y 1.3e-55)
           t_0
           (if (<= y 1.3e-9)
             (/ y (+ y 1.0))
             (if (<= y 1.22e+79) t_0 1.0))))))))

C

double code(double x, double y) {
	double t_0 = x / (y + 1.0);
	double tmp;
	if (y <= -5.4e+76) {
		tmp = 1.0;
	} else if (y <= -1.42e-33) {
		tmp = t_0;
	} else if (y <= -2.15e-94) {
		tmp = y;
	} else if (y <= 1.3e-55) {
		tmp = t_0;
	} else if (y <= 1.3e-9) {
		tmp = y / (y + 1.0);
	} else if (y <= 1.22e+79) {
		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 <= (-5.4d+76)) then
        tmp = 1.0d0
    else if (y <= (-1.42d-33)) then
        tmp = t_0
    else if (y <= (-2.15d-94)) then
        tmp = y
    else if (y <= 1.3d-55) then
        tmp = t_0
    else if (y <= 1.3d-9) then
        tmp = y / (y + 1.0d0)
    else if (y <= 1.22d+79) 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 <= -5.4e+76) {
		tmp = 1.0;
	} else if (y <= -1.42e-33) {
		tmp = t_0;
	} else if (y <= -2.15e-94) {
		tmp = y;
	} else if (y <= 1.3e-55) {
		tmp = t_0;
	} else if (y <= 1.3e-9) {
		tmp = y / (y + 1.0);
	} else if (y <= 1.22e+79) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y + 1.0)
	tmp = 0
	if y <= -5.4e+76:
		tmp = 1.0
	elif y <= -1.42e-33:
		tmp = t_0
	elif y <= -2.15e-94:
		tmp = y
	elif y <= 1.3e-55:
		tmp = t_0
	elif y <= 1.3e-9:
		tmp = y / (y + 1.0)
	elif y <= 1.22e+79:
		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 <= -5.4e+76)
		tmp = 1.0;
	elseif (y <= -1.42e-33)
		tmp = t_0;
	elseif (y <= -2.15e-94)
		tmp = y;
	elseif (y <= 1.3e-55)
		tmp = t_0;
	elseif (y <= 1.3e-9)
		tmp = Float64(y / Float64(y + 1.0));
	elseif (y <= 1.22e+79)
		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 <= -5.4e+76)
		tmp = 1.0;
	elseif (y <= -1.42e-33)
		tmp = t_0;
	elseif (y <= -2.15e-94)
		tmp = y;
	elseif (y <= 1.3e-55)
		tmp = t_0;
	elseif (y <= 1.3e-9)
		tmp = y / (y + 1.0);
	elseif (y <= 1.22e+79)
		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, -5.4e+76], 1.0, If[LessEqual[y, -1.42e-33], t$95$0, If[LessEqual[y, -2.15e-94], y, If[LessEqual[y, 1.3e-55], t$95$0, If[LessEqual[y, 1.3e-9], N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.22e+79], t$95$0, 1.0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -5.3999999999999998e76 or 1.22000000000000002e79 < y

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]

   2. Taylor expanded in y around inf 85.0%

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

   if -5.3999999999999998e76 < y < -1.42000000000000007e-33 or -2.1499999999999999e-94 < y < 1.2999999999999999e-55 or 1.3000000000000001e-9 < y < 1.22000000000000002e79

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]

   2. Taylor expanded in x around inf 82.6%

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

      1. +-commutative 82.6%

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

   4. Simplified 82.6%

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

   if -1.42000000000000007e-33 < y < -2.1499999999999999e-94

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]

   2. Taylor expanded in x around 0 70.6%

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

      1. +-commutative 70.6%

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

   4. Simplified 70.6%

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

   5. Taylor expanded in y around 0 70.6%

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

   if 1.2999999999999999e-55 < y < 1.3000000000000001e-9

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]

   2. Taylor expanded in x around 0 69.3%

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

      1. +-commutative 69.3%

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

   4. Simplified 69.3%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{+76}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.42 \cdot 10^{-33}:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{elif}\;y \leq -2.15 \cdot 10^{-94}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-55}:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-9}:\\ \;\;\;\;\frac{y}{y + 1}\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{+79}:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 86.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y + 1}\\ \mathbf{if}\;y \leq -1900000:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-56}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-8}:\\ \;\;\;\;\frac{y}{y + 1}\\ \mathbf{elif}\;y \leq 10200000000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y 1.0))))
   (if (<= y -1900000.0)
     (+ 1.0 (/ x y))
     (if (<= y 4.2e-56)
       t_0
       (if (<= y 1.1e-8)
         (/ y (+ y 1.0))
         (if (<= y 10200000000000.0) t_0 (+ 1.0 (/ (+ x -1.0) y))))))))

