Data.Colour.SRGB:invTransferFunction from colour-2.3.3

Percentage Accurate: 100.0% → 100.0%

Time: 4.5s

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. Final simplification 100.0%

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

Alternative 2: 86.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y + 1}\\ \mathbf{if}\;y \leq -1.15 \cdot 10^{+61}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-7}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+31}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y 1.0))))
   (if (<= y -1.15e+61)
     1.0
     (if (<= y -9e-8)
       t_0
       (if (<= y 1.15e-7) (+ x y) (if (<= y 2.2e+31) t_0 1.0))))))

C

double code(double x, double y) {
	double t_0 = x / (y + 1.0);
	double tmp;
	if (y <= -1.15e+61) {
		tmp = 1.0;
	} else if (y <= -9e-8) {
		tmp = t_0;
	} else if (y <= 1.15e-7) {
		tmp = x + y;
	} else if (y <= 2.2e+31) {
		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 <= (-1.15d+61)) then
        tmp = 1.0d0
    else if (y <= (-9d-8)) then
        tmp = t_0
    else if (y <= 1.15d-7) then
        tmp = x + y
    else if (y <= 2.2d+31) 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 <= -1.15e+61) {
		tmp = 1.0;
	} else if (y <= -9e-8) {
		tmp = t_0;
	} else if (y <= 1.15e-7) {
		tmp = x + y;
	} else if (y <= 2.2e+31) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y + 1.0)
	tmp = 0
	if y <= -1.15e+61:
		tmp = 1.0
	elif y <= -9e-8:
		tmp = t_0
	elif y <= 1.15e-7:
		tmp = x + y
	elif y <= 2.2e+31:
		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 <= -1.15e+61)
		tmp = 1.0;
	elseif (y <= -9e-8)
		tmp = t_0;
	elseif (y <= 1.15e-7)
		tmp = Float64(x + y);
	elseif (y <= 2.2e+31)
		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 <= -1.15e+61)
		tmp = 1.0;
	elseif (y <= -9e-8)
		tmp = t_0;
	elseif (y <= 1.15e-7)
		tmp = x + y;
	elseif (y <= 2.2e+31)
		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, -1.15e+61], 1.0, If[LessEqual[y, -9e-8], t$95$0, If[LessEqual[y, 1.15e-7], N[(x + y), $MachinePrecision], If[LessEqual[y, 2.2e+31], t$95$0, 1.0]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.15e61 or 2.2000000000000001e31 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 81.4%

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

   if -1.15e61 < y < -8.99999999999999986e-8 or 1.14999999999999997e-7 < y < 2.2000000000000001e31

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 81.7%

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

      1. +-commutative 81.7%

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

   5. Simplified 81.7%

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

   if -8.99999999999999986e-8 < y < 1.14999999999999997e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.4%

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

   4. Taylor expanded in x around 0 98.9%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+61}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-7}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+31}:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y + 1}\\ \mathbf{if}\;y \leq -4.6 \cdot 10^{+62}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -4 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-6}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+32}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y 1.0))))
   (if (<= y -4.6e+62)
     1.0
     (if (<= y -4e-8)
       t_0
       (if (<= y 1.45e-6) (+ x y) (if (<= y 1.2e+32) t_0 (/ y (+ y 1.0))))))))

C

double code(double x, double y) {
	double t_0 = x / (y + 1.0);
	double tmp;
	if (y <= -4.6e+62) {
		tmp = 1.0;
	} else if (y <= -4e-8) {
		tmp = t_0;
	} else if (y <= 1.45e-6) {
		tmp = x + y;
	} else if (y <= 1.2e+32) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = x / (y + 1.0d0)
    if (y <= (-4.6d+62)) then
        tmp = 1.0d0
    else if (y <= (-4d-8)) then
        tmp = t_0
    else if (y <= 1.45d-6) then
        tmp = x + y
    else if (y <= 1.2d+32) then
        tmp = t_0
    else
        tmp = y / (y + 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 <= -4.6e+62) {
		tmp = 1.0;
	} else if (y <= -4e-8) {
		tmp = t_0;
	} else if (y <= 1.45e-6) {
		tmp = x + y;
	} else if (y <= 1.2e+32) {
		tmp = t_0;
	} else {
		tmp = y / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y + 1.0)
	tmp = 0
	if y <= -4.6e+62:
		tmp = 1.0
	elif y <= -4e-8:
		tmp = t_0
	elif y <= 1.45e-6:
		tmp = x + y
	elif y <= 1.2e+32:
		tmp = t_0
	else:
		tmp = y / (y + 1.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y + 1.0))
	tmp = 0.0
	if (y <= -4.6e+62)
		tmp = 1.0;
	elseif (y <= -4e-8)
		tmp = t_0;
	elseif (y <= 1.45e-6)
		tmp = Float64(x + y);
	elseif (y <= 1.2e+32)
		tmp = t_0;
	else
		tmp = Float64(y / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y + 1.0);
	tmp = 0.0;
	if (y <= -4.6e+62)
		tmp = 1.0;
	elseif (y <= -4e-8)
		tmp = t_0;
	elseif (y <= 1.45e-6)
		tmp = x + y;
	elseif (y <= 1.2e+32)
		tmp = t_0;
	else
		tmp = y / (y + 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, -4.6e+62], 1.0, If[LessEqual[y, -4e-8], t$95$0, If[LessEqual[y, 1.45e-6], N[(x + y), $MachinePrecision], If[LessEqual[y, 1.2e+32], t$95$0, N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -4.59999999999999968e62

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 85.9%

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

   if -4.59999999999999968e62 < y < -4.0000000000000001e-8 or 1.4500000000000001e-6 < y < 1.19999999999999996e32

