Data.Colour.SRGB:invTransferFunction from colour-2.3.3

Percentage Accurate: 100.0% → 100.0%

Time: 5.8s

Alternatives: 10

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: 73.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{+82}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -3.7 \cdot 10^{+52}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-86}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-36}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 3.7:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.4e+82)
   1.0
   (if (<= y -3.7e+52)
     (/ x y)
     (if (<= y -1.0)
       1.0
       (if (<= y 3.2e-86) x (if (<= y 1.35e-36) y (if (<= y 3.7) x 1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.4e+82) {
		tmp = 1.0;
	} else if (y <= -3.7e+52) {
		tmp = x / y;
	} else if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 3.2e-86) {
		tmp = x;
	} else if (y <= 1.35e-36) {
		tmp = y;
	} else if (y <= 3.7) {
		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 <= (-4.4d+82)) then
        tmp = 1.0d0
    else if (y <= (-3.7d+52)) then
        tmp = x / y
    else if (y <= (-1.0d0)) then
        tmp = 1.0d0
    else if (y <= 3.2d-86) then
        tmp = x
    else if (y <= 1.35d-36) then
        tmp = y
    else if (y <= 3.7d0) 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 <= -4.4e+82) {
		tmp = 1.0;
	} else if (y <= -3.7e+52) {
		tmp = x / y;
	} else if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 3.2e-86) {
		tmp = x;
	} else if (y <= 1.35e-36) {
		tmp = y;
	} else if (y <= 3.7) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.4e+82:
		tmp = 1.0
	elif y <= -3.7e+52:
		tmp = x / y
	elif y <= -1.0:
		tmp = 1.0
	elif y <= 3.2e-86:
		tmp = x
	elif y <= 1.35e-36:
		tmp = y
	elif y <= 3.7:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.4e+82)
		tmp = 1.0;
	elseif (y <= -3.7e+52)
		tmp = Float64(x / y);
	elseif (y <= -1.0)
		tmp = 1.0;
	elseif (y <= 3.2e-86)
		tmp = x;
	elseif (y <= 1.35e-36)
		tmp = y;
	elseif (y <= 3.7)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.4e+82)
		tmp = 1.0;
	elseif (y <= -3.7e+52)
		tmp = x / y;
	elseif (y <= -1.0)
		tmp = 1.0;
	elseif (y <= 3.2e-86)
		tmp = x;
	elseif (y <= 1.35e-36)
		tmp = y;
	elseif (y <= 3.7)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.4e+82], 1.0, If[LessEqual[y, -3.7e+52], N[(x / y), $MachinePrecision], If[LessEqual[y, -1.0], 1.0, If[LessEqual[y, 3.2e-86], x, If[LessEqual[y, 1.35e-36], y, If[LessEqual[y, 3.7], x, 1.0]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -4.4000000000000002e82 or -3.7e52 < y < -1 or 3.7000000000000002 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 77.1%

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

   if -4.4000000000000002e82 < y < -3.7e52

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 86.7%

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

      1. +-commutative 86.7%

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

   4. Simplified 86.7%

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

   5. Taylor expanded in y around inf 86.7%

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

   if -1 < y < 3.20000000000000006e-86 or 1.35000000000000004e-36 < y < 3.7000000000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 82.7%

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

   if 3.20000000000000006e-86 < y < 1.35000000000000004e-36

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 87.6%

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

      1. +-commutative 87.6%

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

   4. Simplified 87.6%

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

   5. Taylor expanded in y around 0 87.6%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{+82}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -3.7 \cdot 10^{+52}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-86}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-36}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 3.7:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3: 85.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-88}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-37}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 0.0225:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))))
   (if (<= y -1.0)
     t_0
     (if (<= y 5e-88) x (if (<= y 3.8e-37) y (if (<= y 0.0225) 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 <= 5e-88) {
		tmp = x;
	} else if (y <= 3.8e-37) {
		tmp = y;
	} else if (y <= 0.0225) {
		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 <= 5d-88) then
        tmp = x
    else if (y <= 3.8d-37) then
        tmp = y
    else if (y <= 0.0225d0) 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 <= 5e-88) {
		tmp = x;
	} else if (y <= 3.8e-37) {
		tmp = y;
	} else if (y <= 0.0225) {
		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 <= 5e-88:
		tmp = x
	elif y <= 3.8e-37:
		tmp = y
	elif y <= 0.0225:
		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 <= 5e-88)
		tmp = x;
	elseif (y <= 3.8e-37)
		tmp = y;
	elseif (y <= 0.0225)
		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 <= 5e-88)
		tmp = x;
	elseif (y <= 3.8e-37)
		tmp = y;
	elseif (y <= 0.0225)
		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, 5e-88], x, If[LessEqual[y, 3.8e-37], y, If[LessEqual[y, 0.0225], x, t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 0.022499999999999999 < y

