Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.5% → 99.8%

Time: 9.3s

Alternatives: 12

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + \left(y + -1\right)}{y}\\ \mathbf{if}\;x \leq -28500000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 820000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x (+ y -1.0)) y)))
   (if (<= x -28500000000000.0)
     t_0
     (if (<= x 820000000.0) (/ (fma (/ x y) x x) (+ x 1.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = (x + (y + -1.0)) / y;
	double tmp;
	if (x <= -28500000000000.0) {
		tmp = t_0;
	} else if (x <= 820000000.0) {
		tmp = fma((x / y), x, x) / (x + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x + Float64(y + -1.0)) / y)
	tmp = 0.0
	if (x <= -28500000000000.0)
		tmp = t_0;
	elseif (x <= 820000000.0)
		tmp = Float64(fma(Float64(x / y), x, x) / Float64(x + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + N[(y + -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, -28500000000000.0], t$95$0, If[LessEqual[x, 820000000.0], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.85e13 or 8.2e8 < x

   1. Initial program 77.7%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

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

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. rgt-mult-inverse N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt-out N/A

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

      12. rgt-mult-inverse N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. associate--l+ N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{x + \left(\color{blue}{\left(0 - 1\right)} + y\right)}{y} \]

      7. associate--r- N/A

         \[\leadsto \frac{x + \color{blue}{\left(0 - \left(1 - y\right)\right)}}{y} \]

      8. sub-neg N/A

         \[\leadsto \frac{x + \left(0 - \color{blue}{\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)}\right)}{y} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{x + \left(0 - \left(1 + \color{blue}{-1 \cdot y}\right)\right)}{y} \]

      10. neg-sub0 N/A

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

      11. mul-1-neg N/A

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

      12. lower-+.f64 N/A

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

      13. mul-1-neg N/A

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

      14. neg-sub0 N/A

          \[\leadsto \frac{x + \color{blue}{\left(0 - \left(1 + -1 \cdot y\right)\right)}}{y} \]

      15. mul-1-neg N/A

          \[\leadsto \frac{x + \left(0 - \left(1 + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right)}{y} \]

      16. sub-neg N/A

          \[\leadsto \frac{x + \left(0 - \color{blue}{\left(1 - y\right)}\right)}{y} \]

      17. associate--r- N/A

          \[\leadsto \frac{x + \color{blue}{\left(\left(0 - 1\right) + y\right)}}{y} \]

      18. metadata-eval N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -2.85e13 < x < 8.2e8

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

Alternative 2: 84.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 0.001:\\ \;\;\;\;x - x \cdot x\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ 1.0 (/ x y))) (+ x 1.0))))
   (if (<= t_0 -1e+21)
     (/ x y)
     (if (<= t_0 0.001)
       (- x (* x x))
       (if (<= t_0 2.0) (+ 1.0 (/ -1.0 x)) (/ x y))))))

C

double code(double x, double y) {
	double t_0 = (x * (1.0 + (x / y))) / (x + 1.0);
	double tmp;
	if (t_0 <= -1e+21) {
		tmp = x / y;
	} else if (t_0 <= 0.001) {
		tmp = x - (x * x);
	} else if (t_0 <= 2.0) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = x / 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 * (1.0d0 + (x / y))) / (x + 1.0d0)
    if (t_0 <= (-1d+21)) then
        tmp = x / y
    else if (t_0 <= 0.001d0) then
        tmp = x - (x * x)
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * (1.0 + (x / y))) / (x + 1.0);
	double tmp;
	if (t_0 <= -1e+21) {
		tmp = x / y;
	} else if (t_0 <= 0.001) {
		tmp = x - (x * x);
	} else if (t_0 <= 2.0) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * (1.0 + (x / y))) / (x + 1.0)
	tmp = 0
	if t_0 <= -1e+21:
		tmp = x / y
	elif t_0 <= 0.001:
		tmp = x - (x * x)
	elif t_0 <= 2.0:
		tmp = 1.0 + (-1.0 / x)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(1.0 + Float64(x / y))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_0 <= -1e+21)
		tmp = Float64(x / y);
	elseif (t_0 <= 0.001)
		tmp = Float64(x - Float64(x * x));
	elseif (t_0 <= 2.0)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * (1.0 + (x / y))) / (x + 1.0);
	tmp = 0.0;
	if (t_0 <= -1e+21)
		tmp = x / y;
	elseif (t_0 <= 0.001)
		tmp = x - (x * x);
	elseif (t_0 <= 2.0)
		tmp = 1.0 + (-1.0 / x);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+21], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 0.001], N[(x - N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -1e21 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 74.2%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 86.0

