Data.Colour.SRGB:invTransferFunction from colour-2.3.3

Percentage Accurate: 100.0% → 100.0%

Time: 3.7s

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: 74.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y + 1}\\ \mathbf{if}\;y \leq -3.6 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{-53}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-66}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{-20}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+23}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y 1.0))))
   (if (<= y -3.6e+24)
     1.0
     (if (<= y -1.45e-12)
       t_0
       (if (<= y -2.3e-53)
         y
         (if (<= y 3.4e-66)
           x
           (if (<= y 9.8e-20) y (if (<= y 1.8e+23) t_0 1.0))))))))

C

double code(double x, double y) {
	double t_0 = x / (y + 1.0);
	double tmp;
	if (y <= -3.6e+24) {
		tmp = 1.0;
	} else if (y <= -1.45e-12) {
		tmp = t_0;
	} else if (y <= -2.3e-53) {
		tmp = y;
	} else if (y <= 3.4e-66) {
		tmp = x;
	} else if (y <= 9.8e-20) {
		tmp = y;
	} else if (y <= 1.8e+23) {
		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 <= (-3.6d+24)) then
        tmp = 1.0d0
    else if (y <= (-1.45d-12)) then
        tmp = t_0
    else if (y <= (-2.3d-53)) then
        tmp = y
    else if (y <= 3.4d-66) then
        tmp = x
    else if (y <= 9.8d-20) then
        tmp = y
    else if (y <= 1.8d+23) 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 <= -3.6e+24) {
		tmp = 1.0;
	} else if (y <= -1.45e-12) {
		tmp = t_0;
	} else if (y <= -2.3e-53) {
		tmp = y;
	} else if (y <= 3.4e-66) {
		tmp = x;
	} else if (y <= 9.8e-20) {
		tmp = y;
	} else if (y <= 1.8e+23) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y + 1.0)
	tmp = 0
	if y <= -3.6e+24:
		tmp = 1.0
	elif y <= -1.45e-12:
		tmp = t_0
	elif y <= -2.3e-53:
		tmp = y
	elif y <= 3.4e-66:
		tmp = x
	elif y <= 9.8e-20:
		tmp = y
	elif y <= 1.8e+23:
		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 <= -3.6e+24)
		tmp = 1.0;
	elseif (y <= -1.45e-12)
		tmp = t_0;
	elseif (y <= -2.3e-53)
		tmp = y;
	elseif (y <= 3.4e-66)
		tmp = x;
	elseif (y <= 9.8e-20)
		tmp = y;
	elseif (y <= 1.8e+23)
		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 <= -3.6e+24)
		tmp = 1.0;
	elseif (y <= -1.45e-12)
		tmp = t_0;
	elseif (y <= -2.3e-53)
		tmp = y;
	elseif (y <= 3.4e-66)
		tmp = x;
	elseif (y <= 9.8e-20)
		tmp = y;
	elseif (y <= 1.8e+23)
		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, -3.6e+24], 1.0, If[LessEqual[y, -1.45e-12], t$95$0, If[LessEqual[y, -2.3e-53], y, If[LessEqual[y, 3.4e-66], x, If[LessEqual[y, 9.8e-20], y, If[LessEqual[y, 1.8e+23], t$95$0, 1.0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.59999999999999983e24 or 1.7999999999999999e23 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 80.4%

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

   if -3.59999999999999983e24 < y < -1.4500000000000001e-12 or 9.8000000000000003e-20 < y < 1.7999999999999999e23

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 74.8%

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

      1. +-commutative 74.8%

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

   4. Simplified 74.8%

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

   if -1.4500000000000001e-12 < y < -2.3000000000000001e-53 or 3.39999999999999997e-66 < y < 9.8000000000000003e-20