   1. Initial program 100.0%

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

      1. clear-num 99.9%

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

      2. inv-pow 99.9%

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

      3. flip-+ 38.8%

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

      4. associate-/r/ 38.8%

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

      5. metadata-eval 38.8%

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

      6. unpow-prod-down 38.7%

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

      7. metadata-eval 38.7%

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

      8. metadata-eval 38.7%

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

      9. inv-pow 38.7%

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

   3. Applied egg-rr 38.7%

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

      1. associate-*r/ 38.7%

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

      2. *-rgt-identity 38.7%

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

      3. unpow-1 38.7%

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

      4. associate-/r/ 38.7%

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

   5. Simplified 38.7%

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

   6. Taylor expanded in y around inf 72.9%

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

      1. neg-mul-1 72.9%

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

   8. Simplified 72.9%

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

   9. Taylor expanded in y around inf 96.8%

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

   if -1 < y < 5.00000000000000009e-88 or 3.8000000000000004e-37 < y < 0.022499999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 83.2%

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

   if 5.00000000000000009e-88 < y < 3.8000000000000004e-37

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 87.6%

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

      1. +-commutative 87.6%

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

   4. Simplified 87.6%

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

   5. Taylor expanded in y around 0 87.6%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-88}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-37}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 0.0225:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]

Alternative 4: 85.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ t_1 := x - x \cdot y\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-87}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-37}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 0.05:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))) (t_1 (- x (* x y))))
   (if (<= y -1.0)
     t_0
     (if (<= y 8e-87) t_1 (if (<= y 2.9e-37) y (if (<= y 0.05) t_1 t_0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double t_1 = x - (x * y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 8e-87) {
		tmp = t_1;
	} else if (y <= 2.9e-37) {
		tmp = y;
	} else if (y <= 0.05) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (x / y)
    t_1 = x - (x * y)
    if (y <= (-1.0d0)) then
        tmp = t_0
    else if (y <= 8d-87) then
        tmp = t_1
    else if (y <= 2.9d-37) then
        tmp = y
    else if (y <= 0.05d0) then
        tmp = t_1
    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 t_1 = x - (x * y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 8e-87) {
		tmp = t_1;
	} else if (y <= 2.9e-37) {
		tmp = y;
	} else if (y <= 0.05) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x / y)
	t_1 = x - (x * y)
	tmp = 0
	if y <= -1.0:
		tmp = t_0
	elif y <= 8e-87:
		tmp = t_1
	elif y <= 2.9e-37:
		tmp = y
	elif y <= 0.05:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / y))
	t_1 = Float64(x - Float64(x * y))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 8e-87)
		tmp = t_1;
	elseif (y <= 2.9e-37)
		tmp = y;
	elseif (y <= 0.05)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (x / y);
	t_1 = x - (x * y);
	tmp = 0.0;
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 8e-87)
		tmp = t_1;
	elseif (y <= 2.9e-37)
		tmp = y;
	elseif (y <= 0.05)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(x - N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 8e-87], t$95$1, If[LessEqual[y, 2.9e-37], y, If[LessEqual[y, 0.05], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 0.050000000000000003 < y

   1. Initial program 100.0%

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

      1. clear-num 99.9%

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

      2. inv-pow 99.9%

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

      3. flip-+ 38.8%

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

      4. associate-/r/ 38.8%

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

      5. metadata-eval 38.8%

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

      6. unpow-prod-down 38.7%

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

      7. metadata-eval 38.7%

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

      8. metadata-eval 38.7%

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

      9. inv-pow 38.7%

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

   3. Applied egg-rr 38.7%

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

      1. associate-*r/ 38.7%

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

      2. *-rgt-identity 38.7%

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

      3. unpow-1 38.7%

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

      4. associate-/r/ 38.7%

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

   5. Simplified 38.7%

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

   6. Taylor expanded in y around inf 72.9%

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

      1. neg-mul-1 72.9%

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

   8. Simplified 72.9%

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

   9. Taylor expanded in y around inf 96.8%

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

   if -1 < y < 8.00000000000000014e-87 or 2.90000000000000005e-37 < y < 0.050000000000000003