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

   5. Applied rewrites 86.0%

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

   if -1e21 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 1e-3

   1. Initial program 99.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 83.5

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

   5. Applied rewrites 83.5%

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

   6. Taylor expanded in x around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. mul-1-neg N/A

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

      4. distribute-lft-neg-out N/A

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

      5. unpow2 N/A

         \[\leadsto x + \left(\mathsf{neg}\left(\color{blue}{{x}^{2}}\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \color{blue}{x - {x}^{2}} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{x - {x}^{2}} \]

      8. unpow2 N/A

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

      9. lower-*.f64 82.8

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

   8. Applied rewrites 82.8%

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

   if 1e-3 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 94.6

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 91.8

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

   8. Applied rewrites 91.8%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq -1 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 0.001:\\ \;\;\;\;x - x \cdot x\\ \mathbf{elif}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 2:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 0.001:\\ \;\;\;\;x - x \cdot x\\ \mathbf{elif}\;t\_0 \leq 1000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ 1.0 (/ x y))) (+ x 1.0))))
   (if (<= t_0 -1e+21)
     (/ x y)
     (if (<= t_0 0.001) (- x (* x x)) (if (<= t_0 1000000.0) 1.0 (/ x y))))))

C

double code(double x, double y) {
	double t_0 = (x * (1.0 + (x / y))) / (x + 1.0);
	double tmp;
	if (t_0 <= -1e+21) {
		tmp = x / y;
	} else if (t_0 <= 0.001) {
		tmp = x - (x * x);
	} else if (t_0 <= 1000000.0) {
		tmp = 1.0;
	} else {
		tmp = x / 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 * (1.0d0 + (x / y))) / (x + 1.0d0)
    if (t_0 <= (-1d+21)) then
        tmp = x / y
    else if (t_0 <= 0.001d0) then
        tmp = x - (x * x)
    else if (t_0 <= 1000000.0d0) then
        tmp = 1.0d0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * (1.0 + (x / y))) / (x + 1.0);
	double tmp;
	if (t_0 <= -1e+21) {
		tmp = x / y;
	} else if (t_0 <= 0.001) {
		tmp = x - (x * x);
	} else if (t_0 <= 1000000.0) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * (1.0 + (x / y))) / (x + 1.0)
	tmp = 0
	if t_0 <= -1e+21:
		tmp = x / y
	elif t_0 <= 0.001:
		tmp = x - (x * x)
	elif t_0 <= 1000000.0:
		tmp = 1.0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(1.0 + Float64(x / y))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_0 <= -1e+21)
		tmp = Float64(x / y);
	elseif (t_0 <= 0.001)
		tmp = Float64(x - Float64(x * x));
	elseif (t_0 <= 1000000.0)
		tmp = 1.0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * (1.0 + (x / y))) / (x + 1.0);
	tmp = 0.0;
	if (t_0 <= -1e+21)
		tmp = x / y;
	elseif (t_0 <= 0.001)
		tmp = x - (x * x);
	elseif (t_0 <= 1000000.0)
		tmp = 1.0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+21], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 0.001], N[(x - N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1000000.0], 1.0, N[(x / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -1e21 or 1e6 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 74.0%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 86.8

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

   5. Applied rewrites 86.8%

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

   if -1e21 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 1e-3

   1. Initial program 99.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 83.5

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

   5. Applied rewrites 83.5%

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

   6. Taylor expanded in x around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. mul-1-neg N/A

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

      4. distribute-lft-neg-out N/A

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

      5. unpow2 N/A

         \[\leadsto x + \left(\mathsf{neg}\left(\color{blue}{{x}^{2}}\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \color{blue}{x - {x}^{2}} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{x - {x}^{2}} \]

      8. unpow2 N/A

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

      9. lower-*.f64 82.8

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

   8. Applied rewrites 82.8%

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

   if 1e-3 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 1e6

   1. Initial program 100.0%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

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

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. rgt-mult-inverse N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt-out N/A

