Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 87.7% → 99.9%

Time: 6.3s

Alternatives: 11

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: 87.7% 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 90.9%

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

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

   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 99.9

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

   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 99.9

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

   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 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 97.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{x}{y} \cdot x}{x + 1}\\ t_1 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_1 \leq -5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y} - x, x, x\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+149}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (/ x y) x) (+ x 1.0)))
        (t_1 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_1 (- INFINITY))
     (/ x y)
     (if (<= t_1 -5.0)
       t_0
       (if (<= t_1 5e-15)
         (fma (- (/ x y) x) x x)
         (if (<= t_1 2.0)
           (/ x (- x -1.0))
           (if (<= t_1 5e+149) t_0 (/ x y))))))))

C

double code(double x, double y) {
	double t_0 = ((x / y) * x) / (x + 1.0);
	double t_1 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = x / y;
	} else if (t_1 <= -5.0) {
		tmp = t_0;
	} else if (t_1 <= 5e-15) {
		tmp = fma(((x / y) - x), x, x);
	} else if (t_1 <= 2.0) {
		tmp = x / (x - -1.0);
	} else if (t_1 <= 5e+149) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x / y) * x) / Float64(x + 1.0))
	t_1 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(x / y);
	elseif (t_1 <= -5.0)
		tmp = t_0;
	elseif (t_1 <= 5e-15)
		tmp = fma(Float64(Float64(x / y) - x), x, x);
	elseif (t_1 <= 2.0)
		tmp = Float64(x / Float64(x - -1.0));
	elseif (t_1 <= 5e+149)
		tmp = t_0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(x / y), $MachinePrecision], If[LessEqual[t$95$1, -5.0], t$95$0, If[LessEqual[t$95$1, 5e-15], N[(N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision] * x + x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+149], t$95$0, N[(x / y), $MachinePrecision]]]]]]]]

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))) < -inf.0 or 4.9999999999999999e149 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 61.3%

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

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

      6. unpow2 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-lft-in N/A

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 69.8

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

   7. Applied rewrites 69.8%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 99.2

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

   10. Applied rewrites 99.2%

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

   if -inf.0 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -5 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 4.9999999999999999e149

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 95.0

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

   5. Applied rewrites 95.0%

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

   if -5 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 4.99999999999999999e-15

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 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 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.8%

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

   if 4.99999999999999999e-15 < (/.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 95.9

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

   5. Applied rewrites 95.9%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq -\infty:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq -5:\\ \;\;\;\;\frac{\frac{x}{y} \cdot x}{x + 1}\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y} - x, x, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 5 \cdot 10^{+149}:\\ \;\;\;\;\frac{\frac{x}{y} \cdot x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot x}{\mathsf{fma}\left(y, x, y\right)}\\ t_1 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_1 \leq -1000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y} - x, x, x\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+149}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x x) (fma y x y)))
        (t_1 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_1 (- INFINITY))
     (/ x y)
     (if (<= t_1 -1000.0)
       t_0
       (if (<= t_1 5e-15)
         (fma (- (/ x y) x) x x)
         (if (<= t_1 2.0)
           (/ x (- x -1.0))
           (if (<= t_1 5e+149) t_0 (/ x y))))))))

C

double code(double x, double y) {
	double t_0 = (x * x) / fma(y, x, y);
	double t_1 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = x / y;
	} else if (t_1 <= -1000.0) {
		tmp = t_0;
	} else if (t_1 <= 5e-15) {
		tmp = fma(((x / y) - x), x, x);
	} else if (t_1 <= 2.0) {
		tmp = x / (x - -1.0);
	} else if (t_1 <= 5e+149) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * x) / fma(y, x, y))
	t_1 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(x / y);
	elseif (t_1 <= -1000.0)
		tmp = t_0;
	elseif (t_1 <= 5e-15)
		tmp = fma(Float64(Float64(x / y) - x), x, x);
	elseif (t_1 <= 2.0)
		tmp = Float64(x / Float64(x - -1.0));
	elseif (t_1 <= 5e+149)
		tmp = t_0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] / N[(y * x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(x / y), $MachinePrecision], If[LessEqual[t$95$1, -1000.0], t$95$0, If[LessEqual[t$95$1, 5e-15], N[(N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision] * x + x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+149], t$95$0, N[(x / y), $MachinePrecision]]]]]]]]

