Data.Colour.SRGB:invTransferFunction from colour-2.3.3

Percentage Accurate: 100.0% → 100.0%

Time: 5.2s

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. Final simplification 100.0%

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

Alternative 2: 84.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y + 1}\\ t_1 := \frac{y}{y + 1}\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{+16}:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{-86}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{-33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 11000000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1 - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y 1.0))) (t_1 (/ y (+ y 1.0))))
   (if (<= y -2.2e+16)
     (+ 1.0 (/ x y))
     (if (<= y -3.2e-86)
       t_1
       (if (<= y 2.5e-58)
         t_0
         (if (<= y 5.1e-33)
           t_1
           (if (<= y 11000000000.0) t_0 (- 1.0 (/ (- 1.0 x) y)))))))))

C

double code(double x, double y) {
	double t_0 = x / (y + 1.0);
	double t_1 = y / (y + 1.0);
	double tmp;
	if (y <= -2.2e+16) {
		tmp = 1.0 + (x / y);
	} else if (y <= -3.2e-86) {
		tmp = t_1;
	} else if (y <= 2.5e-58) {
		tmp = t_0;
	} else if (y <= 5.1e-33) {
		tmp = t_1;
	} else if (y <= 11000000000.0) {
		tmp = t_0;
	} else {
		tmp = 1.0 - ((1.0 - x) / y);
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = x / (y + 1.0d0)
    t_1 = y / (y + 1.0d0)
    if (y <= (-2.2d+16)) then
        tmp = 1.0d0 + (x / y)
    else if (y <= (-3.2d-86)) then
        tmp = t_1
    else if (y <= 2.5d-58) then
        tmp = t_0
    else if (y <= 5.1d-33) then
        tmp = t_1
    else if (y <= 11000000000.0d0) then
        tmp = t_0
    else
        tmp = 1.0d0 - ((1.0d0 - x) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y + 1.0);
	double t_1 = y / (y + 1.0);
	double tmp;
	if (y <= -2.2e+16) {
		tmp = 1.0 + (x / y);
	} else if (y <= -3.2e-86) {
		tmp = t_1;
	} else if (y <= 2.5e-58) {
		tmp = t_0;
	} else if (y <= 5.1e-33) {
		tmp = t_1;
	} else if (y <= 11000000000.0) {
		tmp = t_0;
	} else {
		tmp = 1.0 - ((1.0 - x) / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y + 1.0)
	t_1 = y / (y + 1.0)
	tmp = 0
	if y <= -2.2e+16:
		tmp = 1.0 + (x / y)
	elif y <= -3.2e-86:
		tmp = t_1
	elif y <= 2.5e-58:
		tmp = t_0
	elif y <= 5.1e-33:
		tmp = t_1
	elif y <= 11000000000.0:
		tmp = t_0
	else:
		tmp = 1.0 - ((1.0 - x) / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y + 1.0))
	t_1 = Float64(y / Float64(y + 1.0))
	tmp = 0.0
	if (y <= -2.2e+16)
		tmp = Float64(1.0 + Float64(x / y));
	elseif (y <= -3.2e-86)
		tmp = t_1;
	elseif (y <= 2.5e-58)
		tmp = t_0;
	elseif (y <= 5.1e-33)
		tmp = t_1;
	elseif (y <= 11000000000.0)
		tmp = t_0;
	else
		tmp = Float64(1.0 - Float64(Float64(1.0 - x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y + 1.0);
	t_1 = y / (y + 1.0);
	tmp = 0.0;
	if (y <= -2.2e+16)
		tmp = 1.0 + (x / y);
	elseif (y <= -3.2e-86)
		tmp = t_1;
	elseif (y <= 2.5e-58)
		tmp = t_0;
	elseif (y <= 5.1e-33)
		tmp = t_1;
	elseif (y <= 11000000000.0)
		tmp = t_0;
	else
		tmp = 1.0 - ((1.0 - x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.2e+16], N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -3.2e-86], t$95$1, If[LessEqual[y, 2.5e-58], t$95$0, If[LessEqual[y, 5.1e-33], t$95$1, If[LessEqual[y, 11000000000.0], t$95$0, N[(1.0 - N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.2e16