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

      12. rgt-mult-inverse N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 93.5

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

   5. Applied rewrites 93.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 88.5%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq -1 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 0.001:\\ \;\;\;\;x - x \cdot x\\ \mathbf{elif}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 1000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1}\\ t_1 := \frac{x + -1}{y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1000000:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ 1.0 (/ x y))) (+ x 1.0))) (t_1 (/ (+ x -1.0) y)))
   (if (<= t_0 -1e+21) t_1 (if (<= t_0 1000000.0) (/ x (+ x 1.0)) t_1))))

C

double code(double x, double y) {
	double t_0 = (x * (1.0 + (x / y))) / (x + 1.0);
	double t_1 = (x + -1.0) / y;
	double tmp;
	if (t_0 <= -1e+21) {
		tmp = t_1;
	} else if (t_0 <= 1000000.0) {
		tmp = x / (x + 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x * (1.0d0 + (x / y))) / (x + 1.0d0)
    t_1 = (x + (-1.0d0)) / y
    if (t_0 <= (-1d+21)) then
        tmp = t_1
    else if (t_0 <= 1000000.0d0) then
        tmp = x / (x + 1.0d0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * (1.0 + (x / y))) / (x + 1.0);
	double t_1 = (x + -1.0) / y;
	double tmp;
	if (t_0 <= -1e+21) {
		tmp = t_1;
	} else if (t_0 <= 1000000.0) {
		tmp = x / (x + 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * (1.0 + (x / y))) / (x + 1.0)
	t_1 = (x + -1.0) / y
	tmp = 0
	if t_0 <= -1e+21:
		tmp = t_1
	elif t_0 <= 1000000.0:
		tmp = x / (x + 1.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(1.0 + Float64(x / y))) / Float64(x + 1.0))
	t_1 = Float64(Float64(x + -1.0) / y)
	tmp = 0.0
	if (t_0 <= -1e+21)
		tmp = t_1;
	elseif (t_0 <= 1000000.0)
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * (1.0 + (x / y))) / (x + 1.0);
	t_1 = (x + -1.0) / y;
	tmp = 0.0;
	if (t_0 <= -1e+21)
		tmp = t_1;
	elseif (t_0 <= 1000000.0)
		tmp = x / (x + 1.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+21], t$95$1, If[LessEqual[t$95$0, 1000000.0], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -1e21 or 1e6 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 74.0%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

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

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. rgt-mult-inverse N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt-out N/A

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

      12. rgt-mult-inverse N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 88.3

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

   5. Applied rewrites 88.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-+.f64 88.1

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

   8. Applied rewrites 88.1%

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

   if -1e21 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 1e6

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 85.4

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

   5. Applied rewrites 85.4%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq -1 \cdot 10^{+21}:\\ \;\;\;\;\frac{x + -1}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 1000000:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + -1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ 1.0 (/ x y))) (+ x 1.0))))
   (if (<= t_0 -1e+21) (/ x y) (if (<= t_0 2.0) (/ x (+ x 1.0)) (/ x y)))))

C

double code(double x, double y) {
	double t_0 = (x * (1.0 + (x / y))) / (x + 1.0);
	double tmp;
	if (t_0 <= -1e+21) {
		tmp = x / y;
	} else if (t_0 <= 2.0) {
		tmp = x / (x + 1.0);
	} else {
		tmp = x / 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 * (1.0d0 + (x / y))) / (x + 1.0d0)
    if (t_0 <= (-1d+21)) then
        tmp = x / y
    else if (t_0 <= 2.0d0) then
        tmp = x / (x + 1.0d0)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * (1.0 + (x / y))) / (x + 1.0);
	double tmp;
	if (t_0 <= -1e+21) {
		tmp = x / y;
	} else if (t_0 <= 2.0) {
		tmp = x / (x + 1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * (1.0 + (x / y))) / (x + 1.0)
	tmp = 0
	if t_0 <= -1e+21:
		tmp = x / y
	elif t_0 <= 2.0:
		tmp = x / (x + 1.0)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(1.0 + Float64(x / y))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_0 <= -1e+21)
		tmp = Float64(x / y);
	elseif (t_0 <= 2.0)
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * (1.0 + (x / y))) / (x + 1.0);
	tmp = 0.0;
	if (t_0 <= -1e+21)
		tmp = x / y;
	elseif (t_0 <= 2.0)
		tmp = x / (x + 1.0);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+21], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -1e21 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 74.2%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 86.0