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 84.9%

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

      1. +-commutative 84.9%

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

   4. Simplified 84.9%

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

   5. Taylor expanded in y around 0 84.8%

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

   if -2.3000000000000001e-53 < y < 3.39999999999999997e-66

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 85.0%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-12}:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{-53}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-66}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{-20}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+23}:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3: 86.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x + -1}{y}\\ t_1 := \frac{x}{y + 1}\\ \mathbf{if}\;y \leq -1150000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{-66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-20}:\\ \;\;\;\;\frac{y}{y + 1}\\ \mathbf{elif}\;y \leq 23000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (+ x -1.0) y))) (t_1 (/ x (+ y 1.0))))
   (if (<= y -1150000000.0)
     t_0
     (if (<= y 5.1e-66)
       t_1
       (if (<= y 3.3e-20) (/ y (+ y 1.0)) (if (<= y 23000.0) t_1 t_0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + ((x + -1.0) / y);
	double t_1 = x / (y + 1.0);
	double tmp;
	if (y <= -1150000000.0) {
		tmp = t_0;
	} else if (y <= 5.1e-66) {
		tmp = t_1;
	} else if (y <= 3.3e-20) {
		tmp = y / (y + 1.0);
	} else if (y <= 23000.0) {
		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 + (-1.0d0)) / y)
    t_1 = x / (y + 1.0d0)
    if (y <= (-1150000000.0d0)) then
        tmp = t_0
    else if (y <= 5.1d-66) then
        tmp = t_1
    else if (y <= 3.3d-20) then
        tmp = y / (y + 1.0d0)
    else if (y <= 23000.0d0) 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 + -1.0) / y);
	double t_1 = x / (y + 1.0);
	double tmp;
	if (y <= -1150000000.0) {
		tmp = t_0;
	} else if (y <= 5.1e-66) {
		tmp = t_1;
	} else if (y <= 3.3e-20) {
		tmp = y / (y + 1.0);
	} else if (y <= 23000.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + ((x + -1.0) / y)
	t_1 = x / (y + 1.0)
	tmp = 0
	if y <= -1150000000.0:
		tmp = t_0
	elif y <= 5.1e-66:
		tmp = t_1
	elif y <= 3.3e-20:
		tmp = y / (y + 1.0)
	elif y <= 23000.0:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(Float64(x + -1.0) / y))
	t_1 = Float64(x / Float64(y + 1.0))
	tmp = 0.0
	if (y <= -1150000000.0)
		tmp = t_0;
	elseif (y <= 5.1e-66)
		tmp = t_1;
	elseif (y <= 3.3e-20)
		tmp = Float64(y / Float64(y + 1.0));
	elseif (y <= 23000.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + ((x + -1.0) / y);
	t_1 = x / (y + 1.0);
	tmp = 0.0;
	if (y <= -1150000000.0)
		tmp = t_0;
	elseif (y <= 5.1e-66)
		tmp = t_1;
	elseif (y <= 3.3e-20)
		tmp = y / (y + 1.0);
	elseif (y <= 23000.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1150000000.0], t$95$0, If[LessEqual[y, 5.1e-66], t$95$1, If[LessEqual[y, 3.3e-20], N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 23000.0], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.15e9 or 23000 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around -inf 99.4%

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

      1. mul-1-neg 99.4%

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

      2. unsub-neg 99.4%

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

      3. mul-1-neg 99.4%

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

      4. sub-neg 99.4%

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

   4. Simplified 99.4%

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

   if -1.15e9 < y < 5.10000000000000022e-66 or 3.3e-20 < y < 23000

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 81.1%

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

      1. +-commutative 81.1%

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

   4. Simplified 81.1%

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

   if 5.10000000000000022e-66 < y < 3.3e-20

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 78.3%

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

      1. +-commutative 78.3%

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

   4. Simplified 78.3%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1150000000:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{-66}:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-20}:\\ \;\;\;\;\frac{y}{y + 1}\\ \mathbf{elif}\;y \leq 23000:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \end{array} \]

Alternative 4: 86.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ t_1 := \frac{x}{y + 1}\\ \mathbf{if}\;y \leq -1900000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-19}:\\ \;\;\;\;\frac{y}{y + 1}\\ \mathbf{elif}\;y \leq 1100000000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))) (t_1 (/ x (+ y 1.0))))
   (if (<= y -1900000000.0)
     t_0
     (if (<= y 2.3e-66)
       t_1
       (if (<= y 2.8e-19)
         (/ y (+ y 1.0))
         (if (<= y 1100000000000.0) 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 <= -1900000000.0) {
		tmp = t_0;
	} else if (y <= 2.3e-66) {
		tmp = t_1;
	} else if (y <= 2.8e-19) {
		tmp = y / (y + 1.0);
	} else if (y <= 1100000000000.0) {
		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 <= (-1900000000.0d0)) then
        tmp = t_0
    else if (y <= 2.3d-66) then
        tmp = t_1
    else if (y <= 2.8d-19) then
        tmp = y / (y + 1.0d0)
    else if (y <= 1100000000000.0d0) 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 <= -1900000000.0) {
		tmp = t_0;
	} else if (y <= 2.3e-66) {
		tmp = t_1;
	} else if (y <= 2.8e-19) {
		tmp = y / (y + 1.0);
	} else if (y <= 1100000000000.0) {
		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 <= -1900000000.0:
		tmp = t_0
	elif y <= 2.3e-66:
		tmp = t_1
	elif y <= 2.8e-19:
		tmp = y / (y + 1.0)
	elif y <= 1100000000000.0:
		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 <= -1900000000.0)
		tmp = t_0;
	elseif (y <= 2.3e-66)
		tmp = t_1;
	elseif (y <= 2.8e-19)
		tmp = Float64(y / Float64(y + 1.0));
	elseif (y <= 1100000000000.0)
		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 <= -1900000000.0)
		tmp = t_0;
	elseif (y <= 2.3e-66)
		tmp = t_1;
	elseif (y <= 2.8e-19)
		tmp = y / (y + 1.0);
	elseif (y <= 1100000000000.0)
		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, -1900000000.0], t$95$0, If[LessEqual[y, 2.3e-66], t$95$1, If[LessEqual[y, 2.8e-19], N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1100000000000.0], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.9e9 or 1.1e12 < 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 99.8%

