Data.Colour.SRGB:invTransferFunction from colour-2.3.3

Percentage Accurate: 100.0% → 100.0%

Time: 5.4s

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. Add Preprocessing
3. Add Preprocessing

Alternative 2: 82.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{-158}:\\ \;\;\;\;y - y \cdot y\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-52}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 460000:\\ \;\;\;\;\frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   (+ 1.0 (/ x y))
   (if (<= y -2.3e-158)
     (- y (* y y))
     (if (<= y 3.8e-52)
       x
       (if (<= y 460000.0) (/ y (+ y 1.0)) (+ 1.0 (/ (+ x -1.0) y)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0 + (x / y);
	} else if (y <= -2.3e-158) {
		tmp = y - (y * y);
	} else if (y <= 3.8e-52) {
		tmp = x;
	} else if (y <= 460000.0) {
		tmp = y / (y + 1.0);
	} else {
		tmp = 1.0 + ((x + -1.0) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = 1.0d0 + (x / y)
    else if (y <= (-2.3d-158)) then
        tmp = y - (y * y)
    else if (y <= 3.8d-52) then
        tmp = x
    else if (y <= 460000.0d0) then
        tmp = y / (y + 1.0d0)
    else
        tmp = 1.0d0 + ((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) {
		tmp = 1.0 + (x / y);
	} else if (y <= -2.3e-158) {
		tmp = y - (y * y);
	} else if (y <= 3.8e-52) {
		tmp = x;
	} else if (y <= 460000.0) {
		tmp = y / (y + 1.0);
	} else {
		tmp = 1.0 + ((x + -1.0) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = 1.0 + (x / y)
	elif y <= -2.3e-158:
		tmp = y - (y * y)
	elif y <= 3.8e-52:
		tmp = x
	elif y <= 460000.0:
		tmp = y / (y + 1.0)
	else:
		tmp = 1.0 + ((x + -1.0) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(1.0 + Float64(x / y));
	elseif (y <= -2.3e-158)
		tmp = Float64(y - Float64(y * y));
	elseif (y <= 3.8e-52)
		tmp = x;
	elseif (y <= 460000.0)
		tmp = Float64(y / Float64(y + 1.0));
	else
		tmp = Float64(1.0 + Float64(Float64(x + -1.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = 1.0 + (x / y);
	elseif (y <= -2.3e-158)
		tmp = y - (y * y);
	elseif (y <= 3.8e-52)
		tmp = x;
	elseif (y <= 460000.0)
		tmp = y / (y + 1.0);
	else
		tmp = 1.0 + ((x + -1.0) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2.3e-158], N[(y - N[(y * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.8e-52], x, If[LessEqual[y, 460000.0], N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -1

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. div-sub 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

   if -1 < y < -2.2999999999999999e-158

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 59.1%

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

      1. +-commutative 59.1%

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

   5. Simplified 59.1%

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

   6. Taylor expanded in y around 0 59.1%

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

      1. neg-mul-1 59.1%

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

      2. distribute-lft-in 59.2%

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

      3. *-rgt-identity 59.2%

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

      4. distribute-rgt-neg-in 59.2%

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

      5. unpow2 59.2%

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

      6. unsub-neg 59.2%

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

   8. Simplified 59.2%

      \[\leadsto \color{blue}{y - {y}^{2}} \]
   9. Step-by-step derivation

      1. unpow2 59.2%

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

   10. Applied egg-rr 59.2%

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

   if -2.2999999999999999e-158 < y < 3.8000000000000003e-52

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 80.7%

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

   if 3.8000000000000003e-52 < y < 4.6e5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 73.5%