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

   5. Applied rewrites 86.0%

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

   if -1e21 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 85.9

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

   5. Applied rewrites 85.9%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq -1 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 2:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 0.001:\\ \;\;\;\;x - x \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* x (+ 1.0 (/ x y))) (+ x 1.0)) 0.001) (- x (* x x)) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((x * (1.0 + (x / y))) / (x + 1.0)) <= 0.001) {
		tmp = x - (x * x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x * (1.0d0 + (x / y))) / (x + 1.0d0)) <= 0.001d0) then
        tmp = x - (x * x)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x * (1.0 + (x / y))) / (x + 1.0)) <= 0.001) {
		tmp = x - (x * x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * (1.0 + (x / y))) / (x + 1.0)) <= 0.001:
		tmp = x - (x * x)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * Float64(1.0 + Float64(x / y))) / Float64(x + 1.0)) <= 0.001)
		tmp = Float64(x - Float64(x * x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x * (1.0 + (x / y))) / (x + 1.0)) <= 0.001)
		tmp = x - (x * x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], 0.001], N[(x - N[(x * x), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 1e-3

   1. Initial program 92.0%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 59.9

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

   5. Applied rewrites 59.9%

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

   6. Taylor expanded in x around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. mul-1-neg N/A

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

      4. distribute-lft-neg-out N/A

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

      5. unpow2 N/A

         \[\leadsto x + \left(\mathsf{neg}\left(\color{blue}{{x}^{2}}\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \color{blue}{x - {x}^{2}} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{x - {x}^{2}} \]

      8. unpow2 N/A

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

      9. lower-*.f64 66.4

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

   8. Applied rewrites 66.4%

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

   if 1e-3 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 85.1%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

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

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. rgt-mult-inverse N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt-out N/A

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

      12. rgt-mult-inverse N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 89.3

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

   5. Applied rewrites 89.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 38.5%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 0.001:\\ \;\;\;\;x - x \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + \left(y + -1\right)}{y}\\ t_1 := \frac{x}{y} \cdot \left(x + y\right)\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -3.9 \cdot 10^{-191}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-196}:\\ \;\;\;\;x - x \cdot x\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x (+ y -1.0)) y)) (t_1 (* (/ x y) (+ x y))))
   (if (<= x -1.0)
     t_0
     (if (<= x -3.9e-191)
       t_1
       (if (<= x 3.1e-196) (- x (* x x)) (if (<= x 1.25) t_1 t_0))))))

C

double code(double x, double y) {
	double t_0 = (x + (y + -1.0)) / y;
	double t_1 = (x / y) * (x + y);
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= -3.9e-191) {
		tmp = t_1;
	} else if (x <= 3.1e-196) {
		tmp = x - (x * x);
	} else if (x <= 1.25) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x + (y + (-1.0d0))) / y
    t_1 = (x / y) * (x + y)
    if (x <= (-1.0d0)) then
        tmp = t_0
    else if (x <= (-3.9d-191)) then
        tmp = t_1
    else if (x <= 3.1d-196) then
        tmp = x - (x * x)
    else if (x <= 1.25d0) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x + (y + -1.0)) / y;
	double t_1 = (x / y) * (x + y);
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= -3.9e-191) {
		tmp = t_1;
	} else if (x <= 3.1e-196) {
		tmp = x - (x * x);
	} else if (x <= 1.25) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x + (y + -1.0)) / y
	t_1 = (x / y) * (x + y)
	tmp = 0
	if x <= -1.0:
		tmp = t_0
	elif x <= -3.9e-191:
		tmp = t_1
	elif x <= 3.1e-196:
		tmp = x - (x * x)
	elif x <= 1.25:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x + Float64(y + -1.0)) / y)
	t_1 = Float64(Float64(x / y) * Float64(x + y))
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= -3.9e-191)
		tmp = t_1;
	elseif (x <= 3.1e-196)
		tmp = Float64(x - Float64(x * x));
	elseif (x <= 1.25)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x + (y + -1.0)) / y;
	t_1 = (x / y) * (x + y);
	tmp = 0.0;
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= -3.9e-191)
		tmp = t_1;
	elseif (x <= 3.1e-196)
		tmp = x - (x * x);
	elseif (x <= 1.25)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + N[(y + -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, -3.9e-191], t$95$1, If[LessEqual[x, 3.1e-196], N[(x - N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.25], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1 or 1.25 < x

