Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.6% → 99.9%

Time: 7.6s

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: 88.6% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 100.0

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 100.0

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 91.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ t_1 := {y}^{-1} \cdot x\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+27}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-36}:\\ \;\;\;\;\left(\frac{x}{y} - x\right) \cdot x + x\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+53}:\\ \;\;\;\;\frac{x \cdot \left(y + x\right)}{\mathsf{fma}\left(y, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))) (t_1 (* (pow y -1.0) x)))
   (if (<= t_0 -5e+27)
     t_1
     (if (<= t_0 5e-36)
       (+ (* (- (/ x y) x) x) x)
       (if (<= t_0 2.0)
         (/ x (- x -1.0))
         (if (<= t_0 5e+53) (/ (* x (+ y x)) (fma y x y)) t_1))))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double t_1 = pow(y, -1.0) * x;
	double tmp;
	if (t_0 <= -5e+27) {
		tmp = t_1;
	} else if (t_0 <= 5e-36) {
		tmp = (((x / y) - x) * x) + x;
	} else if (t_0 <= 2.0) {
		tmp = x / (x - -1.0);
	} else if (t_0 <= 5e+53) {
		tmp = (x * (y + x)) / fma(y, x, y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	t_1 = Float64((y ^ -1.0) * x)
	tmp = 0.0
	if (t_0 <= -5e+27)
		tmp = t_1;
	elseif (t_0 <= 5e-36)
		tmp = Float64(Float64(Float64(Float64(x / y) - x) * x) + x);
	elseif (t_0 <= 2.0)
		tmp = Float64(x / Float64(x - -1.0));
	elseif (t_0 <= 5e+53)
		tmp = Float64(Float64(x * Float64(y + x)) / fma(y, x, y));
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[Power[y, -1.0], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+27], t$95$1, If[LessEqual[t$95$0, 5e-36], N[(N[(N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision] * x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+53], N[(N[(x * N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y * x + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -4.99999999999999979e27 or 5.0000000000000004e53 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 71.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 71.8

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

   4. Applied rewrites 71.8%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{{x}^{2}}{y \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 85.6

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

   7. Applied rewrites 85.6%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{y} \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 94.2%

       \[\leadsto \frac{1}{y} \cdot x \]

   if -4.99999999999999979e27 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 5.00000000000000004e-36

   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

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/r/ N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. distribute-rgt-in N/A

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

      11. *-lft-identity N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. difference-of-squares-rev N/A

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

      15. difference-of-sqr--1-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.9%

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

   9. Applied rewrites 100.0%

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

   if 5.00000000000000004e-36 < (/.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. rgt-mult-inverse N/A

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

      4. cancel-sign-sub N/A

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

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

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

      6. rgt-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 97.0

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

   5. Applied rewrites 97.0%

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

   if 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 5.0000000000000004e53

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-*r/ N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. pow2 N/A

          \[\leadsto \frac{\left(\color{blue}{{\left(\frac{x}{y}\right)}^{2}} - 1\right) \cdot x}{\left(\frac{x}{y} - 1\right) \cdot \left(x + 1\right)} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{\left(\color{blue}{{\left(\frac{x}{y}\right)}^{2}} - 1\right) \cdot x}{\left(\frac{x}{y} - 1\right) \cdot \left(x + 1\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\left({\left(\frac{x}{y}\right)}^{2} - 1\right) \cdot x}{\color{blue}{\left(\frac{x}{y} - 1\right) \cdot \left(x + 1\right)}} \]

      15. lower--.f64 99.2

          \[\leadsto \frac{\left({\left(\frac{x}{y}\right)}^{2} - 1\right) \cdot x}{\color{blue}{\left(\frac{x}{y} - 1\right)} \cdot \left(x + 1\right)} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{\left({\left(\frac{x}{y}\right)}^{2} - 1\right) \cdot x}{\left(\frac{x}{y} - 1\right) \cdot \color{blue}{\left(x + 1\right)}} \]

