Data.Colour.SRGB:invTransferFunction from colour-2.3.3

Percentage Accurate: 100.0% → 100.0%

Time: 6.6s

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 98.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{1 + y}\\ t_1 := \frac{x}{1 + y}\\ \mathbf{if}\;t\_0 \leq -1000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(1 - x, y, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{y}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ y x) (+ 1.0 y))) (t_1 (/ x (+ 1.0 y))))
   (if (<= t_0 -1000000.0)
     t_1
     (if (<= t_0 5e-10)
       (fma (- 1.0 x) y x)
       (if (<= t_0 2.0) (/ y (+ 1.0 y)) t_1)))))

C

double code(double x, double y) {
	double t_0 = (y + x) / (1.0 + y);
	double t_1 = x / (1.0 + y);
	double tmp;
	if (t_0 <= -1000000.0) {
		tmp = t_1;
	} else if (t_0 <= 5e-10) {
		tmp = fma((1.0 - x), y, x);
	} else if (t_0 <= 2.0) {
		tmp = y / (1.0 + y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y + x) / Float64(1.0 + y))
	t_1 = Float64(x / Float64(1.0 + y))
	tmp = 0.0
	if (t_0 <= -1000000.0)
		tmp = t_1;
	elseif (t_0 <= 5e-10)
		tmp = fma(Float64(1.0 - x), y, x);
	elseif (t_0 <= 2.0)
		tmp = Float64(y / Float64(1.0 + y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1000000.0], t$95$1, If[LessEqual[t$95$0, 5e-10], N[(N[(1.0 - x), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(y / N[(1.0 + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < -1e6 or 2 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64)))

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if -1e6 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 5.00000000000000031e-10

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 98.2

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

   5. Applied rewrites 98.2%

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

   if 5.00000000000000031e-10 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 98.5

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

   5. Applied rewrites 98.5%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y + x}{1 + y} \leq -1000000:\\ \;\;\;\;\frac{x}{1 + y}\\ \mathbf{elif}\;\frac{y + x}{1 + y} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(1 - x, y, x\right)\\ \mathbf{elif}\;\frac{y + x}{1 + y} \leq 2:\\ \;\;\;\;\frac{y}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{1 + y}\\ t_1 := \frac{x}{1 + y}\\ \mathbf{if}\;t\_0 \leq -1000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ y x) (+ 1.0 y))) (t_1 (/ x (+ 1.0 y))))
   (if (<= t_0 -1000000.0)
     t_1
     (if (<= t_0 0.2) (fma 1.0 y x) (if (<= t_0 2.0) 1.0 t_1)))))

C

double code(double x, double y) {
	double t_0 = (y + x) / (1.0 + y);
	double t_1 = x / (1.0 + y);
	double tmp;
	if (t_0 <= -1000000.0) {
		tmp = t_1;
	} else if (t_0 <= 0.2) {
		tmp = fma(1.0, y, x);
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y + x) / Float64(1.0 + y))
	t_1 = Float64(x / Float64(1.0 + y))
	tmp = 0.0
	if (t_0 <= -1000000.0)
		tmp = t_1;
	elseif (t_0 <= 0.2)
		tmp = fma(1.0, y, x);
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1000000.0], t$95$1, If[LessEqual[t$95$0, 0.2], N[(1.0 * y + x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < -1e6 or 2 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64)))

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if -1e6 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 0.20000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 97.1

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

   5. Applied rewrites 97.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(1, y, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 97.1%

      \[\leadsto \mathsf{fma}\left(1, y, x\right) \]

   if 0.20000000000000001 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 97.6%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y + x}{1 + y} \leq -1000000:\\ \;\;\;\;\frac{x}{1 + y}\\ \mathbf{elif}\;\frac{y + x}{1 + y} \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \mathbf{elif}\;\frac{y + x}{1 + y} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{1 + y}\\ \mathbf{if}\;t\_0 \leq 0.2 \lor \neg \left(t\_0 \leq 2\right):\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ y x) (+ 1.0 y))))
   (if (or (<= t_0 0.2) (not (<= t_0 2.0))) (fma 1.0 y x) 1.0)))