   1. Initial program 78.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

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

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. rgt-mult-inverse N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt-out N/A

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

      12. rgt-mult-inverse N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. associate--l+ N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{x + \left(\color{blue}{\left(0 - 1\right)} + y\right)}{y} \]

      7. associate--r- N/A

         \[\leadsto \frac{x + \color{blue}{\left(0 - \left(1 - y\right)\right)}}{y} \]

      8. sub-neg N/A

         \[\leadsto \frac{x + \left(0 - \color{blue}{\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)}\right)}{y} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{x + \left(0 - \left(1 + \color{blue}{-1 \cdot y}\right)\right)}{y} \]

      10. neg-sub0 N/A

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

      11. mul-1-neg N/A

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

      12. lower-+.f64 N/A

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

      13. mul-1-neg N/A

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

      14. neg-sub0 N/A

          \[\leadsto \frac{x + \color{blue}{\left(0 - \left(1 + -1 \cdot y\right)\right)}}{y} \]

      15. mul-1-neg N/A

          \[\leadsto \frac{x + \left(0 - \left(1 + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right)}{y} \]

      16. sub-neg N/A

          \[\leadsto \frac{x + \left(0 - \color{blue}{\left(1 - y\right)}\right)}{y} \]

      17. associate--r- N/A

          \[\leadsto \frac{x + \color{blue}{\left(\left(0 - 1\right) + y\right)}}{y} \]

      18. metadata-eval N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 98.5

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

   8. Applied rewrites 98.5%

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

   if -1 < x < -3.8999999999999999e-191 or 3.09999999999999993e-196 < x < 1.25

   1. Initial program 99.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{1 + x} + \frac{{x}^{2}}{1 + x}}{y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{1 + x} + \frac{{x}^{2}}{1 + x}}{y}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{1 + x} + \frac{{x}^{2}}{1 + x}}{y} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{x}{1 + x}} + \frac{{x}^{2}}{1 + x}}{y} \]

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 92.7

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

   5. Applied rewrites 92.7%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. div-inv N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. times-frac N/A

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

      11. *-rgt-identity N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 86.4

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

   7. Applied rewrites 86.4%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 83.4

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

   10. Applied rewrites 83.4%

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

   if -3.8999999999999999e-191 < x < 3.09999999999999993e-196

   1. Initial program 100.0%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 95.5

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

   5. Applied rewrites 95.5%

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

   6. Taylor expanded in x around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. mul-1-neg N/A

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

      4. distribute-lft-neg-out N/A

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

      5. unpow2 N/A

         \[\leadsto x + \left(\mathsf{neg}\left(\color{blue}{{x}^{2}}\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \color{blue}{x - {x}^{2}} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{x - {x}^{2}} \]

      8. unpow2 N/A

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

      9. lower-*.f64 95.5

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

   8. Applied rewrites 95.5%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x + \left(y + -1\right)}{y}\\ \mathbf{elif}\;x \leq -3.9 \cdot 10^{-191}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-196}:\\ \;\;\;\;x - x \cdot x\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\frac{x}{y} \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(y + -1\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + \left(y + -1\right)}{y}\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 70000000:\\ \;\;\;\;x \cdot \frac{x + y}{\mathsf{fma}\left(x, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x (+ y -1.0)) y)))
   (if (<= x -1.25e+16)
     t_0
     (if (<= x 70000000.0) (* x (/ (+ x y) (fma x y y))) t_0))))

C

double code(double x, double y) {
	double t_0 = (x + (y + -1.0)) / y;
	double tmp;
	if (x <= -1.25e+16) {
		tmp = t_0;
	} else if (x <= 70000000.0) {
		tmp = x * ((x + y) / fma(x, y, y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x + Float64(y + -1.0)) / y)
	tmp = 0.0
	if (x <= -1.25e+16)
		tmp = t_0;
	elseif (x <= 70000000.0)
		tmp = Float64(x * Float64(Float64(x + y) / fma(x, y, y)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + N[(y + -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, -1.25e+16], t$95$0, If[LessEqual[x, 70000000.0], N[(x * N[(N[(x + y), $MachinePrecision] / N[(x * y + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.25e16 or 7e7 < x