      17. +-commutative N/A

          \[\leadsto \frac{\left({\left(\frac{x}{y}\right)}^{2} - 1\right) \cdot x}{\left(\frac{x}{y} - 1\right) \cdot \color{blue}{\left(1 + x\right)}} \]

      18. lower-+.f64 99.2

          \[\leadsto \frac{\left({\left(\frac{x}{y}\right)}^{2} - 1\right) \cdot x}{\left(\frac{x}{y} - 1\right) \cdot \color{blue}{\left(1 + x\right)}} \]

   4. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\left({\left(\frac{x}{y}\right)}^{2} - 1\right) \cdot x}{\left(\frac{x}{y} - 1\right) \cdot \left(1 + x\right)}} \]

   5. Taylor expanded in y around 0

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

      1. div-add-rev N/A

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

      2. associate-/l/ N/A

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

      3. *-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. distribute-lft-out N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

      12. lower-fma.f64 99.2

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

   7. Applied rewrites 99.2%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq -5 \cdot 10^{+27}:\\ \;\;\;\;{y}^{-1} \cdot x\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 5 \cdot 10^{-36}:\\ \;\;\;\;\left(\frac{x}{y} - x\right) \cdot x + x\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 5 \cdot 10^{+53}:\\ \;\;\;\;\frac{x \cdot \left(y + x\right)}{\mathsf{fma}\left(y, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;{y}^{-1} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ t_1 := {y}^{-1} \cdot x\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+27}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-36}:\\ \;\;\;\;\left(\frac{x}{y} - x\right) \cdot x + x\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+53}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, x, y\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))) (t_1 (* (pow y -1.0) x)))
   (if (<= t_0 -5e+27)
     t_1
     (if (<= t_0 5e-36)
       (+ (* (- (/ x y) x) x) x)
       (if (<= t_0 2.0)
         (/ x (- x -1.0))
         (if (<= t_0 5e+53) (* (/ x (fma y x y)) x) t_1))))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double t_1 = pow(y, -1.0) * x;
	double tmp;
	if (t_0 <= -5e+27) {
		tmp = t_1;
	} else if (t_0 <= 5e-36) {
		tmp = (((x / y) - x) * x) + x;
	} else if (t_0 <= 2.0) {
		tmp = x / (x - -1.0);
	} else if (t_0 <= 5e+53) {
		tmp = (x / fma(y, x, y)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	t_1 = Float64((y ^ -1.0) * x)
	tmp = 0.0
	if (t_0 <= -5e+27)
		tmp = t_1;
	elseif (t_0 <= 5e-36)
		tmp = Float64(Float64(Float64(Float64(x / y) - x) * x) + x);
	elseif (t_0 <= 2.0)
		tmp = Float64(x / Float64(x - -1.0));
	elseif (t_0 <= 5e+53)
		tmp = Float64(Float64(x / fma(y, x, y)) * x);
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[Power[y, -1.0], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+27], t$95$1, If[LessEqual[t$95$0, 5e-36], N[(N[(N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision] * x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+53], N[(N[(x / N[(y * x + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -4.99999999999999979e27 or 5.0000000000000004e53 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 71.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 71.8

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

   4. Applied rewrites 71.8%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{{x}^{2}}{y \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 85.6

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

   7. Applied rewrites 85.6%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{y} \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 94.2%

       \[\leadsto \frac{1}{y} \cdot x \]

   if -4.99999999999999979e27 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 5.00000000000000004e-36

   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

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/r/ N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. distribute-rgt-in N/A

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

      11. *-lft-identity N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. difference-of-squares-rev N/A

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

      15. difference-of-sqr--1-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.9%

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

   9. Applied rewrites 100.0%

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

   if 5.00000000000000004e-36 < (/.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. rgt-mult-inverse N/A