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

      1. neg-mul-1 99.8%

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

   7. Simplified 99.8%

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

   if -1.9e9 < y < 2.29999999999999992e-66 or 2.80000000000000003e-19 < y < 1.1e12

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 80.8%

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

      1. +-commutative 80.8%

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

   4. Simplified 80.8%

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

   if 2.29999999999999992e-66 < y < 2.80000000000000003e-19

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 78.3%

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

      1. +-commutative 78.3%

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

   4. Simplified 78.3%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1900000000:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-66}:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-19}:\\ \;\;\;\;\frac{y}{y + 1}\\ \mathbf{elif}\;y \leq 1100000000000:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]

Alternative 5: 73.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -550000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.35 \cdot 10^{-49}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-67}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-20}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 4.3:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -550000000.0)
   1.0
   (if (<= y -1.35e-49)
     y
     (if (<= y 1.3e-67) x (if (<= y 1.85e-20) y (if (<= y 4.3) x 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -550000000.0) {
		tmp = 1.0;
	} else if (y <= -1.35e-49) {
		tmp = y;
	} else if (y <= 1.3e-67) {
		tmp = x;
	} else if (y <= 1.85e-20) {
		tmp = y;
	} else if (y <= 4.3) {
		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 <= (-550000000.0d0)) then
        tmp = 1.0d0
    else if (y <= (-1.35d-49)) then
        tmp = y
    else if (y <= 1.3d-67) then
        tmp = x
    else if (y <= 1.85d-20) then
        tmp = y
    else if (y <= 4.3d0) 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 <= -550000000.0) {
		tmp = 1.0;
	} else if (y <= -1.35e-49) {
		tmp = y;
	} else if (y <= 1.3e-67) {
		tmp = x;
	} else if (y <= 1.85e-20) {
		tmp = y;
	} else if (y <= 4.3) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -550000000.0:
		tmp = 1.0
	elif y <= -1.35e-49:
		tmp = y
	elif y <= 1.3e-67:
		tmp = x
	elif y <= 1.85e-20:
		tmp = y
	elif y <= 4.3:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -550000000.0)
		tmp = 1.0;
	elseif (y <= -1.35e-49)
		tmp = y;
	elseif (y <= 1.3e-67)
		tmp = x;
	elseif (y <= 1.85e-20)
		tmp = y;
	elseif (y <= 4.3)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -550000000.0)
		tmp = 1.0;
	elseif (y <= -1.35e-49)
		tmp = y;
	elseif (y <= 1.3e-67)
		tmp = x;
	elseif (y <= 1.85e-20)
		tmp = y;
	elseif (y <= 4.3)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -550000000.0], 1.0, If[LessEqual[y, -1.35e-49], y, If[LessEqual[y, 1.3e-67], x, If[LessEqual[y, 1.85e-20], y, If[LessEqual[y, 4.3], x, 1.0]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.5e8 or 4.29999999999999982 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 75.3%

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

   if -5.5e8 < y < -1.35e-49 or 1.2999999999999999e-67 < y < 1.85e-20

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 67.7%

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

      1. +-commutative 67.7%

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

   4. Simplified 67.7%

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

   5. Taylor expanded in y around 0 67.8%

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

   if -1.35e-49 < y < 1.2999999999999999e-67 or 1.85e-20 < y < 4.29999999999999982

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 83.0%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -550000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.35 \cdot 10^{-49}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-67}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-20}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 4.3:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 74.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+35} \lor \neg \left(x \leq 1.05 \cdot 10^{+24}\right):\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -6e+35) (not (<= x 1.05e+24))) (/ x (+ y 1.0)) (/ y (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -6e+35) || !(x <= 1.05e+24)) {
		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 <= (-6d+35)) .or. (.not. (x <= 1.05d+24))) 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 <= -6e+35) || !(x <= 1.05e+24)) {
		tmp = x / (y + 1.0);
	} else {
		tmp = y / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -6e+35) or not (x <= 1.05e+24):
		tmp = x / (y + 1.0)
	else:
		tmp = y / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -6e+35) || !(x <= 1.05e+24))
		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 <= -6e+35) || ~((x <= 1.05e+24)))
		tmp = x / (y + 1.0);
	else
		tmp = y / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -6e+35], N[Not[LessEqual[x, 1.05e+24]], $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 < -5.99999999999999981e35 or 1.0500000000000001e24 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 86.0%

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

      1. +-commutative 86.0%

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

   4. Simplified 86.0%

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

   if -5.99999999999999981e35 < x < 1.0500000000000001e24

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 74.9%

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

      1. +-commutative 74.9%

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

   4. Simplified 74.9%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+35} \lor \neg \left(x \leq 1.05 \cdot 10^{+24}\right):\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + 1}\\ \end{array} \]

Alternative 7: 74.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1.0) 1.0 (if (<= y 3.5) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 3.5) {
		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 <= 3.5d0) 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 <= 3.5) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = 1.0
	elif y <= 3.5:
		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 <= 3.5)
		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 <= 3.5)
		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, 3.5], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 3.5 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 73.6%

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

   if -1 < y < 3.5

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 75.4%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 38.7% 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 37.3%

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

3. Final simplification 37.3%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023202 
(FPCore (x y)
  :name "Data.Colour.SRGB:invTransferFunction from colour-2.3.3"
  :precision binary64
  (/ (+ x y) (+ y 1.0)))