Data.Colour.SRGB:invTransferFunction from colour-2.3.3

Percentage Accurate: 100.0% → 100.0%

Time: 3.9s

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: 98.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.86\right):\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \left(x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 0.86)))
   (+ 1.0 (/ x y))
   (- x (* y (+ x -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.86)) {
		tmp = 1.0 + (x / y);
	} else {
		tmp = x - (y * (x + -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)) .or. (.not. (y <= 0.86d0))) then
        tmp = 1.0d0 + (x / y)
    else
        tmp = x - (y * (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.86)) {
		tmp = 1.0 + (x / y);
	} else {
		tmp = x - (y * (x + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 0.86):
		tmp = 1.0 + (x / y)
	else:
		tmp = x - (y * (x + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 0.86))
		tmp = Float64(1.0 + Float64(x / y));
	else
		tmp = Float64(x - Float64(y * Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 0.86)))
		tmp = 1.0 + (x / y);
	else
		tmp = x - (y * (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 0.86]], $MachinePrecision]], N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x - N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.859999999999999987 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 97.5%

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

      1. +-commutative 97.5%

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

      2. associate--l+ 97.5%

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

      3. +-commutative 97.5%

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

      4. associate--r- 97.5%

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

      5. div-sub 97.5%

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

   4. Simplified 97.5%

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

   5. Taylor expanded in x around inf 97.4%

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

      1. neg-mul-1 97.4%

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

      2. distribute-neg-frac 97.4%

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

   7. Simplified 97.4%

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

      1. div-inv 97.3%

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

      2. cancel-sign-sub 97.3%

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

      3. div-inv 97.4%

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

      4. +-commutative 97.4%

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

   9. Applied egg-rr 97.4%

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

   if -1 < y < 0.859999999999999987

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 98.9%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.86\right):\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \left(x + -1\right)\\ \end{array} \]

Alternative 3: 98.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 1.0))
		tmp = Float64(1.0 + Float64(Float64(x + -1.0) / y));
	else
		tmp = Float64(x - Float64(y * Float64(x + -1.0)));
	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 + -1.0) / y);
	else
		tmp = x - (y * (x + -1.0));
	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[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x - N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 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 97.5%

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

      1. +-commutative 97.5%

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

      2. associate--l+ 97.5%

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

      3. +-commutative 97.5%

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

      4. associate--r- 97.5%

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

      5. div-sub 97.5%

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

   4. Simplified 97.5%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 98.9%

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

4. Final simplification 98.3%

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

Alternative 4: 73.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-16}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+60}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) 1.0 (if (<= y 3.7e-16) x (if (<= y 2.4e+60) (/ x y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 3.7e-16) {
		tmp = x;
	} else if (y <= 2.4e+60) {
		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 <= 3.7d-16) then
        tmp = x
    else if (y <= 2.4d+60) 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 <= 3.7e-16) {
		tmp = x;
	} else if (y <= 2.4e+60) {
		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 <= 3.7e-16:
		tmp = x
	elif y <= 2.4e+60:
		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 <= 3.7e-16)
		tmp = x;
	elseif (y <= 2.4e+60)
		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 <= 3.7e-16)
		tmp = x;
	elseif (y <= 2.4e+60)
		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, 3.7e-16], x, If[LessEqual[y, 2.4e+60], N[(x / y), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 2.4e60 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 79.5%

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

   if -1 < y < 3.7e-16

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 78.4%

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

   if 3.7e-16 < y < 2.4e60

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 74.0%

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

      1. +-commutative 74.0%

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

      2. associate--l+ 74.0%

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

      3. +-commutative 74.0%

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

      4. associate--r- 74.0%

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

      5. div-sub 74.0%

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

   4. Simplified 74.0%

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

   5. Taylor expanded in x around inf 60.3%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-16}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+60}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 73.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-16}:\\ \;\;\;\;x \cdot \left(1 - y\right)\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+60}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   1.0
   (if (<= y 3.7e-16) (* x (- 1.0 y)) (if (<= y 7.5e+60) (/ x y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 3.7e-16) {
		tmp = x * (1.0 - y);
	} else if (y <= 7.5e+60) {
		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 <= 3.7d-16) then
        tmp = x * (1.0d0 - y)
    else if (y <= 7.5d+60) 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 <= 3.7e-16) {
		tmp = x * (1.0 - y);
	} else if (y <= 7.5e+60) {
		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 <= 3.7e-16:
		tmp = x * (1.0 - y)
	elif y <= 7.5e+60:
		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 <= 3.7e-16)
		tmp = Float64(x * Float64(1.0 - y));
	elseif (y <= 7.5e+60)
		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 <= 3.7e-16)
		tmp = x * (1.0 - y);
	elseif (y <= 7.5e+60)
		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, 3.7e-16], N[(x * N[(1.0 - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.5e+60], N[(x / y), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 7.5e60 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 79.5%

