Data.Colour.SRGB:invTransferFunction from colour-2.3.3

Percentage Accurate: 100.0% → 100.0%

Time: 3.6s

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. Final simplification 100.0%

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

Alternative 2: 73.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y + 1}\\ \mathbf{if}\;x \leq -2.8 \cdot 10^{+78}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3950000000000:\\ \;\;\;\;\frac{y}{y + 1}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+42}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+69}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y 1.0))))
   (if (<= x -2.8e+78)
     t_0
     (if (<= x 3950000000000.0)
       (/ y (+ y 1.0))
       (if (<= x 9e+42) x (if (<= x 8e+69) 1.0 t_0))))))

C

double code(double x, double y) {
	double t_0 = x / (y + 1.0);
	double tmp;
	if (x <= -2.8e+78) {
		tmp = t_0;
	} else if (x <= 3950000000000.0) {
		tmp = y / (y + 1.0);
	} else if (x <= 9e+42) {
		tmp = x;
	} else if (x <= 8e+69) {
		tmp = 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 = x / (y + 1.0d0)
    if (x <= (-2.8d+78)) then
        tmp = t_0
    else if (x <= 3950000000000.0d0) then
        tmp = y / (y + 1.0d0)
    else if (x <= 9d+42) then
        tmp = x
    else if (x <= 8d+69) then
        tmp = 1.0d0
    else
        tmp = t_0
    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 (x <= -2.8e+78) {
		tmp = t_0;
	} else if (x <= 3950000000000.0) {
		tmp = y / (y + 1.0);
	} else if (x <= 9e+42) {
		tmp = x;
	} else if (x <= 8e+69) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y + 1.0)
	tmp = 0
	if x <= -2.8e+78:
		tmp = t_0
	elif x <= 3950000000000.0:
		tmp = y / (y + 1.0)
	elif x <= 9e+42:
		tmp = x
	elif x <= 8e+69:
		tmp = 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y + 1.0))
	tmp = 0.0
	if (x <= -2.8e+78)
		tmp = t_0;
	elseif (x <= 3950000000000.0)
		tmp = Float64(y / Float64(y + 1.0));
	elseif (x <= 9e+42)
		tmp = x;
	elseif (x <= 8e+69)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y + 1.0);
	tmp = 0.0;
	if (x <= -2.8e+78)
		tmp = t_0;
	elseif (x <= 3950000000000.0)
		tmp = y / (y + 1.0);
	elseif (x <= 9e+42)
		tmp = x;
	elseif (x <= 8e+69)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.8e+78], t$95$0, If[LessEqual[x, 3950000000000.0], N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9e+42], x, If[LessEqual[x, 8e+69], 1.0, t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.8000000000000001e78 or 8.0000000000000006e69 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 82.1%

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

      1. +-commutative 82.1%

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

   4. Simplified 82.1%

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

   if -2.8000000000000001e78 < x < 3.95e12

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 79.9%

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

      1. +-commutative 79.9%

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

   4. Simplified 79.9%

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

   if 3.95e12 < x < 9.00000000000000025e42

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 86.8%

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

   if 9.00000000000000025e42 < x < 8.0000000000000006e69

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 100.0%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+78}:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{elif}\;x \leq 3950000000000:\\ \;\;\;\;\frac{y}{y + 1}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+42}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+69}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + 1}\\ \end{array} \]