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 81.7%

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

      1. +-commutative 81.7%

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

   5. Simplified 81.7%

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

   if -4.0000000000000001e-8 < y < 1.4500000000000001e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.4%

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

   4. Taylor expanded in x around 0 98.9%

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

   if 1.19999999999999996e32 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 77.3%

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

      1. +-commutative 77.3%

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

   5. Simplified 77.3%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{+62}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -4 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-6}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+32}:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 310:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+34}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   1.0
   (if (<= y 310.0) (+ x y) (if (<= y 1.3e+34) (/ x y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 310.0) {
		tmp = x + y;
	} else if (y <= 1.3e+34) {
		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 <= (-1.0d0)) then
        tmp = 1.0d0
    else if (y <= 310.0d0) then
        tmp = x + y
    else if (y <= 1.3d+34) 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 <= -1.0) {
		tmp = 1.0;
	} else if (y <= 310.0) {
		tmp = x + y;
	} else if (y <= 1.3e+34) {
		tmp = x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = 1.0
	elif y <= 310.0:
		tmp = x + y
	elif y <= 1.3e+34:
		tmp = x / y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = 1.0;
	elseif (y <= 310.0)
		tmp = Float64(x + y);
	elseif (y <= 1.3e+34)
		tmp = Float64(x / y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = 1.0;
	elseif (y <= 310.0)
		tmp = x + y;
	elseif (y <= 1.3e+34)
		tmp = x / y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], 1.0, If[LessEqual[y, 310.0], N[(x + y), $MachinePrecision], If[LessEqual[y, 1.3e+34], N[(x / y), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 1.29999999999999999e34 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 76.3%

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

   if -1 < y < 310

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 98.7%

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

   4. Taylor expanded in x around 0 97.9%

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

   if 310 < y < 1.29999999999999999e34

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 99.7%

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

      1. +-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in y around inf 98.9%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 310:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+34}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.78\right):\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 0.78))) (+ 1.0 (/ x y)) (+ x (* y (- 1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.78)) {
		tmp = 1.0 + (x / y);
	} else {
		tmp = x + (y * (1.0 - x));
	}
	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)) .or. (.not. (y <= 0.78d0))) then
        tmp = 1.0d0 + (x / y)
    else
        tmp = x + (y * (1.0d0 - x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.78)) {
		tmp = 1.0 + (x / y);
	} else {
		tmp = x + (y * (1.0 - x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 0.78):
		tmp = 1.0 + (x / y)
	else:
		tmp = x + (y * (1.0 - x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 0.78))
		tmp = Float64(1.0 + Float64(x / y));
	else
		tmp = Float64(x + Float64(y * Float64(1.0 - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 0.78)))
		tmp = 1.0 + (x / y);
	else
		tmp = x + (y * (1.0 - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 0.78]], $MachinePrecision]], N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.78000000000000003 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 98.9%

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

      1. +-commutative 98.9%

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

      2. associate--l+ 98.9%

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

      3. +-commutative 98.9%

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

      4. associate--r- 98.9%

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

      5. div-sub 98.9%

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

   5. Simplified 98.9%

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

   6. Taylor expanded in x around inf 98.9%

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

      1. neg-mul-1 98.9%

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

      2. distribute-neg-frac 98.9%

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

   8. Simplified 98.9%

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

   if -1 < y < 0.78000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 98.7%

      \[\leadsto \color{blue}{x + y \cdot \left(1 - x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.78\right):\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = 1.0 + (x / y);
	} else {
		tmp = x + y;
	}
	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)) .or. (.not. (y <= 1.0d0))) then
        tmp = 1.0d0 + (x / y)
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 98.9%

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

      1. +-commutative 98.9%

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

      2. associate--l+ 98.9%

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

      3. +-commutative 98.9%

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

      4. associate--r- 98.9%

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

      5. div-sub 98.9%

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

   5. Simplified 98.9%

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

   6. Taylor expanded in x around inf 98.9%

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

      1. neg-mul-1 98.9%

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

      2. distribute-neg-frac 98.9%

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

   8. Simplified 98.9%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 98.7%

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

   4. Taylor expanded in x around 0 97.9%

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

4. Final simplification 98.4%

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

Alternative 7: 84.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+31}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) 1.0 (if (<= y 1.65e+31) (+ x y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 1.65e+31) {
		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 <= (-1.0d0)) then
        tmp = 1.0d0
    else if (y <= 1.65d+31) 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 <= -1.0) {
		tmp = 1.0;
	} else if (y <= 1.65e+31) {
		tmp = x + y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = 1.0
	elif y <= 1.65e+31:
		tmp = x + y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = 1.0;
	elseif (y <= 1.65e+31)
		tmp = Float64(x + y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = 1.0;
	elseif (y <= 1.65e+31)
		tmp = x + y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], 1.0, If[LessEqual[y, 1.65e+31], N[(x + y), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.64999999999999996e31 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 76.3%

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

   if -1 < y < 1.64999999999999996e31

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 95.0%

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

   4. Taylor expanded in x around 0 94.5%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+31}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+31}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) 1.0 (if (<= y 1.65e+31) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 1.65e+31) {
		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 <= 1.65d+31) 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 <= 1.65e+31) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = 1.0
	elif y <= 1.65e+31:
		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 <= 1.65e+31)
		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 <= 1.65e+31)
		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, 1.65e+31], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.64999999999999996e31 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 76.3%

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

   if -1 < y < 1.64999999999999996e31

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 70.5%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+31}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 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. Add Preprocessing

3. Taylor expanded in y around inf 38.5%

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

4. Final simplification 38.5%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024019 
(FPCore (x y)
  :name "Data.Colour.SRGB:invTransferFunction from colour-2.3.3"
  :precision binary64
  (/ (+ x y) (+ y 1.0)))