C

double code(double x, double y) {
	double t_0 = x / (y + 1.0);
	double tmp;
	if (y <= -1900000.0) {
		tmp = 1.0 + (x / y);
	} else if (y <= 4.2e-56) {
		tmp = t_0;
	} else if (y <= 1.1e-8) {
		tmp = y / (y + 1.0);
	} else if (y <= 10200000000000.0) {
		tmp = t_0;
	} else {
		tmp = 1.0 + ((x + -1.0) / 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 / (y + 1.0d0)
    if (y <= (-1900000.0d0)) then
        tmp = 1.0d0 + (x / y)
    else if (y <= 4.2d-56) then
        tmp = t_0
    else if (y <= 1.1d-8) then
        tmp = y / (y + 1.0d0)
    else if (y <= 10200000000000.0d0) then
        tmp = t_0
    else
        tmp = 1.0d0 + ((x + (-1.0d0)) / y)
    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 <= -1900000.0) {
		tmp = 1.0 + (x / y);
	} else if (y <= 4.2e-56) {
		tmp = t_0;
	} else if (y <= 1.1e-8) {
		tmp = y / (y + 1.0);
	} else if (y <= 10200000000000.0) {
		tmp = t_0;
	} else {
		tmp = 1.0 + ((x + -1.0) / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y + 1.0)
	tmp = 0
	if y <= -1900000.0:
		tmp = 1.0 + (x / y)
	elif y <= 4.2e-56:
		tmp = t_0
	elif y <= 1.1e-8:
		tmp = y / (y + 1.0)
	elif y <= 10200000000000.0:
		tmp = t_0
	else:
		tmp = 1.0 + ((x + -1.0) / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y + 1.0))
	tmp = 0.0
	if (y <= -1900000.0)
		tmp = Float64(1.0 + Float64(x / y));
	elseif (y <= 4.2e-56)
		tmp = t_0;
	elseif (y <= 1.1e-8)
		tmp = Float64(y / Float64(y + 1.0));
	elseif (y <= 10200000000000.0)
		tmp = t_0;
	else
		tmp = Float64(1.0 + Float64(Float64(x + -1.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y + 1.0);
	tmp = 0.0;
	if (y <= -1900000.0)
		tmp = 1.0 + (x / y);
	elseif (y <= 4.2e-56)
		tmp = t_0;
	elseif (y <= 1.1e-8)
		tmp = y / (y + 1.0);
	elseif (y <= 10200000000000.0)
		tmp = t_0;
	else
		tmp = 1.0 + ((x + -1.0) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1900000.0], N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.2e-56], t$95$0, If[LessEqual[y, 1.1e-8], N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 10200000000000.0], t$95$0, N[(1.0 + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.9e6

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]

   2. Taylor expanded in y around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. mul-1-neg 100.0%

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

      4. sub-neg 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. neg-mul-1 100.0%

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

      2. distribute-neg-frac 100.0%

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

   7. Simplified 100.0%

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

   if -1.9e6 < y < 4.20000000000000012e-56 or 1.0999999999999999e-8 < y < 1.02e13

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]

   2. Taylor expanded in x around inf 83.0%

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

      1. +-commutative 83.0%

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

   4. Simplified 83.0%

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

   if 4.20000000000000012e-56 < y < 1.0999999999999999e-8

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]

   2. Taylor expanded in x around 0 69.3%

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

      1. +-commutative 69.3%

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

   4. Simplified 69.3%

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

   if 1.02e13 < y

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]

   2. Taylor expanded in y around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. mul-1-neg 100.0%

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

      4. sub-neg 100.0%

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

   4. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \frac{1 - x}{y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1900000:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-56}:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-8}:\\ \;\;\;\;\frac{y}{y + 1}\\ \mathbf{elif}\;y \leq 10200000000000:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \end{array} \]

Alternative 5: 86.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ t_1 := \frac{x}{y + 1}\\ \mathbf{if}\;y \leq -850:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-10}:\\ \;\;\;\;\frac{y}{y + 1}\\ \mathbf{elif}\;y \leq 10200000000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))) (t_1 (/ x (+ y 1.0))))
   (if (<= y -850.0)
     t_0
     (if (<= y 1.35e-55)
       t_1
       (if (<= y 1.7e-10)
         (/ y (+ y 1.0))
         (if (<= y 10200000000000.0) t_1 t_0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double t_1 = x / (y + 1.0);
	double tmp;
	if (y <= -850.0) {
		tmp = t_0;
	} else if (y <= 1.35e-55) {
		tmp = t_1;
	} else if (y <= 1.7e-10) {
		tmp = y / (y + 1.0);
	} else if (y <= 10200000000000.0) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (x / y)
    t_1 = x / (y + 1.0d0)
    if (y <= (-850.0d0)) then
        tmp = t_0
    else if (y <= 1.35d-55) then
        tmp = t_1
    else if (y <= 1.7d-10) then
        tmp = y / (y + 1.0d0)
    else if (y <= 10200000000000.0d0) then
        tmp = t_1
    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 t_1 = x / (y + 1.0);
	double tmp;
	if (y <= -850.0) {
		tmp = t_0;
	} else if (y <= 1.35e-55) {
		tmp = t_1;
	} else if (y <= 1.7e-10) {
		tmp = y / (y + 1.0);
	} else if (y <= 10200000000000.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x / y)
	t_1 = x / (y + 1.0)
	tmp = 0
	if y <= -850.0:
		tmp = t_0
	elif y <= 1.35e-55:
		tmp = t_1
	elif y <= 1.7e-10:
		tmp = y / (y + 1.0)
	elif y <= 10200000000000.0:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / y))
	t_1 = Float64(x / Float64(y + 1.0))
	tmp = 0.0
	if (y <= -850.0)
		tmp = t_0;
	elseif (y <= 1.35e-55)
		tmp = t_1;
	elseif (y <= 1.7e-10)
		tmp = Float64(y / Float64(y + 1.0));
	elseif (y <= 10200000000000.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (x / y);
	t_1 = x / (y + 1.0);
	tmp = 0.0;
	if (y <= -850.0)
		tmp = t_0;
	elseif (y <= 1.35e-55)
		tmp = t_1;
	elseif (y <= 1.7e-10)
		tmp = y / (y + 1.0);
	elseif (y <= 10200000000000.0)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -850.0], t$95$0, If[LessEqual[y, 1.35e-55], t$95$1, If[LessEqual[y, 1.7e-10], N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 10200000000000.0], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -850 or 1.02e13 < y

