Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.3% → 99.9%

Time: 7.2s

Alternatives: 15

Speedup: 1.1×

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: 88.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(y + x\right) \cdot \frac{x}{1 + x}}{y}\\ \mathbf{if}\;x \leq -5 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 10^{-49}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - y, x, y\right)}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (+ y x) (/ x (+ 1.0 x))) y)))
   (if (<= x -5e-18)
     t_0
     (if (<= x 1e-49) (* (/ (fma (- 1.0 y) x y) y) x) t_0))))

C

double code(double x, double y) {
	double t_0 = ((y + x) * (x / (1.0 + x))) / y;
	double tmp;
	if (x <= -5e-18) {
		tmp = t_0;
	} else if (x <= 1e-49) {
		tmp = (fma((1.0 - y), x, y) / y) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(y + x) * Float64(x / Float64(1.0 + x))) / y)
	tmp = 0.0
	if (x <= -5e-18)
		tmp = t_0;
	elseif (x <= 1e-49)
		tmp = Float64(Float64(fma(Float64(1.0 - y), x, y) / y) * x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(y + x), $MachinePrecision] * N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, -5e-18], t$95$0, If[LessEqual[x, 1e-49], N[(N[(N[(N[(1.0 - y), $MachinePrecision] * x + y), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.00000000000000036e-18 or 9.99999999999999936e-50 < x

   1. Initial program 77.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -5.00000000000000036e-18 < x < 9.99999999999999936e-50

   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. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

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

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 79.2%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(y + x\right) \cdot \frac{x}{1 + x}}{y}\\ \mathbf{elif}\;x \leq 10^{-49}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - y, x, y\right)}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y + x\right) \cdot \frac{x}{1 + x}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+50}:\\ \;\;\;\;\frac{x - 1}{y}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-81}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (+ (/ x y) 1.0) x) (+ 1.0 x))))
   (if (<= t_0 -1e+50)
     (/ (- x 1.0) y)
     (if (<= t_0 2e-81)
       (fma (/ x y) x x)
       (if (<= t_0 2.0) (/ x (+ 1.0 x)) (/ x y))))))

C

double code(double x, double y) {
	double t_0 = (((x / y) + 1.0) * x) / (1.0 + x);
	double tmp;
	if (t_0 <= -1e+50) {
		tmp = (x - 1.0) / y;
	} else if (t_0 <= 2e-81) {
		tmp = fma((x / y), x, x);
	} else if (t_0 <= 2.0) {
		tmp = x / (1.0 + x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(Float64(x / y) + 1.0) * x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_0 <= -1e+50)
		tmp = Float64(Float64(x - 1.0) / y);
	elseif (t_0 <= 2e-81)
		tmp = fma(Float64(x / y), x, x);
	elseif (t_0 <= 2.0)
		tmp = Float64(x / Float64(1.0 + x));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+50], N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$0, 2e-81], N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], 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))) < -1.0000000000000001e50

   1. Initial program 74.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-fma.f64 74.0

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

   4. Applied rewrites 74.0%

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

   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. associate-/r* N/A

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

      2. metadata-eval N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. associate-/r* N/A

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

      11. distribute-neg-frac2 N/A

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

      12. mul-1-neg N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. mul-1-neg N/A

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

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

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

      18. sub-neg N/A

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

      19. metadata-eval N/A

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

      20. distribute-neg-in N/A

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

      21. mul-1-neg N/A

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

      22. remove-double-neg N/A

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

      23. metadata-eval N/A

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

   7. Applied rewrites 77.2%

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

   8. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

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

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. rgt-mult-inverse N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt-out N/A

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

      12. rgt-mult-inverse N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 85.4

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

   10. Applied rewrites 85.4%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 85.6%

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

   if -1.0000000000000001e50 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 1.9999999999999999e-81

   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. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

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

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 97.6

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 97.3%

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

   if 1.9999999999999999e-81 < (/.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. lower-+.f64 94.7

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

   5. Applied rewrites 94.7%

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

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

   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 inf

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

      1. lower-/.f64 80.3

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

   5. Applied rewrites 80.3%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq -1 \cdot 10^{+50}:\\ \;\;\;\;\frac{x - 1}{y}\\ \mathbf{elif}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq 2 \cdot 10^{-81}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, x\right)\\ \mathbf{elif}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq 2:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x}\\ \mathbf{if}\;t\_0 \leq -50000000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;x - x \cdot x\\ \mathbf{elif}\;t\_0 \leq 50:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (+ (/ x y) 1.0) x) (+ 1.0 x))))
   (if (<= t_0 -50000000000.0)
     (/ x y)
     (if (<= t_0 5e-9) (- x (* x x)) (if (<= t_0 50.0) 1.0 (/ x y))))))