   1. Initial program 99.9%

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

   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 100.0%

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

      1. neg-mul-1 100.0%

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

      2. distribute-neg-frac 100.0%

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

   7. Simplified 100.0%

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

   if -2.2e16 < y < -3.20000000000000006e-86 or 2.49999999999999989e-58 < y < 5.10000000000000008e-33

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 76.0%

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

      1. +-commutative 76.0%

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

   4. Simplified 76.0%

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

   if -3.20000000000000006e-86 < y < 2.49999999999999989e-58 or 5.10000000000000008e-33 < y < 1.1e10

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 82.1%

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

      1. +-commutative 82.1%

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

   4. Simplified 82.1%

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

   if 1.1e10 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 99.9%

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

      1. +-commutative 99.9%

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

      2. associate--l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. associate--r- 99.9%

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

      5. div-sub 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+16}:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{-86}:\\ \;\;\;\;\frac{y}{y + 1}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-58}:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{-33}:\\ \;\;\;\;\frac{y}{y + 1}\\ \mathbf{elif}\;y \leq 11000000000:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1 - x}{y}\\ \end{array} \]

Alternative 3: 73.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y + 1}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.08 \cdot 10^{-88}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-35}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y 1.0))))
   (if (<= y -1.0)
     1.0
     (if (<= y -1.08e-88)
       y
       (if (<= y 5.5e-58)
         t_0
         (if (<= y 1.55e-35) y (if (<= y 5e+15) t_0 1.0)))))))

C

double code(double x, double y) {
	double t_0 = x / (y + 1.0);
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= -1.08e-88) {
		tmp = y;
	} else if (y <= 5.5e-58) {
		tmp = t_0;
	} else if (y <= 1.55e-35) {
		tmp = y;
	} else if (y <= 5e+15) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (y + 1.0d0)
    if (y <= (-1.0d0)) then
        tmp = 1.0d0
    else if (y <= (-1.08d-88)) then
        tmp = y
    else if (y <= 5.5d-58) then
        tmp = t_0
    else if (y <= 1.55d-35) then
        tmp = y
    else if (y <= 5d+15) then
        tmp = t_0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y + 1.0);
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= -1.08e-88) {
		tmp = y;
	} else if (y <= 5.5e-58) {
		tmp = t_0;
	} else if (y <= 1.55e-35) {
		tmp = y;
	} else if (y <= 5e+15) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y + 1.0)
	tmp = 0
	if y <= -1.0:
		tmp = 1.0
	elif y <= -1.08e-88:
		tmp = y
	elif y <= 5.5e-58:
		tmp = t_0
	elif y <= 1.55e-35:
		tmp = y
	elif y <= 5e+15:
		tmp = t_0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y + 1.0))
	tmp = 0.0
	if (y <= -1.0)
		tmp = 1.0;
	elseif (y <= -1.08e-88)
		tmp = y;
	elseif (y <= 5.5e-58)
		tmp = t_0;
	elseif (y <= 1.55e-35)
		tmp = y;
	elseif (y <= 5e+15)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y + 1.0);
	tmp = 0.0;
	if (y <= -1.0)
		tmp = 1.0;
	elseif (y <= -1.08e-88)
		tmp = y;
	elseif (y <= 5.5e-58)
		tmp = t_0;
	elseif (y <= 1.55e-35)
		tmp = y;
	elseif (y <= 5e+15)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], 1.0, If[LessEqual[y, -1.08e-88], y, If[LessEqual[y, 5.5e-58], t$95$0, If[LessEqual[y, 1.55e-35], y, If[LessEqual[y, 5e+15], t$95$0, 1.0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 5e15 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 68.5%

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

   if -1 < y < -1.07999999999999995e-88 or 5.49999999999999996e-58 < y < 1.55000000000000006e-35

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 72.8%

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

      1. +-commutative 72.8%

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

   4. Simplified 72.8%

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

   5. Taylor expanded in y around 0 71.3%

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

   if -1.07999999999999995e-88 < y < 5.49999999999999996e-58 or 1.55000000000000006e-35 < y < 5e15

