Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.4% → 99.9%

Time: 6.3s

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: 88.4% 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: 99.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-58}:\\ \;\;\;\;\frac{\frac{x}{1 + x} \cdot \left(y + x\right)}{y}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 1}{y} + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2e-58)
   (/ (* (/ x (+ 1.0 x)) (+ y x)) y)
   (if (<= x 2e+16) (/ (fma (/ x y) x x) (+ x 1.0)) (+ (/ (- x 1.0) y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2e-58) {
		tmp = ((x / (1.0 + x)) * (y + x)) / y;
	} else if (x <= 2e+16) {
		tmp = fma((x / y), x, x) / (x + 1.0);
	} else {
		tmp = ((x - 1.0) / y) + 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2e-58)
		tmp = Float64(Float64(Float64(x / Float64(1.0 + x)) * Float64(y + x)) / y);
	elseif (x <= 2e+16)
		tmp = Float64(fma(Float64(x / y), x, x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(Float64(x - 1.0) / y) + 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2e-58], N[(N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 2e+16], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.0000000000000001e-58

   1. Initial program 86.5%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -2.0000000000000001e-58 < x < 2e16

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-lft1-in N/A

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

      5. 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%

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

   if 2e16 < x

   1. Initial program 78.6%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 100.0%

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

Alternative 2: 85.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -1000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(x - 1, x, 1\right) \cdot x\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 - {x}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -1000.0)
     (/ x y)
     (if (<= t_0 0.5)
       (* (fma (- x 1.0) x 1.0) x)
       (if (<= t_0 2.0) (- 1.0 (pow x -1.0)) (/ x y))))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -1000.0) {
		tmp = x / y;
	} else if (t_0 <= 0.5) {
		tmp = fma((x - 1.0), x, 1.0) * x;
	} else if (t_0 <= 2.0) {
		tmp = 1.0 - pow(x, -1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_0 <= -1000.0)
		tmp = Float64(x / y);
	elseif (t_0 <= 0.5)
		tmp = Float64(fma(Float64(x - 1.0), x, 1.0) * x);
	elseif (t_0 <= 2.0)
		tmp = Float64(1.0 - (x ^ -1.0));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1000.0], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[(N[(N[(x - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(1.0 - N[Power[x, -1.0], $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))) < -1e3 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 79.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 75.7%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 81.9

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

   10. Applied rewrites 81.9%

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

   if -1e3 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.5

   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. lower-+.f64 86.7

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

   5. Applied rewrites 86.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 86.0%

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

   if 0.5 < (/.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. lower-+.f64 97.6

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 93.8%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq -1000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(x - 1, x, 1\right) \cdot x\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 2:\\ \;\;\;\;1 - {x}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 81.9%

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

   7. Applied rewrites 98.4%

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

   if -1 < x < 1

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

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

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. *-rgt-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 98.2

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

   5. Applied rewrites 98.2%

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

   7. Applied rewrites 98.3%

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

4. Final simplification 98.3%

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

Alternative 4: 86.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -1000 \lor \neg \left(t\_0 \leq 2\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (or (<= t_0 -1000.0) (not (<= t_0 2.0))) (/ x y) (/ x (+ 1.0 x)))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if ((t_0 <= -1000.0) || !(t_0 <= 2.0)) {
		tmp = x / y;
	} else {
		tmp = x / (1.0 + x);
	}
	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 * ((x / y) + 1.0d0)) / (x + 1.0d0)
    if ((t_0 <= (-1000.0d0)) .or. (.not. (t_0 <= 2.0d0))) then
        tmp = x / y
    else
        tmp = x / (1.0d0 + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if ((t_0 <= -1000.0) || !(t_0 <= 2.0)) {
		tmp = x / y;
	} else {
		tmp = x / (1.0 + x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0)
	tmp = 0
	if (t_0 <= -1000.0) or not (t_0 <= 2.0):
		tmp = x / y
	else:
		tmp = x / (1.0 + x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if ((t_0 <= -1000.0) || !(t_0 <= 2.0))
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(1.0 + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	tmp = 0.0;
	if ((t_0 <= -1000.0) || ~((t_0 <= 2.0)))
		tmp = x / y;
	else
		tmp = x / (1.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1000.0], N[Not[LessEqual[t$95$0, 2.0]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(x / N[(1.0 + x), $MachinePrecision]), $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))) < -1e3 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 79.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 75.7%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 81.9