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 84.0%

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

      1. +-commutative 84.0%

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

   4. Simplified 84.0%

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

   5. Taylor expanded in y around 0 83.8%

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

      1. +-commutative 83.8%

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

      2. mul-1-neg 83.8%

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

      3. unsub-neg 83.8%

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

      4. *-commutative 83.8%

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

   7. Simplified 83.8%

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

   if 8.00000000000000014e-87 < y < 2.90000000000000005e-37

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 87.6%

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

      1. +-commutative 87.6%

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

   4. Simplified 87.6%

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

   5. Taylor expanded in y around 0 87.6%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-87}:\\ \;\;\;\;x - x \cdot y\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-37}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 0.05:\\ \;\;\;\;x - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]

Alternative 5: 85.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ t_1 := \frac{x}{y + 1}\\ \mathbf{if}\;y \leq -2.8 \cdot 10^{+25}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-85}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-31}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 29000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))) (t_1 (/ x (+ y 1.0))))
   (if (<= y -2.8e+25)
     t_0
     (if (<= y 1.3e-85)
       t_1
       (if (<= y 8.5e-31) y (if (<= y 29000.0) t_1 t_0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double t_1 = x / (y + 1.0);
	double tmp;
	if (y <= -2.8e+25) {
		tmp = t_0;
	} else if (y <= 1.3e-85) {
		tmp = t_1;
	} else if (y <= 8.5e-31) {
		tmp = y;
	} else if (y <= 29000.0) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (x / y)
    t_1 = x / (y + 1.0d0)
    if (y <= (-2.8d+25)) then
        tmp = t_0
    else if (y <= 1.3d-85) then
        tmp = t_1
    else if (y <= 8.5d-31) then
        tmp = y
    else if (y <= 29000.0d0) then
        tmp = t_1
    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 t_1 = x / (y + 1.0);
	double tmp;
	if (y <= -2.8e+25) {
		tmp = t_0;
	} else if (y <= 1.3e-85) {
		tmp = t_1;
	} else if (y <= 8.5e-31) {
		tmp = y;
	} else if (y <= 29000.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x / y)
	t_1 = x / (y + 1.0)
	tmp = 0
	if y <= -2.8e+25:
		tmp = t_0
	elif y <= 1.3e-85:
		tmp = t_1
	elif y <= 8.5e-31:
		tmp = y
	elif y <= 29000.0:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / y))
	t_1 = Float64(x / Float64(y + 1.0))
	tmp = 0.0
	if (y <= -2.8e+25)
		tmp = t_0;
	elseif (y <= 1.3e-85)
		tmp = t_1;
	elseif (y <= 8.5e-31)
		tmp = y;
	elseif (y <= 29000.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (x / y);
	t_1 = x / (y + 1.0);
	tmp = 0.0;
	if (y <= -2.8e+25)
		tmp = t_0;
	elseif (y <= 1.3e-85)
		tmp = t_1;
	elseif (y <= 8.5e-31)
		tmp = y;
	elseif (y <= 29000.0)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.8e+25], t$95$0, If[LessEqual[y, 1.3e-85], t$95$1, If[LessEqual[y, 8.5e-31], y, If[LessEqual[y, 29000.0], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.8000000000000002e25 or 29000 < y

   1. Initial program 100.0%

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

      1. clear-num 99.9%

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

      2. inv-pow 99.9%

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

      3. flip-+ 37.0%

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

      4. associate-/r/ 36.9%

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

      5. metadata-eval 36.9%

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

      6. unpow-prod-down 36.9%

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

      7. metadata-eval 36.9%

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

      8. metadata-eval 36.9%

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

      9. inv-pow 36.9%

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

   3. Applied egg-rr 36.9%

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

      1. associate-*r/ 36.9%

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

      2. *-rgt-identity 36.9%

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

      3. unpow-1 36.9%

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

      4. associate-/r/ 36.9%

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

   5. Simplified 36.9%

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

   6. Taylor expanded in y around inf 75.7%

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

      1. neg-mul-1 75.7%

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

   8. Simplified 75.7%

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

   9. Taylor expanded in y around inf 99.0%

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

   if -2.8000000000000002e25 < y < 1.30000000000000006e-85 or 8.5000000000000007e-31 < y < 29000