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

      4. cancel-sign-sub N/A

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

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

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

      6. rgt-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 97.0

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

   5. Applied rewrites 97.0%

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

   if 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 5.0000000000000004e53

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 91.6

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

   5. Applied rewrites 91.6%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq -5 \cdot 10^{+27}:\\ \;\;\;\;{y}^{-1} \cdot x\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 5 \cdot 10^{-36}:\\ \;\;\;\;\left(\frac{x}{y} - x\right) \cdot x + x\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 5 \cdot 10^{+53}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, x, y\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;{y}^{-1} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ t_1 := {y}^{-1} \cdot x\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+27}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-36}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y} - x, x, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+53}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, x, y\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))) (t_1 (* (pow y -1.0) x)))
   (if (<= t_0 -5e+27)
     t_1
     (if (<= t_0 5e-36)
       (fma (- (/ x y) x) x x)
       (if (<= t_0 2.0)
         (/ x (- x -1.0))
         (if (<= t_0 5e+53) (* (/ x (fma y x y)) x) t_1))))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double t_1 = pow(y, -1.0) * x;
	double tmp;
	if (t_0 <= -5e+27) {
		tmp = t_1;
	} else if (t_0 <= 5e-36) {
		tmp = fma(((x / y) - x), x, x);
	} else if (t_0 <= 2.0) {
		tmp = x / (x - -1.0);
	} else if (t_0 <= 5e+53) {
		tmp = (x / fma(y, x, y)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	t_1 = Float64((y ^ -1.0) * x)
	tmp = 0.0
	if (t_0 <= -5e+27)
		tmp = t_1;
	elseif (t_0 <= 5e-36)
		tmp = fma(Float64(Float64(x / y) - x), x, x);
	elseif (t_0 <= 2.0)
		tmp = Float64(x / Float64(x - -1.0));
	elseif (t_0 <= 5e+53)
		tmp = Float64(Float64(x / fma(y, x, y)) * x);
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[Power[y, -1.0], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+27], t$95$1, If[LessEqual[t$95$0, 5e-36], N[(N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision] * x + x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+53], N[(N[(x / N[(y * x + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -4.99999999999999979e27 or 5.0000000000000004e53 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 71.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 71.8

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

   4. Applied rewrites 71.8%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{{x}^{2}}{y \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 85.6

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

   7. Applied rewrites 85.6%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{y} \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 94.2%

       \[\leadsto \frac{1}{y} \cdot x \]

   if -4.99999999999999979e27 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 5.00000000000000004e-36

   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

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/r/ N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. distribute-rgt-in N/A

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

      11. *-lft-identity N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. difference-of-squares-rev N/A

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

      15. difference-of-sqr--1-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.9%

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

   if 5.00000000000000004e-36 < (/.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. rgt-mult-inverse N/A

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

      4. cancel-sign-sub N/A

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

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

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

      6. rgt-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 97.0

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

   5. Applied rewrites 97.0%

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

   if 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 5.0000000000000004e53

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 91.6

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

   5. Applied rewrites 91.6%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq -5 \cdot 10^{+27}:\\ \;\;\;\;{y}^{-1} \cdot x\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 5 \cdot 10^{-36}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y} - x, x, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 5 \cdot 10^{+53}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, x, y\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;{y}^{-1} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ t_1 := {y}^{-1} \cdot x\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+27}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-36}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y} - x, x, x\right)\\ \mathbf{elif}\;t\_0 \leq 36:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))) (t_1 (* (pow y -1.0) x)))
   (if (<= t_0 -5e+27)
     t_1
     (if (<= t_0 5e-36)
       (fma (- (/ x y) x) x x)
       (if (<= t_0 36.0) (/ x (- x -1.0)) t_1)))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double t_1 = pow(y, -1.0) * x;
	double tmp;
	if (t_0 <= -5e+27) {
		tmp = t_1;
	} else if (t_0 <= 5e-36) {
		tmp = fma(((x / y) - x), x, x);
	} else if (t_0 <= 36.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	t_1 = Float64((y ^ -1.0) * x)
	tmp = 0.0
	if (t_0 <= -5e+27)
		tmp = t_1;
	elseif (t_0 <= 5e-36)
		tmp = fma(Float64(Float64(x / y) - x), x, x);
	elseif (t_0 <= 36.0)
		tmp = Float64(x / Float64(x - -1.0));
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[Power[y, -1.0], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+27], t$95$1, If[LessEqual[t$95$0, 5e-36], N[(N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision] * x + x), $MachinePrecision], If[LessEqual[t$95$0, 36.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