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 81.6%

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

      1. +-commutative 81.6%

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

   4. Simplified 81.6%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.08 \cdot 10^{-88}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-58}:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-35}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y + 1}\\ t_1 := 1 + \frac{x}{y}\\ t_2 := \frac{y}{y + 1}\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{-86}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-36}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 11000000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y 1.0))) (t_1 (+ 1.0 (/ x y))) (t_2 (/ y (+ y 1.0))))
   (if (<= y -2.2e+16)
     t_1
     (if (<= y -3.2e-86)
       t_2
       (if (<= y 2.5e-58)
         t_0
         (if (<= y 7e-36) t_2 (if (<= y 11000000000.0) t_0 t_1)))))))

C

double code(double x, double y) {
	double t_0 = x / (y + 1.0);
	double t_1 = 1.0 + (x / y);
	double t_2 = y / (y + 1.0);
	double tmp;
	if (y <= -2.2e+16) {
		tmp = t_1;
	} else if (y <= -3.2e-86) {
		tmp = t_2;
	} else if (y <= 2.5e-58) {
		tmp = t_0;
	} else if (y <= 7e-36) {
		tmp = t_2;
	} else if (y <= 11000000000.0) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x / (y + 1.0d0)
    t_1 = 1.0d0 + (x / y)
    t_2 = y / (y + 1.0d0)
    if (y <= (-2.2d+16)) then
        tmp = t_1
    else if (y <= (-3.2d-86)) then
        tmp = t_2
    else if (y <= 2.5d-58) then
        tmp = t_0
    else if (y <= 7d-36) then
        tmp = t_2
    else if (y <= 11000000000.0d0) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y + 1.0);
	double t_1 = 1.0 + (x / y);
	double t_2 = y / (y + 1.0);
	double tmp;
	if (y <= -2.2e+16) {
		tmp = t_1;
	} else if (y <= -3.2e-86) {
		tmp = t_2;
	} else if (y <= 2.5e-58) {
		tmp = t_0;
	} else if (y <= 7e-36) {
		tmp = t_2;
	} else if (y <= 11000000000.0) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y + 1.0)
	t_1 = 1.0 + (x / y)
	t_2 = y / (y + 1.0)
	tmp = 0
	if y <= -2.2e+16:
		tmp = t_1
	elif y <= -3.2e-86:
		tmp = t_2
	elif y <= 2.5e-58:
		tmp = t_0
	elif y <= 7e-36:
		tmp = t_2
	elif y <= 11000000000.0:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y + 1.0))
	t_1 = Float64(1.0 + Float64(x / y))
	t_2 = Float64(y / Float64(y + 1.0))
	tmp = 0.0
	if (y <= -2.2e+16)
		tmp = t_1;
	elseif (y <= -3.2e-86)
		tmp = t_2;
	elseif (y <= 2.5e-58)
		tmp = t_0;
	elseif (y <= 7e-36)
		tmp = t_2;
	elseif (y <= 11000000000.0)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y + 1.0);
	t_1 = 1.0 + (x / y);
	t_2 = y / (y + 1.0);
	tmp = 0.0;
	if (y <= -2.2e+16)
		tmp = t_1;
	elseif (y <= -3.2e-86)
		tmp = t_2;
	elseif (y <= 2.5e-58)
		tmp = t_0;
	elseif (y <= 7e-36)
		tmp = t_2;
	elseif (y <= 11000000000.0)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.2e+16], t$95$1, If[LessEqual[y, -3.2e-86], t$95$2, If[LessEqual[y, 2.5e-58], t$95$0, If[LessEqual[y, 7e-36], t$95$2, If[LessEqual[y, 11000000000.0], t$95$0, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.2e16 or 1.1e10 < 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%

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

   if -2.2e16 < y < -3.20000000000000006e-86 or 2.49999999999999989e-58 < y < 6.9999999999999999e-36

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 76.0%

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

      1. +-commutative 76.0%

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

   4. Simplified 76.0%

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

   if -3.20000000000000006e-86 < y < 2.49999999999999989e-58 or 6.9999999999999999e-36 < y < 1.1e10

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 82.1%

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

      1. +-commutative 82.1%

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

   4. Simplified 82.1%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+16}:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{-86}:\\ \;\;\;\;\frac{y}{y + 1}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-58}:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-36}:\\ \;\;\;\;\frac{y}{y + 1}\\ \mathbf{elif}\;y \leq 11000000000:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]