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

   10. Applied rewrites 81.9%

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

   if -1e3 < (/.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. lower-+.f64 89.0

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

   5. Applied rewrites 89.0%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq -1000 \lor \neg \left(\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 2\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -1000 \lor \neg \left(t\_0 \leq 0.9999999999999987\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, x, 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (or (<= t_0 -1000.0) (not (<= t_0 0.9999999999999987)))
     (/ x y)
     (* (fma (- x 1.0) x 1.0) x))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if ((t_0 <= -1000.0) || !(t_0 <= 0.9999999999999987)) {
		tmp = x / y;
	} else {
		tmp = fma((x - 1.0), x, 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if ((t_0 <= -1000.0) || !(t_0 <= 0.9999999999999987))
		tmp = Float64(x / y);
	else
		tmp = Float64(fma(Float64(x - 1.0), x, 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1000.0], N[Not[LessEqual[t$95$0, 0.9999999999999987]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(N[(N[(x - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $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))) < -1e3 or 0.99999999999999867 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 83.7%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 72.3%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 67.2

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

   10. Applied rewrites 67.2%

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

   if -1e3 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.99999999999999867

   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. lower-+.f64 86.8

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

   5. Applied rewrites 86.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 83.3%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq -1000 \lor \neg \left(\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 0.9999999999999987\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, x, 1\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -1000 \lor \neg \left(t\_0 \leq 0.5\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (or (<= t_0 -1000.0) (not (<= t_0 0.5))) (/ x y) (fma (- x) x x))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if ((t_0 <= -1000.0) || !(t_0 <= 0.5)) {
		tmp = x / y;
	} else {
		tmp = fma(-x, x, x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if ((t_0 <= -1000.0) || !(t_0 <= 0.5))
		tmp = Float64(x / y);
	else
		tmp = fma(Float64(-x), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1000.0], N[Not[LessEqual[t$95$0, 0.5]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[((-x) * x + x), $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))) < -1e3 or 0.5 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 84.1%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 72.4%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 65.4

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

   10. Applied rewrites 65.4%

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

   if -1e3 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.5

   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. lower-+.f64 86.7

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

   5. Applied rewrites 86.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 85.9%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq -1000 \lor \neg \left(\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 0.5\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq -2 \cdot 10^{+44}:\\ \;\;\;\;\left(-x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)) -2e+44) (* (- x) x) (* 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (((x * ((x / y) + 1.0)) / (x + 1.0)) <= -2e+44) {
		tmp = -x * x;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x * ((x / y) + 1.0d0)) / (x + 1.0d0)) <= (-2d+44)) then
        tmp = -x * x
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x * ((x / y) + 1.0)) / (x + 1.0)) <= -2e+44) {
		tmp = -x * x;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * ((x / y) + 1.0)) / (x + 1.0)) <= -2e+44:
		tmp = -x * x
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0)) <= -2e+44)
		tmp = Float64(Float64(-x) * x);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x * ((x / y) + 1.0)) / (x + 1.0)) <= -2e+44)
		tmp = -x * x;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], -2e+44], N[((-x) * x), $MachinePrecision], N[(1.0 * x), $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))) < -2.0000000000000002e44

   1. Initial program 76.3%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. lower-+.f64 1.1

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

   5. Applied rewrites 1.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 35.6%

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

   9. Taylor expanded in x around inf

      \[\leadsto -1 \cdot {x}^{\color{blue}{2}} \]
   10. Step-by-step derivation

   11. Applied rewrites 35.7%

       \[\leadsto \left(-x\right) \cdot x \]

   if -2.0000000000000002e44 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 94.3%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. lower-+.f64 61.8