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))) < -4.99999999999999979e27 or 36 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 73.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 73.8

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

   4. Applied rewrites 73.8%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{{x}^{2}}{y \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 86.6

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

   7. Applied rewrites 86.6%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{y} \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 90.0%

       \[\leadsto \frac{1}{y} \cdot x \]

   if -4.99999999999999979e27 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 5.00000000000000004e-36

   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

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/r/ N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. distribute-rgt-in N/A

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

      11. *-lft-identity N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. difference-of-squares-rev N/A

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

      15. difference-of-sqr--1-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.9%

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

   if 5.00000000000000004e-36 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 36

   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. rgt-mult-inverse N/A

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

      4. cancel-sign-sub N/A

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

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

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

      6. rgt-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 92.2

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

   5. Applied rewrites 92.2%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq -5 \cdot 10^{+27}:\\ \;\;\;\;{y}^{-1} \cdot x\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 5 \cdot 10^{-36}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y} - x, x, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 36:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;{y}^{-1} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+27} \lor \neg \left(t\_0 \leq 36\right):\\ \;\;\;\;{y}^{-1} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (or (<= t_0 -5e+27) (not (<= t_0 36.0)))
     (* (pow y -1.0) x)
     (/ x (- x -1.0)))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if ((t_0 <= -5e+27) || !(t_0 <= 36.0)) {
		tmp = pow(y, -1.0) * x;
	} else {
		tmp = x / (x - -1.0);
	}
	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 * ((x / y) + 1.0d0)) / (x + 1.0d0)
    if ((t_0 <= (-5d+27)) .or. (.not. (t_0 <= 36.0d0))) then
        tmp = (y ** (-1.0d0)) * x
    else
        tmp = x / (x - (-1.0d0))
    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 <= -5e+27) || !(t_0 <= 36.0)) {
		tmp = Math.pow(y, -1.0) * x;
	} else {
		tmp = x / (x - -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0)
	tmp = 0
	if (t_0 <= -5e+27) or not (t_0 <= 36.0):
		tmp = math.pow(y, -1.0) * x
	else:
		tmp = x / (x - -1.0)
	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 <= -5e+27) || !(t_0 <= 36.0))
		tmp = Float64((y ^ -1.0) * x);
	else
		tmp = Float64(x / Float64(x - -1.0));
	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 <= -5e+27) || ~((t_0 <= 36.0)))
		tmp = (y ^ -1.0) * x;
	else
		tmp = x / (x - -1.0);
	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, -5e+27], N[Not[LessEqual[t$95$0, 36.0]], $MachinePrecision]], N[(N[Power[y, -1.0], $MachinePrecision] * x), $MachinePrecision], N[(x / N[(x - -1.0), $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))) < -4.99999999999999979e27 or 36 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 73.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 73.8

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

   4. Applied rewrites 73.8%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{{x}^{2}}{y \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 86.6

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

   7. Applied rewrites 86.6%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{y} \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 90.0%

       \[\leadsto \frac{1}{y} \cdot x \]

   if -4.99999999999999979e27 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 36