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

      1. +-commutative 73.5%

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

   5. Simplified 73.5%

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

   if 4.6e5 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 99.9%

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

      1. associate--l+ 99.9%

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

      2. div-sub 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

   5. Simplified 99.9%

      \[\leadsto \color{blue}{1 + \frac{x + -1}{y}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 3: 82.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{-158}:\\ \;\;\;\;y - y \cdot y\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{-52}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 720000000000:\\ \;\;\;\;\frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))))
   (if (<= y -1.0)
     t_0
     (if (<= y -2.3e-158)
       (- y (* y y))
       (if (<= y 5.7e-52) x (if (<= y 720000000000.0) (/ y (+ y 1.0)) t_0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= -2.3e-158) {
		tmp = y - (y * y);
	} else if (y <= 5.7e-52) {
		tmp = x;
	} else if (y <= 720000000000.0) {
		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 + (x / y)
    if (y <= (-1.0d0)) then
        tmp = t_0
    else if (y <= (-2.3d-158)) then
        tmp = y - (y * y)
    else if (y <= 5.7d-52) then
        tmp = x
    else if (y <= 720000000000.0d0) 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 + (x / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= -2.3e-158) {
		tmp = y - (y * y);
	} else if (y <= 5.7e-52) {
		tmp = x;
	} else if (y <= 720000000000.0) {
		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 <= -1.0:
		tmp = t_0
	elif y <= -2.3e-158:
		tmp = y - (y * y)
	elif y <= 5.7e-52:
		tmp = x
	elif y <= 720000000000.0:
		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 <= -1.0)
		tmp = t_0;
	elseif (y <= -2.3e-158)
		tmp = Float64(y - Float64(y * y));
	elseif (y <= 5.7e-52)
		tmp = x;
	elseif (y <= 720000000000.0)
		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 + (x / y);
	tmp = 0.0;
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= -2.3e-158)
		tmp = y - (y * y);
	elseif (y <= 5.7e-52)
		tmp = x;
	elseif (y <= 720000000000.0)
		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[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, -2.3e-158], N[(y - N[(y * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.7e-52], x, If[LessEqual[y, 720000000000.0], N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1 or 7.2e11 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. div-sub 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

   if -1 < y < -2.2999999999999999e-158

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 59.1%

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

      1. +-commutative 59.1%

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

   5. Simplified 59.1%

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

   6. Taylor expanded in y around 0 59.1%

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

      1. neg-mul-1 59.1%

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

      2. distribute-lft-in 59.2%

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

      3. *-rgt-identity 59.2%

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

      4. distribute-rgt-neg-in 59.2%

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

      5. unpow2 59.2%

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

      6. unsub-neg 59.2%

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

   8. Simplified 59.2%

      \[\leadsto \color{blue}{y - {y}^{2}} \]
   9. Step-by-step derivation

      1. unpow2 59.2%

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

   10. Applied egg-rr 59.2%

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

   if -2.2999999999999999e-158 < y < 5.6999999999999997e-52

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 80.7%

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

   if 5.6999999999999997e-52 < y < 7.2e11

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 75.9%

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

      1. +-commutative 75.9%

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

   5. Simplified 75.9%

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

Alternative 4: 82.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{-158}:\\ \;\;\;\;y - y \cdot y\\ \mathbf{elif}\;y \leq 6.7 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))))
   (if (<= y -1.0)
     t_0
     (if (<= y -2.3e-158) (- y (* y y)) (if (<= y 6.7e-6) x t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= -2.3e-158) {
		tmp = y - (y * y);
	} else if (y <= 6.7e-6) {
		tmp = x;
	} 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 <= (-1.0d0)) then
        tmp = t_0
    else if (y <= (-2.3d-158)) then
        tmp = y - (y * y)
    else if (y <= 6.7d-6) then
        tmp = x
    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 <= -1.0) {
		tmp = t_0;
	} else if (y <= -2.3e-158) {
		tmp = y - (y * y);
	} else if (y <= 6.7e-6) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x / y)
	tmp = 0
	if y <= -1.0:
		tmp = t_0
	elif y <= -2.3e-158:
		tmp = y - (y * y)
	elif y <= 6.7e-6:
		tmp = x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / y))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= -2.3e-158)
		tmp = Float64(y - Float64(y * y));
	elseif (y <= 6.7e-6)
		tmp = x;
	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 <= -1.0)
		tmp = t_0;
	elseif (y <= -2.3e-158)
		tmp = y - (y * y);
	elseif (y <= 6.7e-6)
		tmp = x;
	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, -1.0], t$95$0, If[LessEqual[y, -2.3e-158], N[(y - N[(y * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.7e-6], x, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 6.7e-6 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 98.0%

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

      1. associate--l+ 98.0%

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

      2. div-sub 98.0%

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

      3. sub-neg 98.0%

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

      4. metadata-eval 98.0%

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

   5. Simplified 98.0%

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

   6. Taylor expanded in x around inf 96.9%

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

   if -1 < y < -2.2999999999999999e-158

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 59.1%

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

      1. +-commutative 59.1%

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

   5. Simplified 59.1%

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

   6. Taylor expanded in y around 0 59.1%

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

      1. neg-mul-1 59.1%

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

      2. distribute-lft-in 59.2%

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

      3. *-rgt-identity 59.2%

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

      4. distribute-rgt-neg-in 59.2%

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

      5. unpow2 59.2%

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

      6. unsub-neg 59.2%

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

   8. Simplified 59.2%

      \[\leadsto \color{blue}{y - {y}^{2}} \]
   9. Step-by-step derivation

      1. unpow2 59.2%

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

   10. Applied egg-rr 59.2%

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

   if -2.2999999999999999e-158 < y < 6.7e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 75.9%