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))) < -inf.0 or 4.9999999999999999e149 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 61.3%

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

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

      6. unpow2 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-lft-in N/A

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 69.8

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

   7. Applied rewrites 69.8%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 99.2

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

   10. Applied rewrites 99.2%

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

   if -inf.0 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -1e3 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 4.9999999999999999e149

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      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 99.8

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

      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 99.8

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

   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. +-commutative N/A

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

      6. unpow2 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-lft-in N/A

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 84.4

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

   7. Applied rewrites 84.4%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{{x}^{2}}{\mathsf{fma}\left(\color{blue}{y}, x, y\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 81.9%

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

   if -1e3 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 4.99999999999999999e-15

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 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 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 98.9%

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

   if 4.99999999999999999e-15 < (/.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 95.9

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

   5. Applied rewrites 95.9%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq -\infty:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq -1000:\\ \;\;\;\;\frac{x \cdot x}{\mathsf{fma}\left(y, x, y\right)}\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y} - x, x, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 5 \cdot 10^{+149}:\\ \;\;\;\;\frac{x \cdot x}{\mathsf{fma}\left(y, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ t_1 := \frac{x}{\mathsf{fma}\left(y, x, y\right)} \cdot x\\ \mathbf{if}\;t\_0 \leq -1000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y} - x, x, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))
        (t_1 (* (/ x (fma y x y)) x)))
   (if (<= t_0 -1000.0)
     t_1
     (if (<= t_0 5e-15)
       (fma (- (/ 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 / fma(y, x, y)) * x;
	double tmp;
	if (t_0 <= -1000.0) {
		tmp = t_1;
	} else if (t_0 <= 5e-15) {
		tmp = fma(((x / y) - x), x, x);
	} else if (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 / fma(y, x, y)) * x)
	tmp = 0.0
	if (t_0 <= -1000.0)
		tmp = t_1;
	elseif (t_0 <= 5e-15)
		tmp = fma(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

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[(y * x + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$0, -1000.0], t$95$1, If[LessEqual[t$95$0, 5e-15], 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], 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))) < -1e3 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 77.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{{x}^{2}}{y \cdot \left(1 + x\right)}} \]
   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 88.4

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

   5. Applied rewrites 88.4%

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

   if -1e3 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 4.99999999999999999e-15

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 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 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 98.9%

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

   if 4.99999999999999999e-15 < (/.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 95.9

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

   5. Applied rewrites 95.9%

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

Alternative 5: 91.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+44}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y} - x, x, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -4e+44)
     (/ x y)
     (if (<= t_0 5e-15)
       (fma (- (/ x y) x) x x)
       (if (<= t_0 2.0) (/ x (- x -1.0)) (/ x y))))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -4e+44) {
		tmp = x / y;
	} else if (t_0 <= 5e-15) {
		tmp = fma(((x / y) - x), x, x);
	} else if (t_0 <= 2.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_0 <= -4e+44)
		tmp = Float64(x / y);
	elseif (t_0 <= 5e-15)
		tmp = fma(Float64(Float64(x / y) - x), x, x);
	elseif (t_0 <= 2.0)
		tmp = Float64(x / Float64(x - -1.0));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+44], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 5e-15], 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], N[(x / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -4.0000000000000004e44 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 75.5%

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

      1. lift-/.f64 N/A

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

      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 99.9

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

      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 99.9

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

   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. +-commutative N/A

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

      6. unpow2 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-lft-in N/A

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 74.6

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

   7. Applied rewrites 74.6%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 85.0

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

   10. Applied rewrites 85.0%

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

   if -4.0000000000000004e44 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 4.99999999999999999e-15

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      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 94.6

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 94.7%

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

   if 4.99999999999999999e-15 < (/.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 95.9

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

   5. Applied rewrites 95.9%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq -4 \cdot 10^{+44}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y} - x, x, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 5 \cdot 10^{+149}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 5e+149)))
     (/ 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 <= -((double) INFINITY)) || !(t_0 <= 5e+149)) {
		tmp = 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 <= Float64(-Inf)) || !(t_0 <= 5e+149))
		tmp = 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, (-Infinity)], N[Not[LessEqual[t$95$0, 5e+149]], $MachinePrecision]], N[(x / y), $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))) < -inf.0 or 4.9999999999999999e149 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 61.3%