Alternative 3: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{1 - x}{y}\\ \mathbf{if}\;y \leq -150:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 8.3 \cdot 10^{-112}:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{elif}\;y \leq 31000000:\\ \;\;\;\;\frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ (- 1.0 x) y))))
   (if (<= y -150.0)
     t_0
     (if (<= y 8.3e-112)
       (/ x (+ y 1.0))
       (if (<= y 31000000.0) (/ y (+ y 1.0)) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 - ((1.0 - x) / y);
	double tmp;
	if (y <= -150.0) {
		tmp = t_0;
	} else if (y <= 8.3e-112) {
		tmp = x / (y + 1.0);
	} else if (y <= 31000000.0) {
		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 - ((1.0d0 - x) / y)
    if (y <= (-150.0d0)) then
        tmp = t_0
    else if (y <= 8.3d-112) then
        tmp = x / (y + 1.0d0)
    else if (y <= 31000000.0d0) 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 - ((1.0 - x) / y);
	double tmp;
	if (y <= -150.0) {
		tmp = t_0;
	} else if (y <= 8.3e-112) {
		tmp = x / (y + 1.0);
	} else if (y <= 31000000.0) {
		tmp = y / (y + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - ((1.0 - x) / y)
	tmp = 0
	if y <= -150.0:
		tmp = t_0
	elif y <= 8.3e-112:
		tmp = x / (y + 1.0)
	elif y <= 31000000.0:
		tmp = y / (y + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(Float64(1.0 - x) / y))
	tmp = 0.0
	if (y <= -150.0)
		tmp = t_0;
	elseif (y <= 8.3e-112)
		tmp = Float64(x / Float64(y + 1.0));
	elseif (y <= 31000000.0)
		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 - ((1.0 - x) / y);
	tmp = 0.0;
	if (y <= -150.0)
		tmp = t_0;
	elseif (y <= 8.3e-112)
		tmp = x / (y + 1.0);
	elseif (y <= 31000000.0)
		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[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -150.0], t$95$0, If[LessEqual[y, 8.3e-112], N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 31000000.0], N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -150 or 3.1e7 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around -inf 99.6%

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

      1. mul-1-neg 99.6%

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

      2. unsub-neg 99.6%

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

      3. mul-1-neg 99.6%

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

      4. sub-neg 99.6%

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

   4. Simplified 99.6%

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

   if -150 < y < 8.29999999999999961e-112

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 77.8%

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

      1. +-commutative 77.8%

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

   4. Simplified 77.8%

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

   if 8.29999999999999961e-112 < y < 3.1e7

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 69.1%

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

      1. +-commutative 69.1%

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

   4. Simplified 69.1%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -150:\\ \;\;\;\;1 - \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 8.3 \cdot 10^{-112}:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{elif}\;y \leq 31000000:\\ \;\;\;\;\frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1 - x}{y}\\ \end{array} \]

Alternative 4: 72.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-111}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.76:\\ \;\;\;\;y - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   1.0
   (if (<= y 1.7e-111) x (if (<= y 0.76) (- y (* y y)) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 1.7e-111) {
		tmp = x;
	} else if (y <= 0.76) {
		tmp = y - (y * 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 <= (-1.0d0)) then
        tmp = 1.0d0
    else if (y <= 1.7d-111) then
        tmp = x
    else if (y <= 0.76d0) then
        tmp = y - (y * y)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 1.7e-111) {
		tmp = x;
	} else if (y <= 0.76) {
		tmp = y - (y * y);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = 1.0
	elif y <= 1.7e-111:
		tmp = x
	elif y <= 0.76:
		tmp = y - (y * y)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = 1.0;
	elseif (y <= 1.7e-111)
		tmp = x;
	elseif (y <= 0.76)
		tmp = Float64(y - Float64(y * y));
	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 <= 1.7e-111)
		tmp = x;
	elseif (y <= 0.76)
		tmp = y - (y * y);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], 1.0, If[LessEqual[y, 1.7e-111], x, If[LessEqual[y, 0.76], N[(y - N[(y * y), $MachinePrecision]), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 0.76000000000000001 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 75.9%

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

   if -1 < y < 1.69999999999999998e-111

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 76.0%

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

   if 1.69999999999999998e-111 < y < 0.76000000000000001

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 67.4%

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

      1. +-commutative 67.4%

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

   4. Simplified 67.4%

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

   5. Taylor expanded in y around 0 67.6%

      \[\leadsto \color{blue}{-1 \cdot {y}^{2} + y} \]
   6. Step-by-step derivation

      1. neg-mul-1 67.6%

         \[\leadsto \color{blue}{\left(-{y}^{2}\right)} + y \]