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

   5. Applied rewrites 61.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 50.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 47.5%

       \[\leadsto 1 \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+103} \lor \neg \left(x \leq 2 \cdot 10^{+16}\right):\\ \;\;\;\;\frac{x - 1}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1e+103) (not (<= x 2e+16)))
   (+ (/ (- x 1.0) y) 1.0)
   (/ (fma (/ x y) x x) (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1e+103) || !(x <= 2e+16)) {
		tmp = ((x - 1.0) / y) + 1.0;
	} else {
		tmp = fma((x / y), x, x) / (x + 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1e+103) || !(x <= 2e+16))
		tmp = Float64(Float64(Float64(x - 1.0) / y) + 1.0);
	else
		tmp = Float64(fma(Float64(x / y), x, x) / Float64(x + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1e+103], N[Not[LessEqual[x, 2e+16]], $MachinePrecision]], N[(N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1e103 or 2e16 < x

   1. Initial program 78.2%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{x} + \frac{1}{y}\right) - \frac{1}{x \cdot y}\right)} \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 100.0%

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

   if -1e103 < x < 2e16

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-lft1-in N/A

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

      5. 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. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+103} \lor \neg \left(x \leq 2 \cdot 10^{+16}\right):\\ \;\;\;\;\frac{x - 1}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 81.9%

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

   7. Applied rewrites 98.4%

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

   if -1 < x < 1

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

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

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. *-rgt-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 98.2

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

   5. Applied rewrites 98.2%

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

4. Final simplification 98.3%

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

Alternative 10: 86.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+17} \lor \neg \left(x \leq 5200000000000\right):\\ \;\;\;\;\frac{x - 1}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -5.4e+17) (not (<= x 5200000000000.0)))
   (+ (/ (- x 1.0) y) 1.0)
   (/ x (+ 1.0 x))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -5.4e+17) || !(x <= 5200000000000.0)) {
		tmp = ((x - 1.0) / y) + 1.0;
	} else {
		tmp = x / (1.0 + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-5.4d+17)) .or. (.not. (x <= 5200000000000.0d0))) then
        tmp = ((x - 1.0d0) / y) + 1.0d0
    else
        tmp = x / (1.0d0 + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -5.4e+17) || !(x <= 5200000000000.0)) {
		tmp = ((x - 1.0) / y) + 1.0;
	} else {
		tmp = x / (1.0 + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -5.4e+17) or not (x <= 5200000000000.0):
		tmp = ((x - 1.0) / y) + 1.0
	else:
		tmp = x / (1.0 + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -5.4e+17) || !(x <= 5200000000000.0))
		tmp = Float64(Float64(Float64(x - 1.0) / y) + 1.0);
	else
		tmp = Float64(x / Float64(1.0 + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -5.4e+17) || ~((x <= 5200000000000.0)))
		tmp = ((x - 1.0) / y) + 1.0;
	else
		tmp = x / (1.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -5.4e+17], N[Not[LessEqual[x, 5200000000000.0]], $MachinePrecision]], N[(N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision] + 1.0), $MachinePrecision], N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.4e17 or 5.2e12 < x

   1. Initial program 81.2%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{x} + \frac{1}{y}\right) - \frac{1}{x \cdot y}\right)} \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 100.0%

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

   if -5.4e17 < x < 5.2e12

   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. lower-+.f64 76.2

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

   5. Applied rewrites 76.2%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+17} \lor \neg \left(x \leq 5200000000000\right):\\ \;\;\;\;\frac{x - 1}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.6% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y)
	return fma(Float64(-x), x, x)
end

Wolfram

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

Derivation

1. Initial program 91.2%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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. lower-+.f64 51.4

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

5. Applied rewrites 51.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 44.8%

   \[\leadsto \mathsf{fma}\left(-x, \color{blue}{x}, x\right) \]
9. Add Preprocessing

Alternative 12: 38.3% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.2%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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. lower-+.f64 51.4

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

5. Applied rewrites 51.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 44.4%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 39.9%

    \[\leadsto 1 \cdot x \]
12. Add Preprocessing

Developer Target 1: 99.9% 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 2024298 
(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)))