C

double code(double x, double y) {
	double t_0 = (((x / y) + 1.0) * x) / (1.0 + x);
	double tmp;
	if (t_0 <= -50000000000.0) {
		tmp = x / y;
	} else if (t_0 <= 5e-9) {
		tmp = x - (x * x);
	} else if (t_0 <= 50.0) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x / y) + 1.0d0) * x) / (1.0d0 + x)
    if (t_0 <= (-50000000000.0d0)) then
        tmp = x / y
    else if (t_0 <= 5d-9) then
        tmp = x - (x * x)
    else if (t_0 <= 50.0d0) then
        tmp = 1.0d0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (((x / y) + 1.0) * x) / (1.0 + x);
	double tmp;
	if (t_0 <= -50000000000.0) {
		tmp = x / y;
	} else if (t_0 <= 5e-9) {
		tmp = x - (x * x);
	} else if (t_0 <= 50.0) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (((x / y) + 1.0) * x) / (1.0 + x)
	tmp = 0
	if t_0 <= -50000000000.0:
		tmp = x / y
	elif t_0 <= 5e-9:
		tmp = x - (x * x)
	elif t_0 <= 50.0:
		tmp = 1.0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(Float64(x / y) + 1.0) * x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_0 <= -50000000000.0)
		tmp = Float64(x / y);
	elseif (t_0 <= 5e-9)
		tmp = Float64(x - Float64(x * x));
	elseif (t_0 <= 50.0)
		tmp = 1.0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (((x / y) + 1.0) * x) / (1.0 + x);
	tmp = 0.0;
	if (t_0 <= -50000000000.0)
		tmp = x / y;
	elseif (t_0 <= 5e-9)
		tmp = x - (x * x);
	elseif (t_0 <= 50.0)
		tmp = 1.0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -50000000000.0], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 5e-9], N[(x - N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 50.0], 1.0, N[(x / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -5e10 or 50 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 71.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 80.3

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

   5. Applied rewrites 80.3%

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

   if -5e10 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 5.0000000000000001e-9

   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. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

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

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 84.9%

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

   if 5.0000000000000001e-9 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 50

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   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. associate-/r* N/A

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

      2. metadata-eval N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. associate-/r* N/A

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

      11. distribute-neg-frac2 N/A

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

      12. mul-1-neg N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. mul-1-neg N/A

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

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

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

      18. sub-neg N/A

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

      19. metadata-eval N/A

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

      20. distribute-neg-in N/A

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

      21. mul-1-neg N/A

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

      22. remove-double-neg N/A

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

      23. metadata-eval N/A

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

   7. Applied rewrites 5.7%

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

   8. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

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

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. rgt-mult-inverse N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt-out N/A

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

      12. rgt-mult-inverse N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 87.3

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

   10. Applied rewrites 87.3%

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

   11. Taylor expanded in y around inf

       \[\leadsto 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 81.6%

       \[\leadsto 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq -50000000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;x - x \cdot x\\ \mathbf{elif}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq 50:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x}\\ \mathbf{if}\;t\_0 \leq -50000000000:\\ \;\;\;\;\frac{x - 1}{y}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (+ (/ x y) 1.0) x) (+ 1.0 x))))
   (if (<= t_0 -50000000000.0)
     (/ (- x 1.0) y)
     (if (<= t_0 2.0) (/ x (+ 1.0 x)) (/ x y)))))

C

double code(double x, double y) {
	double t_0 = (((x / y) + 1.0) * x) / (1.0 + x);
	double tmp;
	if (t_0 <= -50000000000.0) {
		tmp = (x - 1.0) / y;
	} else if (t_0 <= 2.0) {
		tmp = x / (1.0 + x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x / y) + 1.0d0) * x) / (1.0d0 + x)
    if (t_0 <= (-50000000000.0d0)) then
        tmp = (x - 1.0d0) / y
    else if (t_0 <= 2.0d0) then
        tmp = x / (1.0d0 + x)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (((x / y) + 1.0) * x) / (1.0 + x);
	double tmp;
	if (t_0 <= -50000000000.0) {
		tmp = (x - 1.0) / y;
	} else if (t_0 <= 2.0) {
		tmp = x / (1.0 + x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (((x / y) + 1.0) * x) / (1.0 + x)
	tmp = 0
	if t_0 <= -50000000000.0:
		tmp = (x - 1.0) / y
	elif t_0 <= 2.0:
		tmp = x / (1.0 + x)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(Float64(x / y) + 1.0) * x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_0 <= -50000000000.0)
		tmp = Float64(Float64(x - 1.0) / y);
	elseif (t_0 <= 2.0)
		tmp = Float64(x / Float64(1.0 + x));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (((x / y) + 1.0) * x) / (1.0 + x);
	tmp = 0.0;
	if (t_0 <= -50000000000.0)
		tmp = (x - 1.0) / y;
	elseif (t_0 <= 2.0)
		tmp = x / (1.0 + x);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -50000000000.0], N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -5e10

