Data.Colour.SRGB:invTransferFunction from colour-2.3.3

Percentage Accurate: 100.0% → 100.0%

Time: 8.5s

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: 84.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{y + 1}\\ \mathbf{if}\;t\_0 \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(x, 1 - y, y\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+135}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (+ y 1.0))))
   (if (<= t_0 0.004)
     (fma x (- 1.0 y) y)
     (if (<= t_0 2.0) 1.0 (if (<= t_0 5e+135) (/ x y) (* x (- 1.0 y)))))))

C

double code(double x, double y) {
	double t_0 = (x + y) / (y + 1.0);
	double tmp;
	if (t_0 <= 0.004) {
		tmp = fma(x, (1.0 - y), y);
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else if (t_0 <= 5e+135) {
		tmp = x / y;
	} else {
		tmp = x * (1.0 - y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x + y) / Float64(y + 1.0))
	tmp = 0.0
	if (t_0 <= 0.004)
		tmp = fma(x, Float64(1.0 - y), y);
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	elseif (t_0 <= 5e+135)
		tmp = Float64(x / y);
	else
		tmp = Float64(x * Float64(1.0 - y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.004], N[(x * N[(1.0 - y), $MachinePrecision] + y), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, If[LessEqual[t$95$0, 5e+135], N[(x / y), $MachinePrecision], N[(x * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 0.0040000000000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*r* N/A

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

      6. *-lft-identity N/A

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

      7. associate-+l+ N/A

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

      8. *-rgt-identity N/A

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

      9. mul-1-neg N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. distribute-lft-in N/A

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

      13. lower-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 80.2

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

   8. Applied rewrites 80.2%

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

   if 0.0040000000000000001 < (/.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 96.5%

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

   if 2 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 5.00000000000000029e135

   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. +-commutative N/A

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

      3. lower-+.f64 96.2

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

   5. Applied rewrites 96.2%

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

   6. Taylor expanded in y around inf

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

      1. lower-/.f64 65.7

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

   8. Applied rewrites 65.7%

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

   if 5.00000000000000029e135 < (/.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. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. lower--.f64 90.5

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

   5. Applied rewrites 90.5%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 90.5

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

   8. Applied rewrites 90.5%

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

      1. distribute-rgt-neg-out N/A

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

      2. distribute-lft-neg-in N/A

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

      3. neg-mul-1 N/A

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

      4. distribute-lft1-in N/A

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

      5. +-commutative N/A

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

      6. neg-mul-1 N/A

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

      7. sub-neg N/A

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

      8. lift--.f64 N/A

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

      9. lower-*.f64 90.5

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

   10. Applied rewrites 90.5%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{y + 1} \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(x, 1 - y, y\right)\\ \mathbf{elif}\;\frac{x + y}{y + 1} \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{x + y}{y + 1} \leq 5 \cdot 10^{+135}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{y + 1}\\ \mathbf{if}\;t\_0 \leq 0.004:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t\_0 \leq 10^{+17}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (+ y 1.0))))
   (if (<= t_0 0.004) (+ x y) (if (<= t_0 1e+17) 1.0 (* x (- 1.0 y))))))

C

double code(double x, double y) {
	double t_0 = (x + y) / (y + 1.0);
	double tmp;
	if (t_0 <= 0.004) {
		tmp = x + y;
	} else if (t_0 <= 1e+17) {
		tmp = 1.0;
	} else {
		tmp = x * (1.0 - y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + y) / (y + 1.0d0)
    if (t_0 <= 0.004d0) then
        tmp = x + y
    else if (t_0 <= 1d+17) then
        tmp = 1.0d0
    else
        tmp = x * (1.0d0 - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x + y) / (y + 1.0);
	double tmp;
	if (t_0 <= 0.004) {
		tmp = x + y;
	} else if (t_0 <= 1e+17) {
		tmp = 1.0;
	} else {
		tmp = x * (1.0 - y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x + y) / (y + 1.0)
	tmp = 0
	if t_0 <= 0.004:
		tmp = x + y
	elif t_0 <= 1e+17:
		tmp = 1.0
	else:
		tmp = x * (1.0 - y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x + y) / Float64(y + 1.0))
	tmp = 0.0
	if (t_0 <= 0.004)
		tmp = Float64(x + y);
	elseif (t_0 <= 1e+17)
		tmp = 1.0;
	else
		tmp = Float64(x * Float64(1.0 - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x + y) / (y + 1.0);
	tmp = 0.0;
	if (t_0 <= 0.004)
		tmp = x + y;
	elseif (t_0 <= 1e+17)
		tmp = 1.0;
	else
		tmp = x * (1.0 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.004], N[(x + y), $MachinePrecision], If[LessEqual[t$95$0, 1e+17], 1.0, N[(x * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 0.0040000000000000001