   1. Initial program 77.7%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

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

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. rgt-mult-inverse N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt-out N/A

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

      12. rgt-mult-inverse N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. associate--l+ N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{x + \left(\color{blue}{\left(0 - 1\right)} + y\right)}{y} \]

      7. associate--r- N/A

         \[\leadsto \frac{x + \color{blue}{\left(0 - \left(1 - y\right)\right)}}{y} \]

      8. sub-neg N/A

         \[\leadsto \frac{x + \left(0 - \color{blue}{\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)}\right)}{y} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{x + \left(0 - \left(1 + \color{blue}{-1 \cdot y}\right)\right)}{y} \]

      10. neg-sub0 N/A

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

      11. mul-1-neg N/A

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

      12. lower-+.f64 N/A

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

      13. mul-1-neg N/A

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

      14. neg-sub0 N/A

          \[\leadsto \frac{x + \color{blue}{\left(0 - \left(1 + -1 \cdot y\right)\right)}}{y} \]

      15. mul-1-neg N/A

          \[\leadsto \frac{x + \left(0 - \left(1 + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right)}{y} \]

      16. sub-neg N/A

          \[\leadsto \frac{x + \left(0 - \color{blue}{\left(1 - y\right)}\right)}{y} \]

      17. associate--r- N/A

          \[\leadsto \frac{x + \color{blue}{\left(\left(0 - 1\right) + y\right)}}{y} \]

      18. metadata-eval N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -1.25e16 < x < 7e7

   1. Initial program 99.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{1 + x} + \frac{{x}^{2}}{1 + x}}{y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{1 + x} + \frac{{x}^{2}}{1 + x}}{y}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{1 + x} + \frac{{x}^{2}}{1 + x}}{y} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{x}{1 + x}} + \frac{{x}^{2}}{1 + x}}{y} \]

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 86.1

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

   5. Applied rewrites 86.1%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. div-inv N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. times-frac N/A

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

      11. *-rgt-identity N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 80.4

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

   7. Applied rewrites 80.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*r* N/A

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

      6. div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. clear-num N/A

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

      10. times-frac N/A

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

      11. associate-/l* N/A

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

      12. lift-*.f64 N/A

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

      13. div-inv N/A

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

      14. times-frac N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l/ N/A

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

      17. clear-num N/A

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

      18. /-rgt-identity N/A

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

      19. lower-*.f64 N/A

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

   9. Applied rewrites 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+16}:\\ \;\;\;\;\frac{x + \left(y + -1\right)}{y}\\ \mathbf{elif}\;x \leq 70000000:\\ \;\;\;\;x \cdot \frac{x + y}{\mathsf{fma}\left(x, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(y + -1\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{x + 1} \cdot \left(1 + \frac{x}{y}\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.6%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. remove-double-neg N/A

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

   5. lift-+.f64 N/A

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

   6. remove-double-neg N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-/l* N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

   \[\leadsto \frac{x}{x + 1} \cdot \left(1 + \frac{x}{y}\right) \]
6. Add Preprocessing

Alternative 10: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 78.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

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

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. rgt-mult-inverse N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt-out N/A

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

      12. rgt-mult-inverse N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. associate--l+ N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{x + \left(\color{blue}{\left(0 - 1\right)} + y\right)}{y} \]

      7. associate--r- N/A

         \[\leadsto \frac{x + \color{blue}{\left(0 - \left(1 - y\right)\right)}}{y} \]

      8. sub-neg N/A

         \[\leadsto \frac{x + \left(0 - \color{blue}{\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)}\right)}{y} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{x + \left(0 - \left(1 + \color{blue}{-1 \cdot y}\right)\right)}{y} \]

      10. neg-sub0 N/A

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

      11. mul-1-neg N/A

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

      12. lower-+.f64 N/A

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

      13. mul-1-neg N/A

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

      14. neg-sub0 N/A

          \[\leadsto \frac{x + \color{blue}{\left(0 - \left(1 + -1 \cdot y\right)\right)}}{y} \]

      15. mul-1-neg N/A

          \[\leadsto \frac{x + \left(0 - \left(1 + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right)}{y} \]