   1. Initial program 77.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-fma.f64 77.4

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

   4. Applied rewrites 77.4%

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

   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. associate-/r* N/A

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

      2. metadata-eval N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. associate-/r* N/A

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

      11. distribute-neg-frac2 N/A

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

      12. mul-1-neg N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. mul-1-neg N/A

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

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

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

      18. sub-neg N/A

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

      19. metadata-eval N/A

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

      20. distribute-neg-in N/A

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

      21. mul-1-neg N/A

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

      22. remove-double-neg N/A

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

      23. metadata-eval N/A

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

   7. Applied rewrites 76.4%

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

   8. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

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

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. rgt-mult-inverse N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt-out N/A

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

      12. rgt-mult-inverse N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 79.8

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

   10. Applied rewrites 79.8%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 79.6%

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

   if -5e10 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 86.5

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

   5. Applied rewrites 86.5%

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

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

   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 inf

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

      1. lower-/.f64 80.3

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

   5. Applied rewrites 80.3%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq -50000000000:\\ \;\;\;\;\frac{x - 1}{y}\\ \mathbf{elif}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq 2:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x}\\ \mathbf{if}\;t\_0 \leq -50000000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (+ (/ x y) 1.0) x) (+ 1.0 x))))
   (if (<= t_0 -50000000000.0)
     (/ x y)
     (if (<= t_0 2.0) (/ x (+ 1.0 x)) (/ x y)))))

C

double code(double x, double y) {
	double t_0 = (((x / y) + 1.0) * x) / (1.0 + x);
	double tmp;
	if (t_0 <= -50000000000.0) {
		tmp = x / y;
	} else if (t_0 <= 2.0) {
		tmp = x / (1.0 + x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x / y) + 1.0d0) * x) / (1.0d0 + x)
    if (t_0 <= (-50000000000.0d0)) then
        tmp = x / y
    else if (t_0 <= 2.0d0) then
        tmp = x / (1.0d0 + x)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (((x / y) + 1.0) * x) / (1.0 + x);
	double tmp;
	if (t_0 <= -50000000000.0) {
		tmp = x / y;
	} else if (t_0 <= 2.0) {
		tmp = x / (1.0 + x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (((x / y) + 1.0) * x) / (1.0 + x)
	tmp = 0
	if t_0 <= -50000000000.0:
		tmp = x / y
	elif t_0 <= 2.0:
		tmp = x / (1.0 + x)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(Float64(x / y) + 1.0) * x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_0 <= -50000000000.0)
		tmp = Float64(x / y);
	elseif (t_0 <= 2.0)
		tmp = Float64(x / Float64(1.0 + x));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (((x / y) + 1.0) * x) / (1.0 + x);
	tmp = 0.0;
	if (t_0 <= -50000000000.0)
		tmp = x / y;
	elseif (t_0 <= 2.0)
		tmp = x / (1.0 + x);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -50000000000.0], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -5e10 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 71.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 79.7

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

   5. Applied rewrites 79.7%

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

   if -5e10 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 86.5

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

   5. Applied rewrites 86.5%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq -50000000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq 2:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x}\\ \mathbf{if}\;t\_0 \leq -50000000000:\\ \;\;\;\;\left(-x\right) \cdot x\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (+ (/ x y) 1.0) x) (+ 1.0 x))))
   (if (<= t_0 -50000000000.0) (* (- x) x) (if (<= t_0 0.5) (* 1.0 x) 1.0))))

