Data.Colour.SRGB:invTransferFunction from colour-2.3.3

Percentage Accurate: 100.0% → 100.0%

Time: 6.6s

Alternatives: 9

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{x + y}{y + 1} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (+ x y) (+ y 1.0)))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return (x + y) / (y + 1.0)

Julia

function code(x, y)
	return Float64(Float64(x + y) / Float64(y + 1.0))
end

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(x + y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x + y}{y + 1} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (+ x y) (+ y 1.0)))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return (x + y) / (y + 1.0)

Julia

function code(x, y)
	return Float64(Float64(x + y) / Float64(y + 1.0))
end

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(x + y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x + y}{y + 1} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (+ x y) (+ y 1.0)))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return (x + y) / (y + 1.0)

Julia

function code(x, y)
	return Float64(Float64(x + y) / Float64(y + 1.0))
end

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(x + y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 71.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 - y\right)\\ \mathbf{if}\;y \leq -6800000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{-50}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-135}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.00215:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (- 1.0 y))))
   (if (<= y -6800000.0)
     1.0
     (if (<= y -5.8e-50)
       t_0
       (if (<= y 6.8e-135) x (if (<= y 0.00215) t_0 1.0))))))

C

double code(double x, double y) {
	double t_0 = y * (1.0 - y);
	double tmp;
	if (y <= -6800000.0) {
		tmp = 1.0;
	} else if (y <= -5.8e-50) {
		tmp = t_0;
	} else if (y <= 6.8e-135) {
		tmp = x;
	} else if (y <= 0.00215) {
		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 = y * (1.0d0 - y)
    if (y <= (-6800000.0d0)) then
        tmp = 1.0d0
    else if (y <= (-5.8d-50)) then
        tmp = t_0
    else if (y <= 6.8d-135) then
        tmp = x
    else if (y <= 0.00215d0) 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 = y * (1.0 - y);
	double tmp;
	if (y <= -6800000.0) {
		tmp = 1.0;
	} else if (y <= -5.8e-50) {
		tmp = t_0;
	} else if (y <= 6.8e-135) {
		tmp = x;
	} else if (y <= 0.00215) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (1.0 - y)
	tmp = 0
	if y <= -6800000.0:
		tmp = 1.0
	elif y <= -5.8e-50:
		tmp = t_0
	elif y <= 6.8e-135:
		tmp = x
	elif y <= 0.00215:
		tmp = t_0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(1.0 - y))
	tmp = 0.0
	if (y <= -6800000.0)
		tmp = 1.0;
	elseif (y <= -5.8e-50)
		tmp = t_0;
	elseif (y <= 6.8e-135)
		tmp = x;
	elseif (y <= 0.00215)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (1.0 - y);
	tmp = 0.0;
	if (y <= -6800000.0)
		tmp = 1.0;
	elseif (y <= -5.8e-50)
		tmp = t_0;
	elseif (y <= 6.8e-135)
		tmp = x;
	elseif (y <= 0.00215)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6800000.0], 1.0, If[LessEqual[y, -5.8e-50], t$95$0, If[LessEqual[y, 6.8e-135], x, If[LessEqual[y, 0.00215], t$95$0, 1.0]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.8e6 or 0.00215 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 68.7%

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

   if -6.8e6 < y < -5.80000000000000016e-50 or 6.79999999999999978e-135 < y < 0.00215

   1. Initial program 99.9%

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

      1. clear-num 99.4%

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

      2. associate-/r/ 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around 0 94.3%