C

double code(double x, double y) {
	double t_0 = (y + x) / (1.0 + y);
	double tmp;
	if ((t_0 <= 0.2) || !(t_0 <= 2.0)) {
		tmp = fma(1.0, y, x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y + x) / Float64(1.0 + y))
	tmp = 0.0
	if ((t_0 <= 0.2) || !(t_0 <= 2.0))
		tmp = fma(1.0, y, x);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.2], N[Not[LessEqual[t$95$0, 2.0]], $MachinePrecision]], N[(1.0 * y + x), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 0.20000000000000001 or 2 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64)))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 79.1

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(1, y, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 79.2%

      \[\leadsto \mathsf{fma}\left(1, y, x\right) \]

   if 0.20000000000000001 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 97.6%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y + x}{1 + y} \leq 0.2 \lor \neg \left(\frac{y + x}{1 + y} \leq 2\right):\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 98.9

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

   5. Applied rewrites 98.9%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 97.9

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

   5. Applied rewrites 97.9%

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

4. Final simplification 98.3%

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

Alternative 6: 85.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 26:\\ \;\;\;\;\mathsf{fma}\left(1 - x, y, x\right)\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+79}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   1.0
   (if (<= y 26.0) (fma (- 1.0 x) y x) (if (<= y 9.5e+79) (/ x y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 26.0) {
		tmp = fma((1.0 - x), y, x);
	} else if (y <= 9.5e+79) {
		tmp = x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = 1.0;
	elseif (y <= 26.0)
		tmp = fma(Float64(1.0 - x), y, x);
	elseif (y <= 9.5e+79)
		tmp = Float64(x / y);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], 1.0, If[LessEqual[y, 26.0], N[(N[(1.0 - x), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[y, 9.5e+79], N[(x / y), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 9.49999999999999994e79 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 77.5%

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

   if -1 < y < 26

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 97.3

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

   5. Applied rewrites 97.3%

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

   if 26 < y < 9.49999999999999994e79

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 72.8

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

   5. Applied rewrites 72.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 69.7%

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

Alternative 7: 98.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.8\right):\\ \;\;\;\;1 - \frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - x, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 0.8)))
   (- 1.0 (/ (- x) y))
   (fma (- 1.0 x) y x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.8)) {
		tmp = 1.0 - (-x / y);
	} else {
		tmp = fma((1.0 - x), y, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 0.8))
		tmp = Float64(1.0 - Float64(Float64(-x) / y));
	else
		tmp = fma(Float64(1.0 - x), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 0.8]], $MachinePrecision]], N[(1.0 - N[((-x) / y), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.80000000000000004 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 98.8%

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

   if -1 < y < 0.80000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 97.9

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

   5. Applied rewrites 97.9%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.8\right):\\ \;\;\;\;1 - \frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - x, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y + x}{1 + y} \leq 0.2:\\ \;\;\;\;1 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (+ y x) (+ 1.0 y)) 0.2) (* 1.0 y) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((y + x) / (1.0 + y)) <= 0.2) {
		tmp = 1.0 * 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 + x) / (1.0d0 + y)) <= 0.2d0) then
        tmp = 1.0d0 * y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((y + x) / (1.0 + y)) <= 0.2) {
		tmp = 1.0 * y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((y + x) / (1.0 + y)) <= 0.2:
		tmp = 1.0 * y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(y + x) / Float64(1.0 + y)) <= 0.2)
		tmp = Float64(1.0 * y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((y + x) / (1.0 + y)) <= 0.2)
		tmp = 1.0 * y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(y + x), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision], 0.2], N[(1.0 * y), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 0.20000000000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 22.8

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

   5. Applied rewrites 22.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 22.4%

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

   9. Taylor expanded in y around 0

      \[\leadsto 1 \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 21.3%

       \[\leadsto 1 \cdot y \]

   if 0.20000000000000001 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64)))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 62.8%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y + x}{1 + y} \leq 0.2:\\ \;\;\;\;1 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(1 - x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], 1.0, If[LessEqual[y, 1.0], N[(N[(1.0 - x), $MachinePrecision] * y + x), $MachinePrecision], 1.0]]

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

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

   5. Applied rewrites 71.4%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 97.9

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

   5. Applied rewrites 97.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - x, y, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 38.5% accurate, 18.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around inf

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

5. Applied rewrites 34.8%

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

Reproduce

herbie shell --seed 2024271 
(FPCore (x y)
  :name "Data.Colour.SRGB:invTransferFunction from colour-2.3.3"
  :precision binary64
  (/ (+ x y) (+ y 1.0)))