      2. +-commutative 67.6%

         \[\leadsto \color{blue}{y + \left(-{y}^{2}\right)} \]

      3. unsub-neg 67.6%

         \[\leadsto \color{blue}{y - {y}^{2}} \]

      4. unpow2 67.6%

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

   7. Simplified 67.6%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-111}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.76:\\ \;\;\;\;y - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 73.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -150:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-111}:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{elif}\;y \leq 0.76:\\ \;\;\;\;y - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -150.0)
   1.0
   (if (<= y 1.25e-111) (/ x (+ y 1.0)) (if (<= y 0.76) (- y (* y y)) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -150.0) {
		tmp = 1.0;
	} else if (y <= 1.25e-111) {
		tmp = x / (y + 1.0);
	} else if (y <= 0.76) {
		tmp = y - (y * 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 <= (-150.0d0)) then
        tmp = 1.0d0
    else if (y <= 1.25d-111) then
        tmp = x / (y + 1.0d0)
    else if (y <= 0.76d0) then
        tmp = y - (y * y)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -150.0) {
		tmp = 1.0;
	} else if (y <= 1.25e-111) {
		tmp = x / (y + 1.0);
	} else if (y <= 0.76) {
		tmp = y - (y * y);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -150.0:
		tmp = 1.0
	elif y <= 1.25e-111:
		tmp = x / (y + 1.0)
	elif y <= 0.76:
		tmp = y - (y * y)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -150.0)
		tmp = 1.0;
	elseif (y <= 1.25e-111)
		tmp = Float64(x / Float64(y + 1.0));
	elseif (y <= 0.76)
		tmp = Float64(y - Float64(y * y));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -150.0)
		tmp = 1.0;
	elseif (y <= 1.25e-111)
		tmp = x / (y + 1.0);
	elseif (y <= 0.76)
		tmp = y - (y * y);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -150.0], 1.0, If[LessEqual[y, 1.25e-111], N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.76], N[(y - N[(y * y), $MachinePrecision]), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -150 or 0.76000000000000001 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 75.9%

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

   if -150 < y < 1.2500000000000001e-111

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 77.8%

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

      1. +-commutative 77.8%

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

   4. Simplified 77.8%

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

   if 1.2500000000000001e-111 < y < 0.76000000000000001

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 67.4%

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

      1. +-commutative 67.4%

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

   4. Simplified 67.4%

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

   5. Taylor expanded in y around 0 67.6%

      \[\leadsto \color{blue}{-1 \cdot {y}^{2} + y} \]
   6. Step-by-step derivation

      1. neg-mul-1 67.6%

         \[\leadsto \color{blue}{\left(-{y}^{2}\right)} + y \]

      2. +-commutative 67.6%

         \[\leadsto \color{blue}{y + \left(-{y}^{2}\right)} \]

      3. unsub-neg 67.6%

         \[\leadsto \color{blue}{y - {y}^{2}} \]