C

double code(double x, double y) {
	double t_0 = (((x / y) + 1.0) * x) / (1.0 + x);
	double tmp;
	if (t_0 <= -50000000000.0) {
		tmp = -x * x;
	} else if (t_0 <= 0.5) {
		tmp = 1.0 * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x / y) + 1.0d0) * x) / (1.0d0 + x)
    if (t_0 <= (-50000000000.0d0)) then
        tmp = -x * x
    else if (t_0 <= 0.5d0) then
        tmp = 1.0d0 * x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (((x / y) + 1.0) * x) / (1.0 + x);
	double tmp;
	if (t_0 <= -50000000000.0) {
		tmp = -x * x;
	} else if (t_0 <= 0.5) {
		tmp = 1.0 * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (((x / y) + 1.0) * x) / (1.0 + x)
	tmp = 0
	if t_0 <= -50000000000.0:
		tmp = -x * x
	elif t_0 <= 0.5:
		tmp = 1.0 * x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(Float64(x / y) + 1.0) * x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_0 <= -50000000000.0)
		tmp = Float64(Float64(-x) * x);
	elseif (t_0 <= 0.5)
		tmp = Float64(1.0 * x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (((x / y) + 1.0) * x) / (1.0 + x);
	tmp = 0.0;
	if (t_0 <= -50000000000.0)
		tmp = -x * x;
	elseif (t_0 <= 0.5)
		tmp = 1.0 * x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -50000000000.0], N[((-x) * x), $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[(1.0 * x), $MachinePrecision], 1.0]]]

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))) < -5e10

   1. Initial program 77.4%

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

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

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

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 36.9

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

   5. Applied rewrites 36.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 24.6%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 25.0%

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

   if -5e10 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.5

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing
   3. 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. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 99.9

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. remove-double-div N/A

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

      7. lower-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 99.9

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.9

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-+.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 83.7%

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

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

   1. Initial program 77.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-fma.f64 77.6

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

   4. Applied rewrites 77.6%

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

   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. associate-/r* N/A

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

      2. metadata-eval N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. associate-/r* N/A

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

      11. distribute-neg-frac2 N/A

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

      12. mul-1-neg N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. mul-1-neg N/A

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

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

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

      18. sub-neg N/A

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

      19. metadata-eval N/A

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

      20. distribute-neg-in N/A

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

      21. mul-1-neg N/A

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

      22. remove-double-neg N/A

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

      23. metadata-eval N/A

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

   7. Applied rewrites 36.5%

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

   8. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

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

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. rgt-mult-inverse N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt-out N/A

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

      12. rgt-mult-inverse N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 85.8

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

   10. Applied rewrites 85.8%

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

   11. Taylor expanded in y around inf

       \[\leadsto 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 35.3%

       \[\leadsto 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq -50000000000:\\ \;\;\;\;\left(-x\right) \cdot x\\ \mathbf{elif}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq 0.5:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;x - x \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (+ (/ x y) 1.0) x) (+ 1.0 x)) 5e-9) (- x (* x x)) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((((x / y) + 1.0) * x) / (1.0 + x)) <= 5e-9) {
		tmp = x - (x * x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((((x / y) + 1.0d0) * x) / (1.0d0 + x)) <= 5d-9) then
        tmp = x - (x * x)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((((x / y) + 1.0) * x) / (1.0 + x)) <= 5e-9) {
		tmp = x - (x * x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((((x / y) + 1.0) * x) / (1.0 + x)) <= 5e-9:
		tmp = x - (x * x)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x / y) + 1.0) * x) / Float64(1.0 + x)) <= 5e-9)
		tmp = Float64(x - Float64(x * x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((((x / y) + 1.0) * x) / (1.0 + x)) <= 5e-9)
		tmp = x - (x * x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], 5e-9], N[(x - N[(x * x), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 5.0000000000000001e-9

   1. Initial program 93.1%

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

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

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

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 80.9

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

   5. Applied rewrites 80.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 66.7%

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

   if 5.0000000000000001e-9 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 77.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-fma.f64 77.9

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

   4. Applied rewrites 77.9%

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

   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. associate-/r* N/A

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

      2. metadata-eval N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. associate-/r* N/A

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

      11. distribute-neg-frac2 N/A

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

      12. mul-1-neg N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. mul-1-neg N/A

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

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

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

      18. sub-neg N/A

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

      19. metadata-eval N/A

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

      20. distribute-neg-in N/A

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

      21. mul-1-neg N/A

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

      22. remove-double-neg N/A

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

      23. metadata-eval N/A

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

   7. Applied rewrites 36.0%

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

   8. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

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

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. rgt-mult-inverse N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt-out N/A