      16. sub-neg N/A

          \[\leadsto \frac{x + \left(0 - \color{blue}{\left(1 - y\right)}\right)}{y} \]

      17. associate--r- N/A

          \[\leadsto \frac{x + \color{blue}{\left(\left(0 - 1\right) + y\right)}}{y} \]

      18. metadata-eval N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 98.5

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

   8. Applied rewrites 98.5%

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

   if -1 < x < 1

   1. Initial program 99.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. distribute-rgt-out-- N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 98.5

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

   5. Applied rewrites 98.5%

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

Alternative 11: 86.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + \left(y + -1\right)}{y}\\ \mathbf{if}\;x \leq -6200:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3700000:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x (+ y -1.0)) y)))
   (if (<= x -6200.0) t_0 (if (<= x 3700000.0) (/ x (+ x 1.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = (x + (y + -1.0)) / y;
	double tmp;
	if (x <= -6200.0) {
		tmp = t_0;
	} else if (x <= 3700000.0) {
		tmp = x / (x + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + (y + (-1.0d0))) / y
    if (x <= (-6200.0d0)) then
        tmp = t_0
    else if (x <= 3700000.0d0) then
        tmp = x / (x + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x + (y + -1.0)) / y;
	double tmp;
	if (x <= -6200.0) {
		tmp = t_0;
	} else if (x <= 3700000.0) {
		tmp = x / (x + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x + (y + -1.0)) / y
	tmp = 0
	if x <= -6200.0:
		tmp = t_0
	elif x <= 3700000.0:
		tmp = x / (x + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x + Float64(y + -1.0)) / y)
	tmp = 0.0
	if (x <= -6200.0)
		tmp = t_0;
	elseif (x <= 3700000.0)
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x + (y + -1.0)) / y;
	tmp = 0.0;
	if (x <= -6200.0)
		tmp = t_0;
	elseif (x <= 3700000.0)
		tmp = x / (x + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + N[(y + -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, -6200.0], t$95$0, If[LessEqual[x, 3700000.0], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -6200 or 3.7e6 < x

   1. Initial program 78.4%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

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

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. rgt-mult-inverse N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt-out N/A

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

      12. rgt-mult-inverse N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. associate--l+ N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{x + \left(\color{blue}{\left(0 - 1\right)} + y\right)}{y} \]

      7. associate--r- N/A

         \[\leadsto \frac{x + \color{blue}{\left(0 - \left(1 - y\right)\right)}}{y} \]

      8. sub-neg N/A

         \[\leadsto \frac{x + \left(0 - \color{blue}{\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)}\right)}{y} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{x + \left(0 - \left(1 + \color{blue}{-1 \cdot y}\right)\right)}{y} \]

      10. neg-sub0 N/A

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

      11. mul-1-neg N/A

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

      12. lower-+.f64 N/A

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

      13. mul-1-neg N/A

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

      14. neg-sub0 N/A

          \[\leadsto \frac{x + \color{blue}{\left(0 - \left(1 + -1 \cdot y\right)\right)}}{y} \]

      15. mul-1-neg N/A

          \[\leadsto \frac{x + \left(0 - \left(1 + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right)}{y} \]

      16. sub-neg N/A

          \[\leadsto \frac{x + \left(0 - \color{blue}{\left(1 - y\right)}\right)}{y} \]

      17. associate--r- N/A

          \[\leadsto \frac{x + \color{blue}{\left(\left(0 - 1\right) + y\right)}}{y} \]

      18. metadata-eval N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.5

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

   8. Applied rewrites 99.5%

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

   if -6200 < x < 3.7e6

   1. Initial program 99.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 76.5

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

   5. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\frac{x}{x + 1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 14.6% accurate, 34.0× 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 89.6%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. sub-neg N/A

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

   5. distribute-lft-in N/A

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

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

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

   7. associate-/r* N/A

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

   8. associate-*r/ N/A

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

   9. rgt-mult-inverse N/A

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

   10. neg-mul-1 N/A

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

   11. distribute-rgt-out N/A

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

   12. rgt-mult-inverse N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-+.f64 49.1

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

5. Applied rewrites 49.1%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 15.2%

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

Developer Target 1: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024219 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"
  :precision binary64

  :alt
  (! :herbie-platform default (* (/ x 1) (/ (+ (/ x y) 1) (+ x 1))))

  (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))