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 83.9%

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

      1. +-commutative 83.9%

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

   4. Simplified 83.9%

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

   if 1.30000000000000006e-85 < y < 8.5000000000000007e-31

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 87.6%

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

      1. +-commutative 87.6%

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

   4. Simplified 87.6%

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

   5. Taylor expanded in y around 0 87.6%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+25}:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-85}:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-31}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 29000:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]

Alternative 6: 85.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ t_1 := \frac{x}{y + 1}\\ \mathbf{if}\;y \leq -2.8 \cdot 10^{+25}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-85}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-32}:\\ \;\;\;\;\frac{y}{y + 1}\\ \mathbf{elif}\;y \leq 27000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))) (t_1 (/ x (+ y 1.0))))
   (if (<= y -2.8e+25)
     t_0
     (if (<= y 1.35e-85)
       t_1
       (if (<= y 5.5e-32) (/ y (+ y 1.0)) (if (<= y 27000.0) t_1 t_0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double t_1 = x / (y + 1.0);
	double tmp;
	if (y <= -2.8e+25) {
		tmp = t_0;
	} else if (y <= 1.35e-85) {
		tmp = t_1;
	} else if (y <= 5.5e-32) {
		tmp = y / (y + 1.0);
	} else if (y <= 27000.0) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (x / y)
    t_1 = x / (y + 1.0d0)
    if (y <= (-2.8d+25)) then
        tmp = t_0
    else if (y <= 1.35d-85) then
        tmp = t_1
    else if (y <= 5.5d-32) then
        tmp = y / (y + 1.0d0)
    else if (y <= 27000.0d0) then
        tmp = t_1
    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 t_1 = x / (y + 1.0);
	double tmp;
	if (y <= -2.8e+25) {
		tmp = t_0;
	} else if (y <= 1.35e-85) {
		tmp = t_1;
	} else if (y <= 5.5e-32) {
		tmp = y / (y + 1.0);
	} else if (y <= 27000.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x / y)
	t_1 = x / (y + 1.0)
	tmp = 0
	if y <= -2.8e+25:
		tmp = t_0
	elif y <= 1.35e-85:
		tmp = t_1
	elif y <= 5.5e-32:
		tmp = y / (y + 1.0)
	elif y <= 27000.0:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / y))
	t_1 = Float64(x / Float64(y + 1.0))
	tmp = 0.0
	if (y <= -2.8e+25)
		tmp = t_0;
	elseif (y <= 1.35e-85)
		tmp = t_1;
	elseif (y <= 5.5e-32)
		tmp = Float64(y / Float64(y + 1.0));
	elseif (y <= 27000.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (x / y);
	t_1 = x / (y + 1.0);
	tmp = 0.0;
	if (y <= -2.8e+25)
		tmp = t_0;
	elseif (y <= 1.35e-85)
		tmp = t_1;
	elseif (y <= 5.5e-32)
		tmp = y / (y + 1.0);
	elseif (y <= 27000.0)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.8e+25], t$95$0, If[LessEqual[y, 1.35e-85], t$95$1, If[LessEqual[y, 5.5e-32], N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 27000.0], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.8000000000000002e25 or 27000 < y

   1. Initial program 100.0%

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

      1. clear-num 99.9%

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

      2. inv-pow 99.9%

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

      3. flip-+ 37.0%

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

      4. associate-/r/ 36.9%

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

      5. metadata-eval 36.9%

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

      6. unpow-prod-down 36.9%

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

      7. metadata-eval 36.9%

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

      8. metadata-eval 36.9%

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

      9. inv-pow 36.9%

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

   3. Applied egg-rr 36.9%

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

      1. associate-*r/ 36.9%

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

      2. *-rgt-identity 36.9%

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

      3. unpow-1 36.9%

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

      4. associate-/r/ 36.9%

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

   5. Simplified 36.9%

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

   6. Taylor expanded in y around inf 75.7%

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

      1. neg-mul-1 75.7%

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

   8. Simplified 75.7%

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

   9. Taylor expanded in y around inf 99.0%

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

   if -2.8000000000000002e25 < y < 1.3500000000000001e-85 or 5.50000000000000024e-32 < y < 27000

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 83.9%

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

      1. +-commutative 83.9%

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

   4. Simplified 83.9%

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

   if 1.3500000000000001e-85 < y < 5.50000000000000024e-32

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 87.6%

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

      1. +-commutative 87.6%

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

   4. Simplified 87.6%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+25}:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-85}:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-32}:\\ \;\;\;\;\frac{y}{y + 1}\\ \mathbf{elif}\;y \leq 27000:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]