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

      12. rgt-mult-inverse N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 84.9

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

   10. Applied rewrites 84.9%

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

   11. Taylor expanded in y around inf

       \[\leadsto 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 35.1%

       \[\leadsto 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;x - x \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq 0.5:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (+ (/ x y) 1.0) x) (+ 1.0 x)) 0.5) (* 1.0 x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((((x / y) + 1.0) * x) / (1.0 + x)) <= 0.5) {
		tmp = 1.0 * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((((x / y) + 1.0d0) * x) / (1.0d0 + x)) <= 0.5d0) then
        tmp = 1.0d0 * x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((((x / y) + 1.0) * x) / (1.0 + x)) <= 0.5) {
		tmp = 1.0 * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((((x / y) + 1.0) * x) / (1.0 + x)) <= 0.5:
		tmp = 1.0 * x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x / y) + 1.0) * x) / Float64(1.0 + x)) <= 0.5)
		tmp = Float64(1.0 * x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((((x / y) + 1.0) * x) / (1.0 + x)) <= 0.5)
		tmp = 1.0 * x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], 0.5], N[(1.0 * x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 93.1%

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

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 99.9

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. remove-double-div N/A

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

      7. lower-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 99.9

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.9

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-+.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 59.6%

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

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

   1. Initial program 77.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-fma.f64 77.6

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

   4. Applied rewrites 77.6%

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

   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. associate-/r* N/A

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

      2. metadata-eval N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. associate-/r* N/A

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

      11. distribute-neg-frac2 N/A

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

      12. mul-1-neg N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. mul-1-neg N/A

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

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

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

      18. sub-neg N/A

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

      19. metadata-eval N/A

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

      20. distribute-neg-in N/A

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

      21. mul-1-neg N/A

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

      22. remove-double-neg N/A

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

      23. metadata-eval N/A

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

   7. Applied rewrites 36.5%

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

   8. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

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

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. rgt-mult-inverse N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt-out N/A

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

      12. rgt-mult-inverse N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 85.8

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

   10. Applied rewrites 85.8%

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

   11. Taylor expanded in y around inf

       \[\leadsto 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 35.3%

       \[\leadsto 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.4%

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

Alternative 9: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - 1}{y} + 1\\ \mathbf{if}\;x \leq -1 \cdot 10^{+44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2450000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ (- x 1.0) y) 1.0)))
   (if (<= x -1e+44)
     t_0
     (if (<= x 2450000000000.0) (/ (fma (/ x y) x x) (+ 1.0 x)) t_0))))

C

double code(double x, double y) {
	double t_0 = ((x - 1.0) / y) + 1.0;
	double tmp;
	if (x <= -1e+44) {
		tmp = t_0;
	} else if (x <= 2450000000000.0) {
		tmp = fma((x / y), x, x) / (1.0 + x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - 1.0) / y) + 1.0)
	tmp = 0.0
	if (x <= -1e+44)
		tmp = t_0;
	elseif (x <= 2450000000000.0)
		tmp = Float64(fma(Float64(x / y), x, x) / Float64(1.0 + x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -1e+44], t$95$0, If[LessEqual[x, 2450000000000.0], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.0000000000000001e44 or 2.45e12 < x

   1. Initial program 69.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-fma.f64 69.9

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

   4. Applied rewrites 69.9%

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

   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. associate-/r* N/A

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

      2. metadata-eval N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. associate-/r* N/A

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

      11. distribute-neg-frac2 N/A

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

      12. mul-1-neg N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. mul-1-neg N/A

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

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

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

      18. sub-neg N/A

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

      19. metadata-eval N/A

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

      20. distribute-neg-in N/A

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

      21. mul-1-neg N/A

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

      22. remove-double-neg N/A

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

      23. metadata-eval N/A

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

   7. Applied rewrites 45.4%

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

   8. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

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

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. rgt-mult-inverse N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt-out N/A

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

      12. rgt-mult-inverse N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 99.7

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

   10. Applied rewrites 99.7%

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

   12. Applied rewrites 100.0%

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

   if -1.0000000000000001e44 < x < 2.45e12

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+44}:\\ \;\;\;\;\frac{x - 1}{y} + 1\\ \mathbf{elif}\;x \leq 2450000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 1}{y} + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.5%

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

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 99.9

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 99.9

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 11: 98.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - 1}{y} + 1\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - y, x, y\right)}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ (- x 1.0) y) 1.0)))
   (if (<= x -1.0) t_0 (if (<= x 0.9) (* (/ (fma (- 1.0 y) x y) y) x) t_0))))