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

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

      6. unpow2 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-lft-in N/A

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 69.8

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

   7. Applied rewrites 69.8%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 99.2

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

   10. Applied rewrites 99.2%

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

   if -inf.0 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 4.9999999999999999e149

   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 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq -\infty \lor \neg \left(\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 5 \cdot 10^{+149}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -5 \lor \neg \left(t\_0 \leq 2\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (or (<= t_0 -5.0) (not (<= t_0 2.0))) (/ x y) (/ 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 <= -5.0) || !(t_0 <= 2.0)) {
		tmp = x / y;
	} 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 <= (-5.0d0)) .or. (.not. (t_0 <= 2.0d0))) then
        tmp = x / y
    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 <= -5.0) || !(t_0 <= 2.0)) {
		tmp = x / y;
	} 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 <= -5.0) or not (t_0 <= 2.0):
		tmp = x / y
	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 <= -5.0) || !(t_0 <= 2.0))
		tmp = Float64(x / y);
	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 <= -5.0) || ~((t_0 <= 2.0)))
		tmp = x / y;
	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, -5.0], N[Not[LessEqual[t$95$0, 2.0]], $MachinePrecision]], N[(x / y), $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))) < -5 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 78.0%

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

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

      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 99.9

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

   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. +-commutative N/A

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

      6. unpow2 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-lft-in N/A

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 75.4

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

   7. Applied rewrites 75.4%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 80.2

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

   10. Applied rewrites 80.2%

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

   if -5 < (/.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 84.2

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

   5. Applied rewrites 84.2%

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

4. Final simplification 82.5%

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

Alternative 8: 73.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -5 \lor \neg \left(t\_0 \leq 4 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (or (<= t_0 -5.0) (not (<= t_0 4e-5))) (/ x y) (* (- 1.0 x) x))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if ((t_0 <= -5.0) || !(t_0 <= 4e-5)) {
		tmp = x / y;
	} else {
		tmp = (1.0 - x) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * ((x / y) + 1.0d0)) / (x + 1.0d0)
    if ((t_0 <= (-5.0d0)) .or. (.not. (t_0 <= 4d-5))) then
        tmp = x / y
    else
        tmp = (1.0d0 - x) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if ((t_0 <= -5.0) || !(t_0 <= 4e-5)) {
		tmp = x / y;
	} else {
		tmp = (1.0 - x) * x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0)
	tmp = 0
	if (t_0 <= -5.0) or not (t_0 <= 4e-5):
		tmp = x / y
	else:
		tmp = (1.0 - x) * x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if ((t_0 <= -5.0) || !(t_0 <= 4e-5))
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(1.0 - x) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	tmp = 0.0;
	if ((t_0 <= -5.0) || ~((t_0 <= 4e-5)))
		tmp = x / y;
	else
		tmp = (1.0 - x) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5.0], N[Not[LessEqual[t$95$0, 4e-5]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 83.7%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing
   3. 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 99.9

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

      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 99.9

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

   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. +-commutative N/A

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

      6. unpow2 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-lft-in N/A

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 69.1

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

   7. Applied rewrites 69.1%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 60.4

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

   10. Applied rewrites 60.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 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 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 80.2%

      \[\leadsto \left(1 - x\right) \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.1%

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

Alternative 9: 42.5% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)) -5.0) (* (- x) x) (* 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (((x * ((x / y) + 1.0)) / (x + 1.0)) <= -5.0) {
		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)) <= (-5.0d0)) 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)) <= -5.0) {
		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)) <= -5.0:
		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)) <= -5.0)
		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)) <= -5.0)
		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], -5.0], 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))) < -5

   1. Initial program 81.0%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 31.3

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

   5. Applied rewrites 31.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 31.3%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 18.4%

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

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

   1. Initial program 93.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      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 65.3

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

   5. Applied rewrites 65.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 46.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 47.3%

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

Alternative 10: 41.8% 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 90.9%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   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 57.4

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

5. Applied rewrites 57.4%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 39.8%

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

Alternative 11: 37.5% 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 90.9%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   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 57.4

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

5. Applied rewrites 57.4%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 39.8%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 37.2%

    \[\leadsto 1 \cdot x \]
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 2024320 
(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)))