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

Alternative 5: 82.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{-158}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 6.7 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))))
   (if (<= y -1.0) t_0 (if (<= y -2.3e-158) y (if (<= y 6.7e-6) x t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= -2.3e-158) {
		tmp = y;
	} else if (y <= 6.7e-6) {
		tmp = x;
	} 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 <= (-1.0d0)) then
        tmp = t_0
    else if (y <= (-2.3d-158)) then
        tmp = y
    else if (y <= 6.7d-6) then
        tmp = x
    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 <= -1.0) {
		tmp = t_0;
	} else if (y <= -2.3e-158) {
		tmp = y;
	} else if (y <= 6.7e-6) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x / y)
	tmp = 0
	if y <= -1.0:
		tmp = t_0
	elif y <= -2.3e-158:
		tmp = y
	elif y <= 6.7e-6:
		tmp = x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / y))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= -2.3e-158)
		tmp = y;
	elseif (y <= 6.7e-6)
		tmp = x;
	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 <= -1.0)
		tmp = t_0;
	elseif (y <= -2.3e-158)
		tmp = y;
	elseif (y <= 6.7e-6)
		tmp = x;
	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, -1.0], t$95$0, If[LessEqual[y, -2.3e-158], y, If[LessEqual[y, 6.7e-6], x, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 6.7e-6 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 98.0%

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

      1. associate--l+ 98.0%

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

      2. div-sub 98.0%

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

      3. sub-neg 98.0%

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

      4. metadata-eval 98.0%

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

   5. Simplified 98.0%

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

   6. Taylor expanded in x around inf 96.9%

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

   if -1 < y < -2.2999999999999999e-158

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 59.1%

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

      1. +-commutative 59.1%

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

   5. Simplified 59.1%

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

   6. Taylor expanded in y around 0 59.1%

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

      1. neg-mul-1 59.1%

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

      2. distribute-lft-in 59.2%

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

      3. *-rgt-identity 59.2%

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

      4. distribute-rgt-neg-in 59.2%

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

      5. unpow2 59.2%

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

      6. unsub-neg 59.2%

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

   8. Simplified 59.2%

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

   9. Taylor expanded in y around 0 58.8%

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

   if -2.2999999999999999e-158 < y < 6.7e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 75.9%

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

Alternative 6: 70.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{-158}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 6.7 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) 1.0 (if (<= y -2.3e-158) y (if (<= y 6.7e-6) x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= -2.3e-158) {
		tmp = y;
	} else if (y <= 6.7e-6) {
		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 <= (-2.3d-158)) then
        tmp = y
    else if (y <= 6.7d-6) 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 <= -2.3e-158) {
		tmp = y;
	} else if (y <= 6.7e-6) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = 1.0
	elif y <= -2.3e-158:
		tmp = y
	elif y <= 6.7e-6:
		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 <= -2.3e-158)
		tmp = y;
	elseif (y <= 6.7e-6)
		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 <= -2.3e-158)
		tmp = y;
	elseif (y <= 6.7e-6)
		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, -2.3e-158], y, If[LessEqual[y, 6.7e-6], x, 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 6.7e-6 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 78.0%

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

   if -1 < y < -2.2999999999999999e-158

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 59.1%

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

      1. +-commutative 59.1%

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

   5. Simplified 59.1%

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

   6. Taylor expanded in y around 0 59.1%

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

      1. neg-mul-1 59.1%

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

      2. distribute-lft-in 59.2%

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

      3. *-rgt-identity 59.2%

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

      4. distribute-rgt-neg-in 59.2%

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

      5. unpow2 59.2%

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

      6. unsub-neg 59.2%

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

   8. Simplified 59.2%

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

   9. Taylor expanded in y around 0 58.8%

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

   if -2.2999999999999999e-158 < y < 6.7e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 75.9%

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

Alternative 7: 73.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 6.7 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) 1.0 (if (<= y 6.7e-6) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 6.7e-6) {
		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 <= 6.7d-6) 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 <= 6.7e-6) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = 1.0
	elif y <= 6.7e-6:
		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 <= 6.7e-6)
		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 <= 6.7e-6)
		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, 6.7e-6], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 6.7e-6 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 78.0%

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

   if -1 < y < 6.7e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 68.4%

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

Alternative 8: 38.5% accurate, 7.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around inf 40.3%

   \[\leadsto \color{blue}{1} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024186 
(FPCore (x y)
  :name "Data.Colour.SRGB:invTransferFunction from colour-2.3.3"
  :precision binary64
  (/ (+ x y) (+ y 1.0)))