Data.Colour.SRGB:invTransferFunction from colour-2.3.3

Percentage Accurate: 100.0% → 100.0%

Time: 6.1s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 98.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{y + 1}\\ t_1 := \frac{x}{1 + y}\\ \mathbf{if}\;t\_0 \leq -50000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{y}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (+ y 1.0))) (t_1 (/ x (+ 1.0 y))))
   (if (<= t_0 -50000.0)
     t_1
     (if (<= t_0 4e-5) (fma 1.0 y x) (if (<= t_0 2.0) (/ y (+ 1.0 y)) t_1)))))

C

double code(double x, double y) {
	double t_0 = (x + y) / (y + 1.0);
	double t_1 = x / (1.0 + y);
	double tmp;
	if (t_0 <= -50000.0) {
		tmp = t_1;
	} else if (t_0 <= 4e-5) {
		tmp = fma(1.0, 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(x + y) / Float64(y + 1.0))
	t_1 = Float64(x / Float64(1.0 + y))
	tmp = 0.0
	if (t_0 <= -50000.0)
		tmp = t_1;
	elseif (t_0 <= 4e-5)
		tmp = fma(1.0, 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[(x + y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -50000.0], t$95$1, If[LessEqual[t$95$0, 4e-5], N[(1.0 * 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))) < -5e4 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 98.1

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

   5. Applied rewrites 98.1%

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

   if -5e4 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 4.00000000000000033e-5

   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 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.8%

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

   if 4.00000000000000033e-5 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 2

   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 98.0

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

   5. Applied rewrites 98.0%

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

Alternative 3: 98.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{y + 1}\\ t_1 := \frac{x}{1 + y}\\ \mathbf{if}\;t\_0 \leq -50000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(1 - x, y, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 - \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (+ y 1.0))) (t_1 (/ x (+ 1.0 y))))
   (if (<= t_0 -50000.0)
     t_1
     (if (<= t_0 0.02)
       (fma (- 1.0 x) y x)
       (if (<= t_0 2.0) (- 1.0 (/ 1.0 y)) t_1)))))

C

double code(double x, double y) {
	double t_0 = (x + y) / (y + 1.0);
	double t_1 = x / (1.0 + y);
	double tmp;
	if (t_0 <= -50000.0) {
		tmp = t_1;
	} else if (t_0 <= 0.02) {
		tmp = fma((1.0 - x), y, x);
	} else if (t_0 <= 2.0) {
		tmp = 1.0 - (1.0 / y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x + y) / Float64(y + 1.0))
	t_1 = Float64(x / Float64(1.0 + y))
	tmp = 0.0
	if (t_0 <= -50000.0)
		tmp = t_1;
	elseif (t_0 <= 0.02)
		tmp = fma(Float64(1.0 - x), y, x);
	elseif (t_0 <= 2.0)
		tmp = Float64(1.0 - Float64(1.0 / y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -50000.0], t$95$1, If[LessEqual[t$95$0, 0.02], N[(N[(1.0 - x), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(1.0 - 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))) < -5e4 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 98.1

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

   5. Applied rewrites 98.1%

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

   if -5e4 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 0.0200000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + y \cdot \left(1 - x\right)} \]
   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 95.9

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

   5. Applied rewrites 95.9%

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

   if 0.0200000000000000004 < (/.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

      \[\leadsto \color{blue}{\left(1 + \frac{x}{y}\right) - \frac{1}{y}} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.8%

      \[\leadsto 1 - \frac{1}{y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{1}{y}\\ \mathbf{if}\;y \leq -2.4 \cdot 10^{+52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 24000000:\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ 1.0 y))))
   (if (<= y -2.4e+52)
     t_0
     (if (<= y -1.0) (/ x y) (if (<= y 24000000.0) (fma 1.0 y x) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (1.0 / y);
	double tmp;
	if (y <= -2.4e+52) {
		tmp = t_0;
	} else if (y <= -1.0) {
		tmp = x / y;
	} else if (y <= 24000000.0) {
		tmp = fma(1.0, y, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(1.0 / y))
	tmp = 0.0
	if (y <= -2.4e+52)
		tmp = t_0;
	elseif (y <= -1.0)
		tmp = Float64(x / y);
	elseif (y <= 24000000.0)
		tmp = fma(1.0, y, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(1.0 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.4e+52], t$95$0, If[LessEqual[y, -1.0], N[(x / y), $MachinePrecision], If[LessEqual[y, 24000000.0], N[(1.0 * y + x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.4e52 or 2.4e7 < 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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 80.8%

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

   if -2.4e52 < y < -1

   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 96.1

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

   5. Applied rewrites 96.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 72.6%

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

   if -1 < y < 2.4e7

   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 96.5

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

   5. Applied rewrites 96.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 96.6%

      \[\leadsto \mathsf{fma}\left(1, y, x\right) \]
3. Recombined 3 regimes into one program.
4. 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 98.8

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

   5. Applied rewrites 98.8%

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

   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 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 98.4%

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

Alternative 6: 98.1% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.82)) {
		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 <= 0.82))
		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, 0.82]], $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 0.819999999999999951 < 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 98.8

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

   5. Applied rewrites 98.8%

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

   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 < 0.819999999999999951

   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 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.82\right):\\ \;\;\;\;1 - \frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - x, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.75\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - x, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.75))) (/ x y) (fma (- 1.0 x) y x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.75)) {
		tmp = 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 <= 1.75))
		tmp = 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, 1.75]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.75 < 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 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 25.7%

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

   if -1 < y < 1.75

   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 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.75\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - x, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(1, y, x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma 1.0 y x))

C

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

Julia

function code(x, y)
	return fma(1.0, y, x)
end

Wolfram

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

Derivation

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 49.8

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

5. Applied rewrites 49.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 50.2%

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

Reproduce

herbie shell --seed 2024298 
(FPCore (x y)
  :name "Data.Colour.SRGB:invTransferFunction from colour-2.3.3"
  :precision binary64
  (/ (+ x y) (+ y 1.0)))