   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. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. lower--.f64 80.2

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

   5. Applied rewrites 80.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 80.1%

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

      1. *-rgt-identity N/A

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

      2. lower-+.f64 80.1

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

   10. Applied rewrites 80.1%

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

   if 0.0040000000000000001 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 1e17

   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 94.1%

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

   if 1e17 < (/.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. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. lower--.f64 70.7

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

   5. Applied rewrites 70.7%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 70.7

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

   8. Applied rewrites 70.7%

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

      1. distribute-rgt-neg-out N/A

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

      2. distribute-lft-neg-in N/A

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

      3. neg-mul-1 N/A

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

      4. distribute-lft1-in N/A

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

      5. +-commutative N/A

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

      6. neg-mul-1 N/A

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

      7. sub-neg N/A

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

      8. lift--.f64 N/A

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

      9. lower-*.f64 70.7

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

   10. Applied rewrites 70.7%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{y + 1} \leq 0.004:\\ \;\;\;\;x + y\\ \mathbf{elif}\;\frac{x + y}{y + 1} \leq 10^{+17}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

      \[\leadsto \color{blue}{\left(1 + \frac{x}{y}\right) - \frac{1}{y}} \]
   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. unsub-neg N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. metadata-eval N/A

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

      15. associate-*r* N/A

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 98.3

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

   5. Applied rewrites 98.3%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*r* N/A

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

      6. *-lft-identity N/A

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

      7. associate-+l+ N/A

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

      8. *-rgt-identity N/A

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

      9. mul-1-neg N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. distribute-lft-in N/A

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

      13. lower-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 98.1

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

   8. Applied rewrites 98.1%

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

Alternative 5: 98.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(x, 1 - y, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))))
   (if (<= y -1.0) t_0 (if (<= y 0.8) (fma x (- 1.0 y) y) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 0.8) {
		tmp = fma(x, (1.0 - y), y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / y))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 0.8)
		tmp = fma(x, Float64(1.0 - y), y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 0.8], N[(x * N[(1.0 - y), $MachinePrecision] + y), $MachinePrecision], t$95$0]]]

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. unsub-neg N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. metadata-eval N/A

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

      15. associate-*r* N/A

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 97.2

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

   8. Applied rewrites 97.2%

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

   if -1 < y < 0.80000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*r* N/A

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

      6. *-lft-identity N/A

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

      7. associate-+l+ N/A

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

      8. *-rgt-identity N/A

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

      9. mul-1-neg N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. distribute-lft-in N/A

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

      13. lower-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 98.1

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

   8. Applied rewrites 98.1%

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

Alternative 6: 86.4% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) 1.0 (if (<= y 1.0) (fma x (- 1.0 y) y) 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(x, (1.0 - y), y);
	} 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(x, Float64(1.0 - y), y);
	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[(x * N[(1.0 - y), $MachinePrecision] + y), $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 73.3%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*r* N/A

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

      6. *-lft-identity N/A

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

      7. associate-+l+ N/A

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

      8. *-rgt-identity N/A

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

      9. mul-1-neg N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. distribute-lft-in N/A

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

      13. lower-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 98.1

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

   8. Applied rewrites 98.1%

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

Alternative 7: 86.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5.5:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) 1.0 (if (<= y 5.5) (+ x y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 5.5) {
		tmp = x + y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = 1.0d0
    else if (y <= 5.5d0) then
        tmp = x + y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 5.5) {
		tmp = x + y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = 1.0
	elif y <= 5.5:
		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 <= 5.5)
		tmp = Float64(x + y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = 1.0;
	elseif (y <= 5.5)
		tmp = x + y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], 1.0, If[LessEqual[y, 5.5], N[(x + y), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 5.5 < 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 73.3%

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

   if -1 < y < 5.5

   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. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. lower--.f64 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.4%

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

      1. *-rgt-identity N/A

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

      2. lower-+.f64 97.4

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

   10. Applied rewrites 97.4%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5.5:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 38.7% 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 40.5%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (x y)
  :name "Data.Colour.SRGB:invTransferFunction from colour-2.3.3"
  :precision binary64
  (/ (+ x y) (+ y 1.0)))