Alternative 7: 98.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.82\right):\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 0.82))) (+ 1.0 (/ x y)) (+ x (* y (- 1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.82)) {
		tmp = 1.0 + (x / y);
	} else {
		tmp = x + (y * (1.0 - x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.0d0)) .or. (.not. (y <= 0.82d0))) then
        tmp = 1.0d0 + (x / y)
    else
        tmp = x + (y * (1.0d0 - x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.82)) {
		tmp = 1.0 + (x / y);
	} else {
		tmp = x + (y * (1.0 - x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 0.82):
		tmp = 1.0 + (x / y)
	else:
		tmp = x + (y * (1.0 - x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 0.82))
		tmp = Float64(1.0 + Float64(x / y));
	else
		tmp = Float64(x + Float64(y * Float64(1.0 - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 0.82)))
		tmp = 1.0 + (x / y);
	else
		tmp = x + (y * (1.0 - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 0.82]], $MachinePrecision]], N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.819999999999999951 < y

   1. Initial program 100.0%

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

      1. clear-num 99.9%

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

      2. inv-pow 99.9%

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

      3. flip-+ 38.8%

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

      4. associate-/r/ 38.8%

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

      5. metadata-eval 38.8%

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

      6. unpow-prod-down 38.7%

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

      7. metadata-eval 38.7%

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

      8. metadata-eval 38.7%

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

      9. inv-pow 38.7%

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

   3. Applied egg-rr 38.7%

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

      1. associate-*r/ 38.7%

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

      2. *-rgt-identity 38.7%

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

      3. unpow-1 38.7%

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

      4. associate-/r/ 38.7%

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

   5. Simplified 38.7%

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

   6. Taylor expanded in y around inf 72.9%

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

      1. neg-mul-1 72.9%

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

   8. Simplified 72.9%

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

   9. Taylor expanded in y around inf 96.8%

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

   if -1 < y < 0.819999999999999951

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.1%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.82\right):\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - x\right)\\ \end{array} \]

Alternative 8: 73.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-85}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-32}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 7.8:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   1.0
   (if (<= y 1.35e-85) x (if (<= y 3.6e-32) y (if (<= y 7.8) x 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 1.35e-85) {
		tmp = x;
	} else if (y <= 3.6e-32) {
		tmp = y;
	} else if (y <= 7.8) {
		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 <= 1.35d-85) then
        tmp = x
    else if (y <= 3.6d-32) then
        tmp = y
    else if (y <= 7.8d0) 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 <= 1.35e-85) {
		tmp = x;
	} else if (y <= 3.6e-32) {
		tmp = y;
	} else if (y <= 7.8) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = 1.0
	elif y <= 1.35e-85:
		tmp = x
	elif y <= 3.6e-32:
		tmp = y
	elif y <= 7.8:
		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 <= 1.35e-85)
		tmp = x;
	elseif (y <= 3.6e-32)
		tmp = y;
	elseif (y <= 7.8)
		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 <= 1.35e-85)
		tmp = x;
	elseif (y <= 3.6e-32)
		tmp = y;
	elseif (y <= 7.8)
		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, 1.35e-85], x, If[LessEqual[y, 3.6e-32], y, If[LessEqual[y, 7.8], x, 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 7.79999999999999982 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 73.5%

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

   if -1 < y < 1.3500000000000001e-85 or 3.59999999999999993e-32 < y < 7.79999999999999982

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 82.7%

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

   if 1.3500000000000001e-85 < y < 3.59999999999999993e-32

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 87.6%

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

      1. +-commutative 87.6%

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

   4. Simplified 87.6%

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

   5. Taylor expanded in y around 0 87.6%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-85}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-32}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 7.8:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9: 74.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.4:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1.0) 1.0 (if (<= y 2.4) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 2.4) {
		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 <= 2.4d0) 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 <= 2.4) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = 1.0
	elif y <= 2.4:
		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 <= 2.4)
		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 <= 2.4)
		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, 2.4], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 2.39999999999999991 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 73.5%

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

   if -1 < y < 2.39999999999999991

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 78.8%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.4:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10: 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. Taylor expanded in y around inf 36.4%

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

3. Final simplification 36.4%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023174 
(FPCore (x y)
  :name "Data.Colour.SRGB:invTransferFunction from colour-2.3.3"
  :precision binary64
  (/ (+ x y) (+ y 1.0)))