   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. rgt-mult-inverse N/A

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

      4. cancel-sign-sub N/A

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

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

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

      6. rgt-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 86.2

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

   5. Applied rewrites 86.2%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq -5 \cdot 10^{+27} \lor \neg \left(\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 36\right):\\ \;\;\;\;{y}^{-1} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ t_1 := \frac{x}{1 + x} \cdot \frac{x}{y}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+27}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-36}:\\ \;\;\;\;\left(\frac{x}{y} - x\right) \cdot x + x\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))
        (t_1 (* (/ x (+ 1.0 x)) (/ x y))))
   (if (<= t_0 -5e+27)
     t_1
     (if (<= t_0 5e-36)
       (+ (* (- (/ x y) x) x) x)
       (if (<= t_0 2.0) (/ x (- x -1.0)) t_1)))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double t_1 = (x / (1.0 + x)) * (x / y);
	double tmp;
	if (t_0 <= -5e+27) {
		tmp = t_1;
	} else if (t_0 <= 5e-36) {
		tmp = (((x / y) - x) * x) + x;
	} else if (t_0 <= 2.0) {
		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 * ((x / y) + 1.0d0)) / (x + 1.0d0)
    t_1 = (x / (1.0d0 + x)) * (x / y)
    if (t_0 <= (-5d+27)) then
        tmp = t_1
    else if (t_0 <= 5d-36) then
        tmp = (((x / y) - x) * x) + x
    else if (t_0 <= 2.0d0) 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 * ((x / y) + 1.0)) / (x + 1.0);
	double t_1 = (x / (1.0 + x)) * (x / y);
	double tmp;
	if (t_0 <= -5e+27) {
		tmp = t_1;
	} else if (t_0 <= 5e-36) {
		tmp = (((x / y) - x) * x) + x;
	} else if (t_0 <= 2.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0)
	t_1 = (x / (1.0 + x)) * (x / y)
	tmp = 0
	if t_0 <= -5e+27:
		tmp = t_1
	elif t_0 <= 5e-36:
		tmp = (((x / y) - x) * x) + x
	elif t_0 <= 2.0:
		tmp = x / (x - -1.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	t_1 = Float64(Float64(x / Float64(1.0 + x)) * Float64(x / y))
	tmp = 0.0
	if (t_0 <= -5e+27)
		tmp = t_1;
	elseif (t_0 <= 5e-36)
		tmp = Float64(Float64(Float64(Float64(x / y) - x) * x) + x);
	elseif (t_0 <= 2.0)
		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 * ((x / y) + 1.0)) / (x + 1.0);
	t_1 = (x / (1.0 + x)) * (x / y);
	tmp = 0.0;
	if (t_0 <= -5e+27)
		tmp = t_1;
	elseif (t_0 <= 5e-36)
		tmp = (((x / y) - x) * x) + x;
	elseif (t_0 <= 2.0)
		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[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+27], t$95$1, If[LessEqual[t$95$0, 5e-36], N[(N[(N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision] * x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

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))) < -4.99999999999999979e27 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 74.2

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

   4. Applied rewrites 74.2%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{{x}^{2}}{y \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 86.1

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

   7. Applied rewrites 86.1%

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

   9. Applied rewrites 99.2%

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

   if -4.99999999999999979e27 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 5.00000000000000004e-36

   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

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/r/ N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. distribute-rgt-in N/A

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

      11. *-lft-identity N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. difference-of-squares-rev N/A

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

      15. difference-of-sqr--1-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.9%

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

   9. Applied rewrites 100.0%

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

   if 5.00000000000000004e-36 < (/.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. rgt-mult-inverse N/A

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

      4. cancel-sign-sub N/A

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

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

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

      6. rgt-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 97.0

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

   5. Applied rewrites 97.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq -5 \cdot 10^{+27}:\\ \;\;\;\;\frac{x}{1 + x} \cdot \frac{x}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 5 \cdot 10^{-36}:\\ \;\;\;\;\left(\frac{x}{y} - x\right) \cdot x + x\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+132} \lor \neg \left(t\_0 \leq 4\right):\\ \;\;\;\;\frac{x}{1 + x} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (or (<= t_0 -1e+132) (not (<= t_0 4.0)))
     (* (/ x (+ 1.0 x)) (/ x y))
     (/ (fma (/ x y) x x) (+ x 1.0)))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if ((t_0 <= -1e+132) || !(t_0 <= 4.0)) {
		tmp = (x / (1.0 + x)) * (x / y);
	} else {
		tmp = fma((x / y), x, x) / (x + 1.0);
	}
	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 <= -1e+132) || !(t_0 <= 4.0))
		tmp = Float64(Float64(x / Float64(1.0 + x)) * Float64(x / y));
	else
		tmp = Float64(fma(Float64(x / y), x, x) / Float64(x + 1.0));
	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, -1e+132], N[Not[LessEqual[t$95$0, 4.0]], $MachinePrecision]], N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] * N[(x / y), $MachinePrecision]), $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 (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -9.99999999999999991e131 or 4 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 71.5%