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]

   2. Taylor expanded in y around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. mul-1-neg 100.0%

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

      4. sub-neg 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around inf 99.9%

      \[\leadsto 1 - \color{blue}{-1 \cdot \frac{x}{y}} \]
   6. 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%

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

   7. Simplified 99.9%

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

   if -850 < y < 1.35000000000000002e-55 or 1.70000000000000007e-10 < y < 1.02e13

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]

   2. Taylor expanded in x around inf 83.0%

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

      1. +-commutative 83.0%

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

   4. Simplified 83.0%

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

   if 1.35000000000000002e-55 < y < 1.70000000000000007e-10

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]

   2. Taylor expanded in x around 0 69.3%

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

      1. +-commutative 69.3%

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

   4. Simplified 69.3%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -850:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-55}:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-10}:\\ \;\;\;\;\frac{y}{y + 1}\\ \mathbf{elif}\;y \leq 10200000000000:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]

Alternative 6: 73.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 10^{-55}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-8}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 10200000000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   1.0
   (if (<= y 1e-55)
     x
     (if (<= y 3.1e-8) y (if (<= y 10200000000000.0) x 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 1e-55) {
		tmp = x;
	} else if (y <= 3.1e-8) {
		tmp = y;
	} else if (y <= 10200000000000.0) {
		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 <= 1d-55) then
        tmp = x
    else if (y <= 3.1d-8) then
        tmp = y
    else if (y <= 10200000000000.0d0) 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 <= 1e-55) {
		tmp = x;
	} else if (y <= 3.1e-8) {
		tmp = y;
	} else if (y <= 10200000000000.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = 1.0
	elif y <= 1e-55:
		tmp = x
	elif y <= 3.1e-8:
		tmp = y
	elif y <= 10200000000000.0:
		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 <= 1e-55)
		tmp = x;
	elseif (y <= 3.1e-8)
		tmp = y;
	elseif (y <= 10200000000000.0)
		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 <= 1e-55)
		tmp = x;
	elseif (y <= 3.1e-8)
		tmp = y;
	elseif (y <= 10200000000000.0)
		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, 1e-55], x, If[LessEqual[y, 3.1e-8], y, If[LessEqual[y, 10200000000000.0], x, 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 1.02e13 < y

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]

   2. Taylor expanded in y around inf 73.9%

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

   if -1 < y < 9.99999999999999995e-56 or 3.1e-8 < y < 1.02e13

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]

   2. Taylor expanded in y around 0 79.8%

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

   if 9.99999999999999995e-56 < y < 3.1e-8

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]

   2. Taylor expanded in x around 0 69.3%

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

      1. +-commutative 69.3%

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

   4. Simplified 69.3%

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

   5. Taylor expanded in y around 0 66.2%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 10^{-55}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-8}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 10200000000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7: 73.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 10200000000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) 1.0 (if (<= y 10200000000000.0) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 10200000000000.0) {
		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 <= 10200000000000.0d0) 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 <= 10200000000000.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = 1.0
	elif y <= 10200000000000.0:
		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 <= 10200000000000.0)
		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 <= 10200000000000.0)
		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, 10200000000000.0], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.02e13 < y

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]

   2. Taylor expanded in y around inf 73.9%

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

   if -1 < y < 1.02e13

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]

   2. Taylor expanded in y around 0 75.7%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 10200000000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

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. Taylor expanded in y around inf 37.2%

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

3. Final simplification 37.2%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023271 
(FPCore (x y)
  :name "Data.Colour.SRGB:invTransferFunction from colour-2.3.3"
  :precision binary64
  (/ (+ x y) (+ y 1.0)))