Data.Colour.SRGB:invTransferFunction from colour-2.3.3

Percentage Accurate: 100.0% → 100.0%

Time: 7.0s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

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: 98.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{1 + y}\\ t_1 := \frac{x}{1 + y}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(-y, y - 1, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{y}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ y x) (+ 1.0 y))) (t_1 (/ x (+ 1.0 y))))
   (if (<= t_0 -5e+17)
     t_1
     (if (<= t_0 5e-11)
       (fma (- y) (- y 1.0) 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 <= -5e+17) {
		tmp = t_1;
	} else if (t_0 <= 5e-11) {
		tmp = fma(-y, (y - 1.0), 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 <= -5e+17)
		tmp = t_1;
	elseif (t_0 <= 5e-11)
		tmp = fma(Float64(-y), Float64(y - 1.0), 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, -5e+17], t$95$1, If[LessEqual[t$95$0, 5e-11], N[((-y) * N[(y - 1.0), $MachinePrecision] + 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))) < -5e17 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 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 100.0

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

   5. Applied rewrites 100.0%

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

   if -5e17 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 5.00000000000000018e-11

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. associate-+l+ N/A

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

      6. distribute-lft-in N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. associate-*l* N/A

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

      10. metadata-eval N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft-in N/A

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

      13. distribute-rgt-in N/A

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

      14. metadata-eval N/A

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

      15. sub-neg N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.4%

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

   if 5.00000000000000018e-11 < (/.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 99.0

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

   5. Applied rewrites 99.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y + x}{1 + y} \leq -5 \cdot 10^{+17}:\\ \;\;\;\;\frac{x}{1 + y}\\ \mathbf{elif}\;\frac{y + x}{1 + y} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(-y, y - 1, x\right)\\ \mathbf{elif}\;\frac{y + x}{1 + y} \leq 2:\\ \;\;\;\;\frac{y}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{1 + y}\\ t_1 := \frac{x}{1 + y}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.0005:\\ \;\;\;\;\mathsf{fma}\left(-y, y - 1, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ y x) (+ 1.0 y))) (t_1 (/ x (+ 1.0 y))))
   (if (<= t_0 -5e+17)
     t_1
     (if (<= t_0 0.0005) (fma (- y) (- y 1.0) 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 <= -5e+17) {
		tmp = t_1;
	} else if (t_0 <= 0.0005) {
		tmp = fma(-y, (y - 1.0), 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 <= -5e+17)
		tmp = t_1;
	elseif (t_0 <= 0.0005)
		tmp = fma(Float64(-y), Float64(y - 1.0), 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, -5e+17], t$95$1, If[LessEqual[t$95$0, 0.0005], N[((-y) * N[(y - 1.0), $MachinePrecision] + 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))) < -5e17 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 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 100.0

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

   5. Applied rewrites 100.0%

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

   if -5e17 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 5.0000000000000001e-4

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. associate-+l+ N/A

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

      6. distribute-lft-in N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. associate-*l* N/A

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

      10. metadata-eval N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft-in N/A

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

      13. distribute-rgt-in N/A

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

      14. metadata-eval N/A

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

      15. sub-neg N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.9%

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

   if 5.0000000000000001e-4 < (/.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 97.7%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y + x}{1 + y} \leq -5 \cdot 10^{+17}:\\ \;\;\;\;\frac{x}{1 + y}\\ \mathbf{elif}\;\frac{y + x}{1 + y} \leq 0.0005:\\ \;\;\;\;\mathsf{fma}\left(-y, y - 1, x\right)\\ \mathbf{elif}\;\frac{y + x}{1 + y} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+56}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 1.3:\\ \;\;\;\;\mathsf{fma}\left(-y, y - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.5e+56)
   1.0
   (if (<= y -1.0) (/ x y) (if (<= y 1.3) (fma (- y) (- y 1.0) x) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.5e+56) {
		tmp = 1.0;
	} else if (y <= -1.0) {
		tmp = x / y;
	} else if (y <= 1.3) {
		tmp = fma(-y, (y - 1.0), x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.5e+56)
		tmp = 1.0;
	elseif (y <= -1.0)
		tmp = Float64(x / y);
	elseif (y <= 1.3)
		tmp = fma(Float64(-y), Float64(y - 1.0), x);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.5e+56], 1.0, If[LessEqual[y, -1.0], N[(x / y), $MachinePrecision], If[LessEqual[y, 1.3], N[((-y) * N[(y - 1.0), $MachinePrecision] + x), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.50000000000000012e56 or 1.30000000000000004 < 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 80.8%

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

   if -2.50000000000000012e56 < y < -1

   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 88.3

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

   5. Applied rewrites 88.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 78.9%

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

   if -1 < y < 1.30000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. associate-+l+ N/A

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

      6. distribute-lft-in N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. associate-*l* N/A

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

      10. metadata-eval N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft-in N/A

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

      13. distribute-rgt-in N/A

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

      14. metadata-eval N/A

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

      15. sub-neg N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.3%

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

Alternative 5: 48.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y + x}{1 + y} \leq 0.0005:\\ \;\;\;\;1 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (+ y x) (+ 1.0 y)) 0.0005) (* 1.0 y) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((y + x) / (1.0 + y)) <= 0.0005) {
		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)) <= 0.0005d0) 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)) <= 0.0005) {
		tmp = 1.0 * y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((y + x) / (1.0 + y)) <= 0.0005:
		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)) <= 0.0005)
		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)) <= 0.0005)
		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], 0.0005], 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))) < 5.0000000000000001e-4

   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 36.9

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

   5. Applied rewrites 36.9%

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

   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 36.6%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 35.8%

       \[\leadsto 1 \cdot y \]

   if 5.0000000000000001e-4 < (/.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 72.5%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y + x}{1 + y} \leq 0.0005:\\ \;\;\;\;1 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+56}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 9:\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.5e+56)
   1.0
   (if (<= y -1.0) (/ x y) (if (<= y 9.0) (fma 1.0 y x) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.5e+56) {
		tmp = 1.0;
	} else if (y <= -1.0) {
		tmp = x / y;
	} else if (y <= 9.0) {
		tmp = fma(1.0, y, x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.5e+56)
		tmp = 1.0;
	elseif (y <= -1.0)
		tmp = Float64(x / y);
	elseif (y <= 9.0)
		tmp = fma(1.0, y, x);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.5e+56], 1.0, If[LessEqual[y, -1.0], N[(x / y), $MachinePrecision], If[LessEqual[y, 9.0], N[(1.0 * y + x), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.50000000000000012e56 or 9 < 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 80.8%

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

   if -2.50000000000000012e56 < y < -1

   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 88.3

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

   5. Applied rewrites 88.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 78.9%

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

   if -1 < y < 9

   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 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.7%

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

Alternative 7: 86.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 9:\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) 1.0 (if (<= y 9.0) (fma 1.0 y x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 9.0) {
		tmp = fma(1.0, 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 <= 9.0)
		tmp = fma(1.0, y, x);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], 1.0, If[LessEqual[y, 9.0], N[(1.0 * y + x), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 9 < 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 76.5%

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

   if -1 < y < 9

   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 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.7%

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

Alternative 8: 38.2% 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 37.8%

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

Reproduce

herbie shell --seed 2024235 
(FPCore (x y)
  :name "Data.Colour.SRGB:invTransferFunction from colour-2.3.3"
  :precision binary64
  (/ (+ x y) (+ y 1.0)))