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

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 71.5

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

   4. Applied rewrites 71.5%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{{x}^{2}}{y \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 88.3

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

   7. Applied rewrites 88.3%

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

   9. Applied rewrites 100.0%

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

   if -9.99999999999999991e131 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 4

   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. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq -1 \cdot 10^{+132} \lor \neg \left(\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 4\right):\\ \;\;\;\;\frac{x}{1 + x} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq -5 \cdot 10^{+27}:\\ \;\;\;\;\left(-x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)) -5e+27)
   (* (- x) x)
   (/ x (- x -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (((x * ((x / y) + 1.0)) / (x + 1.0)) <= -5e+27) {
		tmp = -x * x;
	} else {
		tmp = x / (x - -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 * ((x / y) + 1.0d0)) / (x + 1.0d0)) <= (-5d+27)) then
        tmp = -x * x
    else
        tmp = x / (x - (-1.0d0))
    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)) <= -5e+27) {
		tmp = -x * x;
	} else {
		tmp = x / (x - -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * ((x / y) + 1.0)) / (x + 1.0)) <= -5e+27:
		tmp = -x * x
	else:
		tmp = x / (x - -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0)) <= -5e+27)
		tmp = Float64(Float64(-x) * x);
	else
		tmp = Float64(x / Float64(x - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x * ((x / y) + 1.0)) / (x + 1.0)) <= -5e+27)
		tmp = -x * x;
	else
		tmp = x / (x - -1.0);
	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], -5e+27], N[((-x) * x), $MachinePrecision], N[(x / N[(x - -1.0), $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))) < -4.99999999999999979e27

   1. Initial program 69.3%

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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 28.9

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

   5. Applied rewrites 28.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 32.7%

      \[\leadsto \left(1 - x\right) \cdot x \]

   9. Taylor expanded in x around inf

      \[\leadsto \left(-1 \cdot x\right) \cdot x \]
   10. Step-by-step derivation

   11. Applied rewrites 32.7%

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

   if -4.99999999999999979e27 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 95.4%

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

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

      4. cancel-sign-sub N/A

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

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

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

      6. rgt-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 67.1

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

   5. Applied rewrites 67.1%

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

Alternative 10: 43.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq -1 \cdot 10^{+114}:\\ \;\;\;\;\left(-x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)) -1e+114) (* (- x) x) (* 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (((x * ((x / y) + 1.0)) / (x + 1.0)) <= -1e+114) {
		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)) <= (-1d+114)) 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)) <= -1e+114) {
		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)) <= -1e+114:
		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)) <= -1e+114)
		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)) <= -1e+114)
		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], -1e+114], 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))) < -1e114

   1. Initial program 64.6%

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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 28.7

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

   5. Applied rewrites 28.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 36.9%

      \[\leadsto \left(1 - x\right) \cdot x \]

   9. Taylor expanded in x around inf

      \[\leadsto \left(-1 \cdot x\right) \cdot x \]
   10. Step-by-step derivation

   11. Applied rewrites 37.0%

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

   if -1e114 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 95.6%

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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 66.3

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

   5. Applied rewrites 66.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 48.4%

      \[\leadsto \left(1 - x\right) \cdot x \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 49.8%

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

Alternative 11: 42.8% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.6%

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

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-/.f64 57.8

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

5. Applied rewrites 57.8%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 45.8%

   \[\leadsto \left(1 - x\right) \cdot x \]
9. Add Preprocessing

Alternative 12: 38.8% 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 88.6%

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

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-/.f64 57.8

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

5. Applied rewrites 57.8%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 45.8%

   \[\leadsto \left(1 - x\right) \cdot x \]

9. Taylor expanded in x around 0

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

11. Applied rewrites 39.1%

    \[\leadsto 1 \cdot x \]
12. 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 2024337 
(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)))