Data.Colour.SRGB:invTransferFunction from colour-2.3.3

Percentage Accurate: 100.0% → 100.0%

Time: 5.5s

Alternatives: 7

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: 85.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.65 \cdot 10^{+87}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{+71}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 0.0052:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+93}:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.65e+87)
   1.0
   (if (<= y -5.8e+71)
     (/ x y)
     (if (<= y -1.0)
       1.0
       (if (<= y 0.0052) (+ x y) (if (<= y 1.7e+93) (/ x (+ y 1.0)) 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.65e+87) {
		tmp = 1.0;
	} else if (y <= -5.8e+71) {
		tmp = x / y;
	} else if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 0.0052) {
		tmp = x + y;
	} else if (y <= 1.7e+93) {
		tmp = x / (y + 1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.65d+87)) then
        tmp = 1.0d0
    else if (y <= (-5.8d+71)) then
        tmp = x / y
    else if (y <= (-1.0d0)) then
        tmp = 1.0d0
    else if (y <= 0.0052d0) then
        tmp = x + y
    else if (y <= 1.7d+93) then
        tmp = x / (y + 1.0d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.65e+87) {
		tmp = 1.0;
	} else if (y <= -5.8e+71) {
		tmp = x / y;
	} else if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 0.0052) {
		tmp = x + y;
	} else if (y <= 1.7e+93) {
		tmp = x / (y + 1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.65e+87:
		tmp = 1.0
	elif y <= -5.8e+71:
		tmp = x / y
	elif y <= -1.0:
		tmp = 1.0
	elif y <= 0.0052:
		tmp = x + y
	elif y <= 1.7e+93:
		tmp = x / (y + 1.0)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.65e+87)
		tmp = 1.0;
	elseif (y <= -5.8e+71)
		tmp = Float64(x / y);
	elseif (y <= -1.0)
		tmp = 1.0;
	elseif (y <= 0.0052)
		tmp = Float64(x + y);
	elseif (y <= 1.7e+93)
		tmp = Float64(x / Float64(y + 1.0));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.65e+87)
		tmp = 1.0;
	elseif (y <= -5.8e+71)
		tmp = x / y;
	elseif (y <= -1.0)
		tmp = 1.0;
	elseif (y <= 0.0052)
		tmp = x + y;
	elseif (y <= 1.7e+93)
		tmp = x / (y + 1.0);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.65e+87], 1.0, If[LessEqual[y, -5.8e+71], N[(x / y), $MachinePrecision], If[LessEqual[y, -1.0], 1.0, If[LessEqual[y, 0.0052], N[(x + y), $MachinePrecision], If[LessEqual[y, 1.7e+93], N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], 1.0]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.65000000000000002e87 or -5.80000000000000014e71 < y < -1 or 1.7e93 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 79.1%

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

   if -2.65000000000000002e87 < y < -5.80000000000000014e71

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. +-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around inf 100.0%

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

   if -1 < y < 0.0051999999999999998

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 99.8%

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

      2. associate-/r/ 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. flip-+ 99.9%

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

      2. associate-/r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. fma-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around 0 98.2%

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

   if 0.0051999999999999998 < y < 1.7e93

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 86.2%

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

      1. +-commutative 86.2%

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

   5. Simplified 86.2%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.65 \cdot 10^{+87}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{+71}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 0.0052:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+93}:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+87}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{+71}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq -7 \cdot 10^{-12} \lor \neg \left(y \leq 1.1 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.9e+87)
   1.0
   (if (<= y -5.8e+71)
     (/ x y)
     (if (or (<= y -7e-12) (not (<= y 1.1e-12))) (/ y (+ y 1.0)) (+ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.9e+87) {
		tmp = 1.0;
	} else if (y <= -5.8e+71) {
		tmp = x / y;
	} else if ((y <= -7e-12) || !(y <= 1.1e-12)) {
		tmp = y / (y + 1.0);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.9d+87)) then
        tmp = 1.0d0
    else if (y <= (-5.8d+71)) then
        tmp = x / y
    else if ((y <= (-7d-12)) .or. (.not. (y <= 1.1d-12))) then
        tmp = y / (y + 1.0d0)
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.9e+87) {
		tmp = 1.0;
	} else if (y <= -5.8e+71) {
		tmp = x / y;
	} else if ((y <= -7e-12) || !(y <= 1.1e-12)) {
		tmp = y / (y + 1.0);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.9e+87:
		tmp = 1.0
	elif y <= -5.8e+71:
		tmp = x / y
	elif (y <= -7e-12) or not (y <= 1.1e-12):
		tmp = y / (y + 1.0)
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.9e+87)
		tmp = 1.0;
	elseif (y <= -5.8e+71)
		tmp = Float64(x / y);
	elseif ((y <= -7e-12) || !(y <= 1.1e-12))
		tmp = Float64(y / Float64(y + 1.0));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.9e+87)
		tmp = 1.0;
	elseif (y <= -5.8e+71)
		tmp = x / y;
	elseif ((y <= -7e-12) || ~((y <= 1.1e-12)))
		tmp = y / (y + 1.0);
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.9e+87], 1.0, If[LessEqual[y, -5.8e+71], N[(x / y), $MachinePrecision], If[Or[LessEqual[y, -7e-12], N[Not[LessEqual[y, 1.1e-12]], $MachinePrecision]], N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.8999999999999998e87

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 81.5%

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

   if -2.8999999999999998e87 < y < -5.80000000000000014e71

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. +-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around inf 100.0%

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

   if -5.80000000000000014e71 < y < -7.0000000000000001e-12 or 1.09999999999999996e-12 < y