C

double code(double x, double y) {
	double t_0 = ((x - 1.0) / y) + 1.0;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 0.9) {
		tmp = (fma((1.0 - y), x, y) / y) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - 1.0) / y) + 1.0)
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 0.9)
		tmp = Float64(Float64(fma(Float64(1.0 - y), x, y) / y) * x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, 0.9], N[(N[(N[(N[(1.0 - y), $MachinePrecision] * x + y), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.900000000000000022 < x

   1. Initial program 74.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-fma.f64 74.1

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

   4. Applied rewrites 74.1%

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

   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. associate-/r* N/A

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

      2. metadata-eval N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. associate-/r* N/A

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

      11. distribute-neg-frac2 N/A

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

      12. mul-1-neg N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. mul-1-neg N/A

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

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

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

      18. sub-neg N/A

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

      19. metadata-eval N/A

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

      20. distribute-neg-in N/A

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

      21. mul-1-neg N/A

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

      22. remove-double-neg N/A

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

      23. metadata-eval N/A

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

   7. Applied rewrites 45.4%

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

   8. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

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

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. rgt-mult-inverse N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt-out N/A

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

      12. rgt-mult-inverse N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 95.9

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

   10. Applied rewrites 95.9%

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

   12. Applied rewrites 96.1%

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

   if -1 < x < 0.900000000000000022

   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. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

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

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 74.6%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 99.3%

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

Alternative 12: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.5%

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

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.9

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 99.9

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 13: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - 1}{y} + 1\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y} - x, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ (- x 1.0) y) 1.0)))
   (if (<= x -1.0) t_0 (if (<= x 0.9) (fma (- (/ x y) x) x x) t_0))))

C

double code(double x, double y) {
	double t_0 = ((x - 1.0) / y) + 1.0;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 0.9) {
		tmp = fma(((x / y) - x), x, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - 1.0) / y) + 1.0)
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 0.9)
		tmp = fma(Float64(Float64(x / y) - x), x, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, 0.9], N[(N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision] * x + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.900000000000000022 < x

   1. Initial program 74.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-fma.f64 74.1

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

   4. Applied rewrites 74.1%

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

   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. associate-/r* N/A

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

      2. metadata-eval N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. associate-/r* N/A

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

      11. distribute-neg-frac2 N/A

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

      12. mul-1-neg N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. mul-1-neg N/A

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

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

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

      18. sub-neg N/A

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

      19. metadata-eval N/A

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

      20. distribute-neg-in N/A

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

      21. mul-1-neg N/A

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

      22. remove-double-neg N/A

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

      23. metadata-eval N/A

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

   7. Applied rewrites 45.4%

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

   8. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

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

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. rgt-mult-inverse N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt-out N/A

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

      12. rgt-mult-inverse N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 95.9

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

   10. Applied rewrites 95.9%

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

   12. Applied rewrites 96.1%

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

   if -1 < x < 0.900000000000000022

   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. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

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

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 99.3

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

   5. Applied rewrites 99.3%

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

Alternative 14: 98.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - 1}{y} + 1\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ (- x 1.0) y) 1.0)))
   (if (<= x -1.0) t_0 (if (<= x 1.3) (fma (/ x y) x x) t_0))))

C

double code(double x, double y) {
	double t_0 = ((x - 1.0) / y) + 1.0;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 1.3) {
		tmp = fma((x / y), x, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - 1.0) / y) + 1.0)
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 1.3)
		tmp = fma(Float64(x / y), x, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, 1.3], N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.30000000000000004 < x

   1. Initial program 73.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-fma.f64 73.9

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

   4. Applied rewrites 73.9%

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

   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. associate-/r* N/A

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

      2. metadata-eval N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. associate-/r* N/A

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

      11. distribute-neg-frac2 N/A

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

      12. mul-1-neg N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. mul-1-neg N/A

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

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

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

      18. sub-neg N/A

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

      19. metadata-eval N/A

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

      20. distribute-neg-in N/A

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

      21. mul-1-neg N/A

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

      22. remove-double-neg N/A

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

      23. metadata-eval N/A

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

   7. Applied rewrites 45.7%

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

   8. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

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

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. rgt-mult-inverse N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt-out N/A

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

      12. rgt-mult-inverse N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 96.6

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

   10. Applied rewrites 96.6%

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

   12. Applied rewrites 96.8%

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

   if -1 < x < 1.30000000000000004

   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. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \left(\frac{1}{y} - 1\right), x, x\right)} \]

      5. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{y} \cdot x - 1 \cdot x}, x, x\right) \]