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

      1. neg-mul-1 94.3%

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

      2. sub-neg 94.3%

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

   7. Simplified 94.3%

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

   8. Taylor expanded in x around 0 60.5%

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

   if -5.80000000000000016e-50 < y < 6.79999999999999978e-135

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 90.5%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6800000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{-50}:\\ \;\;\;\;y \cdot \left(1 - y\right)\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-135}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.00215:\\ \;\;\;\;y \cdot \left(1 - y\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ t_1 := \frac{x}{y + 1}\\ \mathbf{if}\;y \leq -10500000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-135}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-24}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 13000000000:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))) (t_1 (/ x (+ y 1.0))))
   (if (<= y -10500000.0)
     t_0
     (if (<= y 6.8e-135)
       t_1
       (if (<= y 2.9e-24) y (if (<= y 13000000000.0) 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 <= -10500000.0) {
		tmp = t_0;
	} else if (y <= 6.8e-135) {
		tmp = t_1;
	} else if (y <= 2.9e-24) {
		tmp = y;
	} else if (y <= 13000000000.0) {
		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 <= (-10500000.0d0)) then
        tmp = t_0
    else if (y <= 6.8d-135) then
        tmp = t_1
    else if (y <= 2.9d-24) then
        tmp = y
    else if (y <= 13000000000.0d0) 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 <= -10500000.0) {
		tmp = t_0;
	} else if (y <= 6.8e-135) {
		tmp = t_1;
	} else if (y <= 2.9e-24) {
		tmp = y;
	} else if (y <= 13000000000.0) {
		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 <= -10500000.0:
		tmp = t_0
	elif y <= 6.8e-135:
		tmp = t_1
	elif y <= 2.9e-24:
		tmp = y
	elif y <= 13000000000.0:
		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 <= -10500000.0)
		tmp = t_0;
	elseif (y <= 6.8e-135)
		tmp = t_1;
	elseif (y <= 2.9e-24)
		tmp = y;
	elseif (y <= 13000000000.0)
		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 <= -10500000.0)
		tmp = t_0;
	elseif (y <= 6.8e-135)
		tmp = t_1;
	elseif (y <= 2.9e-24)
		tmp = y;
	elseif (y <= 13000000000.0)
		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, -10500000.0], t$95$0, If[LessEqual[y, 6.8e-135], t$95$1, If[LessEqual[y, 2.9e-24], y, If[LessEqual[y, 13000000000.0], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.05e7 or 1.3e10 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 100.0%

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

      1. +-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%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 99.3%

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

      1. neg-mul-1 99.3%

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

      2. distribute-neg-frac 99.3%

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

   8. Simplified 99.3%

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

      1. div-inv 99.2%

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

      2. cancel-sign-sub 99.2%

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

      3. div-inv 99.3%

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

      4. +-commutative 99.3%

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

   10. Applied egg-rr 99.3%

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

   if -1.05e7 < y < 6.79999999999999978e-135 or 2.8999999999999999e-24 < y < 1.3e10

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 85.2%

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

      1. +-commutative 85.2%

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

   5. Simplified 85.2%

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

   if 6.79999999999999978e-135 < y < 2.8999999999999999e-24

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 66.2%

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

      1. +-commutative 66.2%

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

   5. Simplified 66.2%

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

   6. Taylor expanded in y around 0 66.2%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -10500000:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-135}:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-24}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 13000000000:\\ \;\;\;\;\frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-135}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-11}:\\ \;\;\;\;y \cdot \left(1 - y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))))
   (if (<= y -1.0)
     t_0
     (if (<= y 1.45e-135) x (if (<= y 5.2e-11) (* y (- 1.0 y)) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.45e-135) {
		tmp = x;
	} else if (y <= 5.2e-11) {
		tmp = y * (1.0 - y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (x / y)
    if (y <= (-1.0d0)) then
        tmp = t_0
    else if (y <= 1.45d-135) then
        tmp = x
    else if (y <= 5.2d-11) then
        tmp = y * (1.0d0 - y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.45e-135) {
		tmp = x;
	} else if (y <= 5.2e-11) {
		tmp = y * (1.0 - y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x / y)
	tmp = 0
	if y <= -1.0:
		tmp = t_0
	elif y <= 1.45e-135:
		tmp = x
	elif y <= 5.2e-11:
		tmp = y * (1.0 - y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / y))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 1.45e-135)
		tmp = x;
	elseif (y <= 5.2e-11)
		tmp = Float64(y * Float64(1.0 - y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (x / y);
	tmp = 0.0;
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 1.45e-135)
		tmp = x;
	elseif (y <= 5.2e-11)
		tmp = y * (1.0 - y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 1.45e-135], x, If[LessEqual[y, 5.2e-11], N[(y * N[(1.0 - y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 5.2000000000000001e-11 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 95.4%