Alternative 5: 73.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{-86}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-59}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-34}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   1.0
   (if (<= y -3.2e-86)
     y
     (if (<= y 3.6e-59) x (if (<= y 6e-34) y (if (<= y 2.45e-5) x 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= -3.2e-86) {
		tmp = y;
	} else if (y <= 3.6e-59) {
		tmp = x;
	} else if (y <= 6e-34) {
		tmp = y;
	} else if (y <= 2.45e-5) {
		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 <= (-3.2d-86)) then
        tmp = y
    else if (y <= 3.6d-59) then
        tmp = x
    else if (y <= 6d-34) then
        tmp = y
    else if (y <= 2.45d-5) 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 <= -3.2e-86) {
		tmp = y;
	} else if (y <= 3.6e-59) {
		tmp = x;
	} else if (y <= 6e-34) {
		tmp = y;
	} else if (y <= 2.45e-5) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = 1.0
	elif y <= -3.2e-86:
		tmp = y
	elif y <= 3.6e-59:
		tmp = x
	elif y <= 6e-34:
		tmp = y
	elif y <= 2.45e-5:
		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 <= -3.2e-86)
		tmp = y;
	elseif (y <= 3.6e-59)
		tmp = x;
	elseif (y <= 6e-34)
		tmp = y;
	elseif (y <= 2.45e-5)
		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 <= -3.2e-86)
		tmp = y;
	elseif (y <= 3.6e-59)
		tmp = x;
	elseif (y <= 6e-34)
		tmp = y;
	elseif (y <= 2.45e-5)
		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, -3.2e-86], y, If[LessEqual[y, 3.6e-59], x, If[LessEqual[y, 6e-34], y, If[LessEqual[y, 2.45e-5], x, 1.0]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 2.45e-5 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 65.6%

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

   if -1 < y < -3.20000000000000006e-86 or 3.6e-59 < y < 6e-34

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 72.8%

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

      1. +-commutative 72.8%

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

   4. Simplified 72.8%

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

   5. Taylor expanded in y around 0 71.3%

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

   if -3.20000000000000006e-86 < y < 3.6e-59 or 6e-34 < y < 2.45e-5

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 83.0%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{-86}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-59}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-34}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 75.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-34} \lor \neg \left(x \leq 1.4 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3.5e-34) (not (<= x 1.4e+15)))
   (/ x (+ y 1.0))
   (/ y (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3.5e-34) || !(x <= 1.4e+15)) {
		tmp = x / (y + 1.0);
	} else {
		tmp = y / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-3.5d-34)) .or. (.not. (x <= 1.4d+15))) then
        tmp = x / (y + 1.0d0)
    else
        tmp = y / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -3.5e-34) || !(x <= 1.4e+15)) {
		tmp = x / (y + 1.0);
	} else {
		tmp = y / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -3.5e-34) or not (x <= 1.4e+15):
		tmp = x / (y + 1.0)
	else:
		tmp = y / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -3.5e-34) || !(x <= 1.4e+15))
		tmp = Float64(x / Float64(y + 1.0));
	else
		tmp = Float64(y / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -3.5e-34) || ~((x <= 1.4e+15)))
		tmp = x / (y + 1.0);
	else
		tmp = y / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -3.5e-34], N[Not[LessEqual[x, 1.4e+15]], $MachinePrecision]], N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.5e-34 or 1.4e15 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 83.1%

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

      1. +-commutative 83.1%

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

   4. Simplified 83.1%

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

   if -3.5e-34 < x < 1.4e15

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 74.2%

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

      1. +-commutative 74.2%

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

   4. Simplified 74.2%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-34} \lor \neg \left(x \leq 1.4 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + 1}\\ \end{array} \]

Alternative 7: 74.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) 1.0 (if (<= y 2.45e-5) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 2.45e-5) {
		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 <= 2.45d-5) 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 <= 2.45e-5) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = 1.0
	elif y <= 2.45e-5:
		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 <= 2.45e-5)
		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 <= 2.45e-5)
		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, 2.45e-5], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 2.45e-5 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 65.6%

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

   if -1 < y < 2.45e-5

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 73.7%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 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. Taylor expanded in y around inf 31.3%

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

3. Final simplification 31.3%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023308 
(FPCore (x y)
  :name "Data.Colour.SRGB:invTransferFunction from colour-2.3.3"
  :precision binary64
  (/ (+ x y) (+ y 1.0)))