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

   if -1 < y < 3.7e-16

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.3%

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

   3. Taylor expanded in x around inf 78.9%

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

      1. neg-mul-1 78.9%

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

      2. unsub-neg 78.9%

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

   5. Simplified 78.9%

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

   if 3.7e-16 < y < 7.5e60

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 74.0%

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

      1. +-commutative 74.0%

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

      2. associate--l+ 74.0%

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

      3. +-commutative 74.0%

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

      4. associate--r- 74.0%

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

      5. div-sub 74.0%

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

   4. Simplified 74.0%

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

   5. Taylor expanded in x around inf 60.3%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-16}:\\ \;\;\;\;x \cdot \left(1 - y\right)\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+60}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 86.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 3.7 \cdot 10^{-16}\right):\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 3.7e-16))) (+ 1.0 (/ x y)) (* x (- 1.0 y))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 3.7e-16)) {
		tmp = 1.0 + (x / y);
	} else {
		tmp = 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) :: tmp
    if ((y <= (-1.0d0)) .or. (.not. (y <= 3.7d-16))) then
        tmp = 1.0d0 + (x / y)
    else
        tmp = x * (1.0d0 - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 3.7e-16)) {
		tmp = 1.0 + (x / y);
	} else {
		tmp = x * (1.0 - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 3.7e-16):
		tmp = 1.0 + (x / y)
	else:
		tmp = x * (1.0 - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 3.7e-16))
		tmp = Float64(1.0 + Float64(x / y));
	else
		tmp = Float64(x * Float64(1.0 - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 3.7e-16)))
		tmp = 1.0 + (x / y);
	else
		tmp = x * (1.0 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 3.7e-16]], $MachinePrecision]], N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 3.7e-16 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 95.0%

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

      1. +-commutative 95.0%

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

      2. associate--l+ 95.0%

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

      3. +-commutative 95.0%

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

      4. associate--r- 95.0%

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

      5. div-sub 95.0%

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

   4. Simplified 95.0%

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

   5. Taylor expanded in x around inf 95.1%

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

      1. neg-mul-1 95.1%

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

      2. distribute-neg-frac 95.1%

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

   7. Simplified 95.1%

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

      1. div-inv 95.0%

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

      2. cancel-sign-sub 95.0%

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

      3. div-inv 95.1%

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

      4. +-commutative 95.1%

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

   9. Applied egg-rr 95.1%

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

   if -1 < y < 3.7e-16

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.3%

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

   3. Taylor expanded in x around inf 78.9%

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

      1. neg-mul-1 78.9%

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

      2. unsub-neg 78.9%

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

   5. Simplified 78.9%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 3.7 \cdot 10^{-16}\right):\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y\right)\\ \end{array} \]

Alternative 7: 87.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4300000000 \lor \neg \left(y \leq 1950000000\right):\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4300000000.0) (not (<= y 1950000000.0)))
   (+ 1.0 (/ x y))
   (/ x (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4300000000.0) || !(y <= 1950000000.0)) {
		tmp = 1.0 + (x / y);
	} else {
		tmp = x / (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 ((y <= (-4300000000.0d0)) .or. (.not. (y <= 1950000000.0d0))) then
        tmp = 1.0d0 + (x / y)
    else
        tmp = x / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -4300000000.0) || !(y <= 1950000000.0)) {
		tmp = 1.0 + (x / y);
	} else {
		tmp = x / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4300000000.0) or not (y <= 1950000000.0):
		tmp = 1.0 + (x / y)
	else:
		tmp = x / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4300000000.0) || !(y <= 1950000000.0))
		tmp = Float64(1.0 + Float64(x / y));
	else
		tmp = Float64(x / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -4300000000.0) || ~((y <= 1950000000.0)))
		tmp = 1.0 + (x / y);
	else
		tmp = x / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4300000000.0], N[Not[LessEqual[y, 1950000000.0]], $MachinePrecision]], N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.3e9 or 1.95e9 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 100.0%

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

      1. +-commutative 100.0%

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

      2. associate--l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. associate--r- 100.0%

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

      5. div-sub 100.0%

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

   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%

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

      2. distribute-neg-frac 99.8%

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

   7. Simplified 99.8%

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

      1. div-inv 99.7%

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

      2. cancel-sign-sub 99.7%

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

      3. div-inv 99.8%

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

      4. +-commutative 99.8%

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

   9. Applied egg-rr 99.8%

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

   if -4.3e9 < y < 1.95e9

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 78.8%

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

      1. +-commutative 78.8%

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

   4. Simplified 78.8%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4300000000 \lor \neg \left(y \leq 1950000000\right):\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + 1}\\ \end{array} \]

Alternative 8: 73.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1950000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) 1.0 (if (<= y 1950000000.0) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 1950000000.0) {
		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 <= 1950000000.0d0) 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 <= 1950000000.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = 1.0
	elif y <= 1950000000.0:
		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 <= 1950000000.0)
		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 <= 1950000000.0)
		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, 1950000000.0], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.95e9 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 72.9%

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

   if -1 < y < 1.95e9

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 75.2%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1950000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9: 39.0% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in y around inf 32.7%

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

3. Final simplification 32.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023322 
(FPCore (x y)
  :name "Data.Colour.SRGB:invTransferFunction from colour-2.3.3"
  :precision binary64
  (/ (+ x y) (+ y 1.0)))