Data.Colour.SRGB:invTransferFunction from colour-2.3.3

Percentage Accurate: 100.0% → 100.0%

Time: 6.5s

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. Add Preprocessing

Alternative 2: 83.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 245:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+151}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   1.0
   (if (<= y 245.0) (+ x y) (if (<= y 2.7e+151) (/ x y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 245.0) {
		tmp = x + y;
	} else if (y <= 2.7e+151) {
		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 <= 245.0d0) then
        tmp = x + y
    else if (y <= 2.7d+151) 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 <= 245.0) {
		tmp = x + y;
	} else if (y <= 2.7e+151) {
		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 <= 245.0:
		tmp = x + y
	elif y <= 2.7e+151:
		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 <= 245.0)
		tmp = Float64(x + y);
	elseif (y <= 2.7e+151)
		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 <= 245.0)
		tmp = x + y;
	elseif (y <= 2.7e+151)
		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, 245.0], N[(x + y), $MachinePrecision], If[LessEqual[y, 2.7e+151], N[(x / y), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 2.7000000000000001e151 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 70.9%

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

   if -1 < y < 245

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), \color{blue}{1}\right) \]
   4. Step-by-step derivation

   5. Simplified 97.9%

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

      1. /-rgt-identity N/A

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 97.9%

         \[\leadsto \mathsf{+.f64}\left(y, \color{blue}{x}\right) \]

   7. Applied egg-rr 97.9%

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

   if 245 < y < 2.7000000000000001e151

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + y\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(y + \color{blue}{1}\right)\right) \]

      3. +-lowering-+.f64 61.9%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right) \]

   5. Simplified 61.9%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{y}\right) \]
   7. Step-by-step derivation

   8. Simplified 58.7%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 245:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+151}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x + -1}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.2:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (+ x -1.0) y))))
   (if (<= y -1.0) t_0 (if (<= y 1.2) (+ x y) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + ((x + -1.0) / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.2) {
		tmp = x + y;
	} 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 + (-1.0d0)) / y)
    if (y <= (-1.0d0)) then
        tmp = t_0
    else if (y <= 1.2d0) then
        tmp = x + y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + ((x + -1.0) / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.2) {
		tmp = x + y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + ((x + -1.0) / y)
	tmp = 0
	if y <= -1.0:
		tmp = t_0
	elif y <= 1.2:
		tmp = x + y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(Float64(x + -1.0) / y))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 1.2)
		tmp = Float64(x + y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + ((x + -1.0) / y);
	tmp = 0.0;
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 1.2)
		tmp = x + y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 1.2], N[(x + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.19999999999999996 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. mul-1-neg N/A

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

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(-1 \cdot \frac{1 + -1 \cdot x}{y}\right)}\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{-1 \cdot \left(1 + -1 \cdot x\right)}{\color{blue}{y}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(-1 \cdot \left(1 + -1 \cdot x\right)\right), \color{blue}{y}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(-1 \cdot \left(-1 \cdot x + 1\right)\right), y\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(-1 \cdot \left(-1 \cdot x\right) + -1 \cdot 1\right), y\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(-1 \cdot -1\right) \cdot x + -1 \cdot 1\right), y\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(1 \cdot x + -1 \cdot 1\right), y\right)\right) \]

      17. *-lft-identity N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(x + -1 \cdot 1\right), y\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(x + -1\right), y\right)\right) \]

      19. +-lowering-+.f64 98.1%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), y\right)\right) \]

   5. Simplified 98.1%

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

   if -1 < y < 1.19999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), \color{blue}{1}\right) \]
   4. Step-by-step derivation

   5. Simplified 98.5%

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

      1. /-rgt-identity N/A

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 98.5%

         \[\leadsto \mathsf{+.f64}\left(y, \color{blue}{x}\right) \]

   7. Applied egg-rr 98.5%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{elif}\;y \leq 1.2:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.0005:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{-98}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-19}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -0.0005) 1.0 (if (<= y -5.8e-98) y (if (<= y 6.2e-19) x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -0.0005) {
		tmp = 1.0;
	} else if (y <= -5.8e-98) {
		tmp = y;
	} else if (y <= 6.2e-19) {
		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 <= (-0.0005d0)) then
        tmp = 1.0d0
    else if (y <= (-5.8d-98)) then
        tmp = y
    else if (y <= 6.2d-19) 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 <= -0.0005) {
		tmp = 1.0;
	} else if (y <= -5.8e-98) {
		tmp = y;
	} else if (y <= 6.2e-19) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -0.0005:
		tmp = 1.0
	elif y <= -5.8e-98:
		tmp = y
	elif y <= 6.2e-19:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -0.0005)
		tmp = 1.0;
	elseif (y <= -5.8e-98)
		tmp = y;
	elseif (y <= 6.2e-19)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -0.0005)
		tmp = 1.0;
	elseif (y <= -5.8e-98)
		tmp = y;
	elseif (y <= 6.2e-19)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -0.0005], 1.0, If[LessEqual[y, -5.8e-98], y, If[LessEqual[y, 6.2e-19], x, 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.0000000000000001e-4 or 6.1999999999999998e-19 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 60.4%