      4. unpow2 67.6%

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

   7. Simplified 67.6%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -150:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-111}:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{elif}\;y \leq 0.76:\\ \;\;\;\;y - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 84.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;y \leq -150:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-111}:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+14}:\\ \;\;\;\;\frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))))
   (if (<= y -150.0)
     t_0
     (if (<= y 1.7e-111)
       (/ x (+ y 1.0))
       (if (<= y 1.65e+14) (/ y (+ y 1.0)) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (y <= -150.0) {
		tmp = t_0;
	} else if (y <= 1.7e-111) {
		tmp = x / (y + 1.0);
	} else if (y <= 1.65e+14) {
		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 / y)
    if (y <= (-150.0d0)) then
        tmp = t_0
    else if (y <= 1.7d-111) then
        tmp = x / (y + 1.0d0)
    else if (y <= 1.65d+14) 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 / y);
	double tmp;
	if (y <= -150.0) {
		tmp = t_0;
	} else if (y <= 1.7e-111) {
		tmp = x / (y + 1.0);
	} else if (y <= 1.65e+14) {
		tmp = y / (y + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x / y)
	tmp = 0
	if y <= -150.0:
		tmp = t_0
	elif y <= 1.7e-111:
		tmp = x / (y + 1.0)
	elif y <= 1.65e+14:
		tmp = y / (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 <= -150.0)
		tmp = t_0;
	elseif (y <= 1.7e-111)
		tmp = Float64(x / Float64(y + 1.0));
	elseif (y <= 1.65e+14)
		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 / y);
	tmp = 0.0;
	if (y <= -150.0)
		tmp = t_0;
	elseif (y <= 1.7e-111)
		tmp = x / (y + 1.0);
	elseif (y <= 1.65e+14)
		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[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -150.0], t$95$0, If[LessEqual[y, 1.7e-111], N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.65e+14], N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -150 or 1.65e14 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around -inf 99.6%

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

      1. mul-1-neg 99.6%

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

      2. unsub-neg 99.6%

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

      3. mul-1-neg 99.6%

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

      4. sub-neg 99.6%

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

   4. Simplified 99.6%

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

   5. Taylor expanded in x around inf 99.0%

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

      1. neg-mul-1 99.0%

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

      2. distribute-neg-frac 99.0%

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

   7. Simplified 99.0%

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

   if -150 < y < 1.69999999999999998e-111

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 77.8%

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

      1. +-commutative 77.8%

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

   4. Simplified 77.8%

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

   if 1.69999999999999998e-111 < y < 1.65e14

   1. Initial program 99.8%

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

   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%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -150:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-111}:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+14}:\\ \;\;\;\;\frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]

Alternative 7: 72.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-111}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) 1.0 (if (<= y 1.25e-111) x (if (<= y 1.0) y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 1.25e-111) {
		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 <= (-1.0d0)) then
        tmp = 1.0d0
    else if (y <= 1.25d-111) 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 <= -1.0) {
		tmp = 1.0;
	} else if (y <= 1.25e-111) {
		tmp = x;
	} else if (y <= 1.0) {
		tmp = y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = 1.0
	elif y <= 1.25e-111:
		tmp = x
	elif y <= 1.0:
		tmp = y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = 1.0;
	elseif (y <= 1.25e-111)
		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 <= -1.0)
		tmp = 1.0;
	elseif (y <= 1.25e-111)
		tmp = x;
	elseif (y <= 1.0)
		tmp = y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], 1.0, If[LessEqual[y, 1.25e-111], x, If[LessEqual[y, 1.0], y, 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 1 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 75.9%

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

   if -1 < y < 1.2500000000000001e-111

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 76.0%

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

   if 1.2500000000000001e-111 < y < 1

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 67.4%

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

      1. +-commutative 67.4%

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

   4. Simplified 67.4%

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

   5. Taylor expanded in y around 0 65.0%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-111}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 74.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-9}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) 1.0 (if (<= y 3.5e-9) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 3.5e-9) {
		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 <= 3.5d-9) 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 <= 3.5e-9) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = 1.0
	elif y <= 3.5e-9:
		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 <= 3.5e-9)
		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 <= 3.5e-9)
		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, 3.5e-9], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 3.4999999999999999e-9 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 75.4%

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

   if -1 < y < 3.4999999999999999e-9

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 70.3%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-9}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9: 39.3% 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 43.2%

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

3. Final simplification 43.2%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023196 
(FPCore (x y)
  :name "Data.Colour.SRGB:invTransferFunction from colour-2.3.3"
  :precision binary64
  (/ (+ x y) (+ y 1.0)))