      6. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1 \cdot x}{y}} - 1 \cdot x, x, x\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{x}}{y} - 1 \cdot x, x, x\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{y} - \color{blue}{x}, x, x\right) \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{y} - x}, x, x\right) \]

      10. lower-/.f64 98.7

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{y}} - x, x, x\right) \]

   5. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y} - x, x, x\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\frac{x}{y}, x, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 98.2%

      \[\leadsto \mathsf{fma}\left(\frac{x}{y}, x, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 14.5% accurate, 34.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 88.5%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{x}{y} + 1\right) \cdot x}}{x + 1} \]

   3. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{x}{y} + 1\right)} \cdot x}{x + 1} \]

   4. distribute-lft1-in N/A

      \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x + x}}{x + 1} \]

   5. lower-fma.f64 88.5

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}}{x + 1} \]

4. Applied rewrites 88.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}}{x + 1} \]

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. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{{x}^{2}}{y}}{1 + x}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\frac{{x}^{2}}{y}}{\color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} + x} \]

   3. remove-double-neg N/A

      \[\leadsto \frac{\frac{{x}^{2}}{y}}{\left(\mathsf{neg}\left(-1\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)}} \]

   4. mul-1-neg N/A

      \[\leadsto \frac{\frac{{x}^{2}}{y}}{\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(\color{blue}{-1 \cdot x}\right)\right)} \]

   5. distribute-neg-in N/A

      \[\leadsto \frac{\frac{{x}^{2}}{y}}{\color{blue}{\mathsf{neg}\left(\left(-1 + -1 \cdot x\right)\right)}} \]

   6. +-commutative N/A

      \[\leadsto \frac{\frac{{x}^{2}}{y}}{\mathsf{neg}\left(\color{blue}{\left(-1 \cdot x + -1\right)}\right)} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\frac{{x}^{2}}{y}}{\mathsf{neg}\left(\left(-1 \cdot x + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)} \]

   8. sub-neg N/A

      \[\leadsto \frac{\frac{{x}^{2}}{y}}{\mathsf{neg}\left(\color{blue}{\left(-1 \cdot x - 1\right)}\right)} \]

   9. distribute-neg-frac2 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\frac{{x}^{2}}{y}}{-1 \cdot x - 1}\right)} \]

   10. associate-/r* N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{x}^{2}}{y \cdot \left(-1 \cdot x - 1\right)}}\right) \]

   11. distribute-neg-frac2 N/A

       \[\leadsto \color{blue}{\frac{{x}^{2}}{\mathsf{neg}\left(y \cdot \left(-1 \cdot x - 1\right)\right)}} \]

   12. mul-1-neg N/A

       \[\leadsto \frac{{x}^{2}}{\color{blue}{-1 \cdot \left(y \cdot \left(-1 \cdot x - 1\right)\right)}} \]

   13. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{{x}^{2}}{-1 \cdot \left(y \cdot \left(-1 \cdot x - 1\right)\right)}} \]

   14. unpow2 N/A

       \[\leadsto \frac{\color{blue}{x \cdot x}}{-1 \cdot \left(y \cdot \left(-1 \cdot x - 1\right)\right)} \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{\color{blue}{x \cdot x}}{-1 \cdot \left(y \cdot \left(-1 \cdot x - 1\right)\right)} \]

   16. mul-1-neg N/A

       \[\leadsto \frac{x \cdot x}{\color{blue}{\mathsf{neg}\left(y \cdot \left(-1 \cdot x - 1\right)\right)}} \]

   17. distribute-rgt-neg-in N/A

       \[\leadsto \frac{x \cdot x}{\color{blue}{y \cdot \left(\mathsf{neg}\left(\left(-1 \cdot x - 1\right)\right)\right)}} \]

   18. sub-neg N/A

       \[\leadsto \frac{x \cdot x}{y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot x + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right)} \]

   19. metadata-eval N/A

       \[\leadsto \frac{x \cdot x}{y \cdot \left(\mathsf{neg}\left(\left(-1 \cdot x + \color{blue}{-1}\right)\right)\right)} \]

   20. distribute-neg-in N/A

       \[\leadsto \frac{x \cdot x}{y \cdot \color{blue}{\left(\left(\mathsf{neg}\left(-1 \cdot x\right)\right) + \left(\mathsf{neg}\left(-1\right)\right)\right)}} \]

   21. mul-1-neg N/A

       \[\leadsto \frac{x \cdot x}{y \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)\right) + \left(\mathsf{neg}\left(-1\right)\right)\right)} \]