   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%

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

   if -7.0000000000000001e-12 < y < 1.09999999999999996e-12

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 99.8%

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

      2. associate-/r/ 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. flip-+ 100.0%

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

      2. associate-/r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. fma-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around 0 99.8%

      \[\leadsto \color{blue}{1} \cdot \left(x + y\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+87}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{+71}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq -7 \cdot 10^{-12} \lor \neg \left(y \leq 1.1 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+89}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{+71}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+45}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.6e+89)
   1.0
   (if (<= y -5.8e+71)
     (/ x y)
     (if (<= y -1.0) 1.0 (if (<= y 1.85e+45) (+ x y) 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.6e+89) {
		tmp = 1.0;
	} else if (y <= -5.8e+71) {
		tmp = x / y;
	} else if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 1.85e+45) {
		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 <= (-1.6d+89)) then
        tmp = 1.0d0
    else if (y <= (-5.8d+71)) then
        tmp = x / y
    else if (y <= (-1.0d0)) then
        tmp = 1.0d0
    else if (y <= 1.85d+45) 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 <= -1.6e+89) {
		tmp = 1.0;
	} else if (y <= -5.8e+71) {
		tmp = x / y;
	} else if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 1.85e+45) {
		tmp = x + y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.6e+89:
		tmp = 1.0
	elif y <= -5.8e+71:
		tmp = x / y
	elif y <= -1.0:
		tmp = 1.0
	elif y <= 1.85e+45:
		tmp = x + y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.6e+89)
		tmp = 1.0;
	elseif (y <= -5.8e+71)
		tmp = Float64(x / y);
	elseif (y <= -1.0)
		tmp = 1.0;
	elseif (y <= 1.85e+45)
		tmp = Float64(x + y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.6e+89)
		tmp = 1.0;
	elseif (y <= -5.8e+71)
		tmp = x / y;
	elseif (y <= -1.0)
		tmp = 1.0;
	elseif (y <= 1.85e+45)
		tmp = x + y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.6e+89], 1.0, If[LessEqual[y, -5.8e+71], N[(x / y), $MachinePrecision], If[LessEqual[y, -1.0], 1.0, If[LessEqual[y, 1.85e+45], N[(x + y), $MachinePrecision], 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.59999999999999994e89 or -5.80000000000000014e71 < y < -1 or 1.84999999999999989e45 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 77.9%

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

   if -1.59999999999999994e89 < y < -5.80000000000000014e71

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. +-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around inf 100.0%

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

   if -1 < y < 1.84999999999999989e45

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 99.7%

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

      2. associate-/r/ 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. flip-+ 99.9%

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

      2. associate-/r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. fma-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around 0 95.6%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+89}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{+71}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+45}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+87}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{+71}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-12}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.5e+87)
   1.0
   (if (<= y -5.8e+71)
     (/ x y)
     (if (<= y -1.0) 1.0 (if (<= y 1.1e-12) x 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.5e+87) {
		tmp = 1.0;
	} else if (y <= -5.8e+71) {
		tmp = x / y;
	} else if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 1.1e-12) {
		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 <= (-2.5d+87)) then
        tmp = 1.0d0
    else if (y <= (-5.8d+71)) then
        tmp = x / y
    else if (y <= (-1.0d0)) then
        tmp = 1.0d0
    else if (y <= 1.1d-12) 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 <= -2.5e+87) {
		tmp = 1.0;
	} else if (y <= -5.8e+71) {
		tmp = x / y;
	} else if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 1.1e-12) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.5e+87:
		tmp = 1.0
	elif y <= -5.8e+71:
		tmp = x / y
	elif y <= -1.0:
		tmp = 1.0
	elif y <= 1.1e-12:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.5e+87)
		tmp = 1.0;
	elseif (y <= -5.8e+71)
		tmp = Float64(x / y);
	elseif (y <= -1.0)
		tmp = 1.0;
	elseif (y <= 1.1e-12)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.5e+87)
		tmp = 1.0;
	elseif (y <= -5.8e+71)
		tmp = x / y;
	elseif (y <= -1.0)
		tmp = 1.0;
	elseif (y <= 1.1e-12)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.5e+87], 1.0, If[LessEqual[y, -5.8e+71], N[(x / y), $MachinePrecision], If[LessEqual[y, -1.0], 1.0, If[LessEqual[y, 1.1e-12], x, 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.4999999999999999e87 or -5.80000000000000014e71 < y < -1 or 1.09999999999999996e-12 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 73.8%

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

   if -2.4999999999999999e87 < y < -5.80000000000000014e71

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. +-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around inf 100.0%

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

   if -1 < y < 1.09999999999999996e-12

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 76.3%

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

Alternative 6: 73.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-12}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) 1.0 (if (<= y 1.1e-12) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 1.1e-12) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = 1.0d0
    else if (y <= 1.1d-12) then
        tmp = x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 1.1e-12) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = 1.0
	elif y <= 1.1e-12:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = 1.0;
	elseif (y <= 1.1e-12)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = 1.0;
	elseif (y <= 1.1e-12)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], 1.0, If[LessEqual[y, 1.1e-12], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.09999999999999996e-12 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 70.5%

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

   if -1 < y < 1.09999999999999996e-12

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 76.3%

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

Alternative 7: 38.6% 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 36.6%

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

Reproduce

herbie shell --seed 2024103 
(FPCore (x y)
  :name "Data.Colour.SRGB:invTransferFunction from colour-2.3.3"
  :precision binary64
  (/ (+ x y) (+ y 1.0)))