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

      1. +-commutative 95.4%

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

      2. associate--l+ 95.4%

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

      3. +-commutative 95.4%

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

      4. associate--r- 95.4%

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

      5. div-sub 95.4%

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

   5. Simplified 95.4%

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

   6. Taylor expanded in x around inf 94.7%

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

      1. neg-mul-1 94.7%

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

      2. distribute-neg-frac 94.7%

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

   8. Simplified 94.7%

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

      1. div-inv 94.5%

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

      2. cancel-sign-sub 94.5%

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

      3. div-inv 94.7%

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

      4. +-commutative 94.7%

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

   10. Applied egg-rr 94.7%

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

   if -1 < y < 1.4500000000000001e-135

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 86.5%

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

   if 1.4500000000000001e-135 < y < 5.2000000000000001e-11

   1. Initial program 99.9%

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

      1. clear-num 99.6%

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

      2. associate-/r/ 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. sub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 61.4%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-135}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-11}:\\ \;\;\;\;y \cdot \left(1 - y\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.76\right):\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot \left(1 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 0.76))) (+ 1.0 (/ x y)) (* (+ x y) (- 1.0 y))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.76)) {
		tmp = 1.0 + (x / y);
	} else {
		tmp = (x + y) * (1.0 - y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.0d0)) .or. (.not. (y <= 0.76d0))) then
        tmp = 1.0d0 + (x / y)
    else
        tmp = (x + y) * (1.0d0 - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.76)) {
		tmp = 1.0 + (x / y);
	} else {
		tmp = (x + y) * (1.0 - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 0.76):
		tmp = 1.0 + (x / y)
	else:
		tmp = (x + y) * (1.0 - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 0.76))
		tmp = Float64(1.0 + Float64(x / y));
	else
		tmp = Float64(Float64(x + y) * Float64(1.0 - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 0.76)))
		tmp = 1.0 + (x / y);
	else
		tmp = (x + y) * (1.0 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 0.76]], $MachinePrecision]], N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(x + y), $MachinePrecision] * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.76000000000000001 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 98.2%

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

      1. +-commutative 98.2%

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

      2. associate--l+ 98.2%

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

      3. +-commutative 98.2%

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

      4. associate--r- 98.2%

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

      5. div-sub 98.2%

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

   5. Simplified 98.2%

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

   6. Taylor expanded in x around inf 97.5%

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

      1. neg-mul-1 97.5%

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

      2. distribute-neg-frac 97.5%

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

   8. Simplified 97.5%

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

      1. div-inv 97.3%

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

      2. cancel-sign-sub 97.3%

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

      3. div-inv 97.5%

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

      4. +-commutative 97.5%

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

   10. Applied egg-rr 97.5%

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

   if -1 < y < 0.76000000000000001

   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. Taylor expanded in y around 0 98.7%

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

      1. neg-mul-1 98.7%

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

      2. sub-neg 98.7%

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

   7. Simplified 98.7%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.76\right):\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot \left(1 - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 98.2%

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

      1. +-commutative 98.2%

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

      2. associate--l+ 98.2%

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

      3. +-commutative 98.2%

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

      4. associate--r- 98.2%

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

      5. div-sub 98.2%

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

   5. Simplified 98.2%

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

   if -1 < y < 1

   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. Taylor expanded in y around 0 98.7%

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

      1. neg-mul-1 98.7%

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

      2. sub-neg 98.7%

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

   7. Simplified 98.7%

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

4. Final simplification 98.5%

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

Alternative 7: 71.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-135}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.00215:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) 1.0 (if (<= y 6.8e-135) x (if (<= y 0.00215) y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 6.8e-135) {
		tmp = x;
	} else if (y <= 0.00215) {
		tmp = y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = 1.0d0
    else if (y <= 6.8d-135) then
        tmp = x
    else if (y <= 0.00215d0) then
        tmp = y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 6.8e-135) {
		tmp = x;
	} else if (y <= 0.00215) {
		tmp = y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = 1.0
	elif y <= 6.8e-135:
		tmp = x
	elif y <= 0.00215:
		tmp = y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = 1.0;
	elseif (y <= 6.8e-135)
		tmp = x;
	elseif (y <= 0.00215)
		tmp = y;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = 1.0;
	elseif (y <= 6.8e-135)
		tmp = x;
	elseif (y <= 0.00215)
		tmp = y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], 1.0, If[LessEqual[y, 6.8e-135], x, If[LessEqual[y, 0.00215], y, 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 0.00215 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 67.6%

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

   if -1 < y < 6.79999999999999978e-135

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 86.5%

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

   if 6.79999999999999978e-135 < y < 0.00215

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 59.1%

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

      1. +-commutative 59.1%

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

   5. Simplified 59.1%

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

   6. Taylor expanded in y around 0 58.9%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-135}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.00215:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.3:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1.0) 1.0 (if (<= y 2.3) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 2.3) {
		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 <= 2.3d0) 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 <= 2.3) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = 1.0
	elif y <= 2.3:
		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 <= 2.3)
		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 <= 2.3)
		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, 2.3], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 2.2999999999999998 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 69.3%

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

   if -1 < y < 2.2999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 76.1%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.3:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 39.1% 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 33.9%

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

4. Final simplification 33.9%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024036 
(FPCore (x y)
  :name "Data.Colour.SRGB:invTransferFunction from colour-2.3.3"
  :precision binary64
  (/ (+ x y) (+ y 1.0)))