   22. remove-double-neg N/A

       \[\leadsto \frac{x \cdot x}{y \cdot \left(\color{blue}{x} + \left(\mathsf{neg}\left(-1\right)\right)\right)} \]

   23. metadata-eval N/A

       \[\leadsto \frac{x \cdot x}{y \cdot \left(x + \color{blue}{1}\right)} \]

7. Applied rewrites 32.3%

   \[\leadsto \color{blue}{\frac{x \cdot x}{\mathsf{fma}\left(y, x, y\right)}} \]

8. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{x} + \frac{1}{y}\right) - \frac{1}{x \cdot y}\right)} \]
9. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{x} + \left(\frac{1}{y} - \frac{1}{x \cdot y}\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{y} - \frac{1}{x \cdot y}\right) + \frac{1}{x}\right)} \]

   3. distribute-lft-in N/A

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{y} - \frac{1}{x \cdot y}\right) + x \cdot \frac{1}{x}} \]

   4. sub-neg N/A

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{y} + \left(\mathsf{neg}\left(\frac{1}{x \cdot y}\right)\right)\right)} + x \cdot \frac{1}{x} \]

   5. distribute-lft-in N/A

      \[\leadsto \color{blue}{\left(x \cdot \frac{1}{y} + x \cdot \left(\mathsf{neg}\left(\frac{1}{x \cdot y}\right)\right)\right)} + x \cdot \frac{1}{x} \]

   6. distribute-rgt-neg-out N/A

      \[\leadsto \left(x \cdot \frac{1}{y} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{x \cdot y}\right)\right)}\right) + x \cdot \frac{1}{x} \]

   7. associate-/r* N/A

      \[\leadsto \left(x \cdot \frac{1}{y} + \left(\mathsf{neg}\left(x \cdot \color{blue}{\frac{\frac{1}{x}}{y}}\right)\right)\right) + x \cdot \frac{1}{x} \]

   8. associate-*r/ N/A

      \[\leadsto \left(x \cdot \frac{1}{y} + \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot \frac{1}{x}}{y}}\right)\right)\right) + x \cdot \frac{1}{x} \]

   9. rgt-mult-inverse N/A

      \[\leadsto \left(x \cdot \frac{1}{y} + \left(\mathsf{neg}\left(\frac{\color{blue}{1}}{y}\right)\right)\right) + x \cdot \frac{1}{x} \]

   10. neg-mul-1 N/A

       \[\leadsto \left(x \cdot \frac{1}{y} + \color{blue}{-1 \cdot \frac{1}{y}}\right) + x \cdot \frac{1}{x} \]

   11. distribute-rgt-out N/A

       \[\leadsto \color{blue}{\frac{1}{y} \cdot \left(x + -1\right)} + x \cdot \frac{1}{x} \]

   12. rgt-mult-inverse N/A

       \[\leadsto \frac{1}{y} \cdot \left(x + -1\right) + \color{blue}{1} \]

   13. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{y}, x + -1, 1\right)} \]

   14. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{y}}, x + -1, 1\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{y}, x + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

   16. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{y}, \color{blue}{x - 1}, 1\right) \]

   17. lower--.f64 43.9

       \[\leadsto \mathsf{fma}\left(\frac{1}{y}, \color{blue}{x - 1}, 1\right) \]

10. Applied rewrites 43.9%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{y}, x - 1, 1\right)} \]

11. Taylor expanded in y around inf

    \[\leadsto 1 \]
12. Step-by-step derivation

13. Applied rewrites 12.6%

    \[\leadsto 1 \]
14. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (/ x 1.0) (/ (+ (/ x y) 1.0) (+ x 1.0))))

C

double code(double x, double y) {
	return (x / 1.0) * (((x / y) + 1.0) / (x + 1.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x / 1.0d0) * (((x / y) + 1.0d0) / (x + 1.0d0))
end function

Java

public static double code(double x, double y) {
	return (x / 1.0) * (((x / y) + 1.0) / (x + 1.0));
}

Python

def code(x, y):
	return (x / 1.0) * (((x / y) + 1.0) / (x + 1.0))

Julia

function code(x, y)
	return Float64(Float64(x / 1.0) * Float64(Float64(Float64(x / y) + 1.0) / Float64(x + 1.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = (x / 1.0) * (((x / y) + 1.0) / (x + 1.0));
end

Wolfram

code[x_, y_] := N[(N[(x / 1.0), $MachinePrecision] * N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024240 
(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)))