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

   if -5.0000000000000001e-4 < y < -5.8e-98

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), \color{blue}{1}\right) \]
   4. Step-by-step derivation

   5. Simplified 97.0%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 69.3%

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

   if -5.8e-98 < y < 6.1999999999999998e-19

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 80.4%

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

Alternative 5: 98.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Derivation

1. Split input into 3 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

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. mul-1-neg N/A

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

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(-1 \cdot \frac{1 + -1 \cdot x}{y}\right)}\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{-1 \cdot \left(1 + -1 \cdot x\right)}{\color{blue}{y}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(-1 \cdot \left(1 + -1 \cdot x\right)\right), \color{blue}{y}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(-1 \cdot \left(-1 \cdot x + 1\right)\right), y\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(-1 \cdot \left(-1 \cdot x\right) + -1 \cdot 1\right), y\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(-1 \cdot -1\right) \cdot x + -1 \cdot 1\right), y\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(1 \cdot x + -1 \cdot 1\right), y\right)\right) \]

      17. *-lft-identity N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(x + -1 \cdot 1\right), y\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(x + -1\right), y\right)\right) \]

      19. +-lowering-+.f64 98.6%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), y\right)\right) \]

   5. Simplified 98.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{x}{y}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 98.0%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right) \]

   8. Simplified 98.0%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), \color{blue}{1}\right) \]
   4. Step-by-step derivation

   5. Simplified 98.5%

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

      1. /-rgt-identity N/A

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 98.5%

         \[\leadsto \mathsf{+.f64}\left(y, \color{blue}{x}\right) \]

   7. Applied egg-rr 98.5%

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

   if 1 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), \color{blue}{y}\right) \]
   4. Step-by-step derivation

   5. Simplified 97.4%

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

4. Final simplification 98.1%

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

Alternative 6: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))))
   (if (<= y -1.0) t_0 (if (<= y 1.0) (+ x y) 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 <= 1.0) {
		tmp = x + y;
	} 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 <= 1.0d0) then
        tmp = x + y
    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 <= 1.0) {
		tmp = x + y;
	} 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 <= 1.0:
		tmp = x + y
	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 <= 1.0)
		tmp = Float64(x + y);
	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 <= 1.0)
		tmp = x + y;
	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, 1.0], N[(x + y), $MachinePrecision], t$95$0]]]

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

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. mul-1-neg N/A

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

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(-1 \cdot \frac{1 + -1 \cdot x}{y}\right)}\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{-1 \cdot \left(1 + -1 \cdot x\right)}{\color{blue}{y}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(-1 \cdot \left(1 + -1 \cdot x\right)\right), \color{blue}{y}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(-1 \cdot \left(-1 \cdot x + 1\right)\right), y\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(-1 \cdot \left(-1 \cdot x\right) + -1 \cdot 1\right), y\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(-1 \cdot -1\right) \cdot x + -1 \cdot 1\right), y\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(1 \cdot x + -1 \cdot 1\right), y\right)\right) \]

      17. *-lft-identity N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(x + -1 \cdot 1\right), y\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(x + -1\right), y\right)\right) \]

      19. +-lowering-+.f64 98.1%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), y\right)\right) \]

   5. Simplified 98.1%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{x}{y}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 97.7%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right) \]

   8. Simplified 97.7%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), \color{blue}{1}\right) \]
   4. Step-by-step derivation

   5. Simplified 98.5%

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

      1. /-rgt-identity N/A

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 98.5%

         \[\leadsto \mathsf{+.f64}\left(y, \color{blue}{x}\right) \]

   7. Applied egg-rr 98.5%

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

4. Final simplification 98.1%

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

Alternative 7: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+25}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) 1.0 (if (<= y 3.6e+25) (+ x y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 3.6e+25) {
		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 <= 3.6d+25) 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 <= 3.6e+25) {
		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 <= 3.6e+25:
		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 <= 3.6e+25)
		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 <= 3.6e+25)
		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, 3.6e+25], N[(x + y), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 3.60000000000000015e25 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 66.5%

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

   if -1 < y < 3.60000000000000015e25

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), \color{blue}{1}\right) \]
   4. Step-by-step derivation

   5. Simplified 95.0%

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

      1. /-rgt-identity N/A

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 95.0%

         \[\leadsto \mathsf{+.f64}\left(y, \color{blue}{x}\right) \]

   7. Applied egg-rr 95.0%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+25}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-19}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) 1.0 (if (<= y 6.2e-19) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 6.2e-19) {
		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 <= 6.2d-19) 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 <= 6.2e-19) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = 1.0
	elif y <= 6.2e-19:
		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 <= 6.2e-19)
		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 <= 6.2e-19)
		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, 6.2e-19], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 6.1999999999999998e-19 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 60.9%

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

   if -1 < y < 6.1999999999999998e-19

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 73.8%

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

Alternative 9: 38.8% 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

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

5. Simplified 29.2%

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

Reproduce

herbie shell --seed 2024152 
(FPCore (x y)
  :name "Data.Colour.SRGB:invTransferFunction from colour-2.3.3"
  :precision binary64
  (/ (+ x y) (+ y 1.0)))