Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, C

Percentage Accurate: 100.0% → 100.0%

Time: 7.3s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\frac{x - y}{1 - y} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 98.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ t_1 := \frac{x}{1 - y}\\ \mathbf{if}\;t\_0 \leq -10:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x + -1, \mathsf{fma}\left(y, y, y\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))) (t_1 (/ x (- 1.0 y))))
   (if (<= t_0 -10.0)
     t_1
     (if (<= t_0 4e-7)
       (fma (+ x -1.0) (fma y y y) x)
       (if (<= t_0 2.0) (/ y (+ y -1.0)) t_1)))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double t_1 = x / (1.0 - y);
	double tmp;
	if (t_0 <= -10.0) {
		tmp = t_1;
	} else if (t_0 <= 4e-7) {
		tmp = fma((x + -1.0), fma(y, y, y), x);
	} else if (t_0 <= 2.0) {
		tmp = y / (y + -1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(1.0 - y))
	t_1 = Float64(x / Float64(1.0 - y))
	tmp = 0.0
	if (t_0 <= -10.0)
		tmp = t_1;
	elseif (t_0 <= 4e-7)
		tmp = fma(Float64(x + -1.0), fma(y, y, y), x);
	elseif (t_0 <= 2.0)
		tmp = Float64(y / Float64(y + -1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -10.0], t$95$1, If[LessEqual[t$95$0, 4e-7], N[(N[(x + -1.0), $MachinePrecision] * N[(y * y + y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(y / N[(y + -1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < -10 or 2 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if -10 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 3.9999999999999998e-7

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. mul-1-neg N/A

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

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

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. remove-double-neg N/A

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

      14. lower-+.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-fma.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if 3.9999999999999998e-7 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 2

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. neg-sub0 N/A

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

      5. associate--r- N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq -10:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{elif}\;\frac{x - y}{1 - y} \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x + -1, \mathsf{fma}\left(y, y, y\right), x\right)\\ \mathbf{elif}\;\frac{x - y}{1 - y} \leq 2:\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ t_1 := \frac{x}{1 - y}\\ \mathbf{if}\;t\_0 \leq -10:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\left(y - x\right) \cdot -1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))) (t_1 (/ x (- 1.0 y))))
   (if (<= t_0 -10.0)
     t_1
     (if (<= t_0 4e-7)
       (* (- y x) -1.0)
       (if (<= t_0 2.0) (/ y (+ y -1.0)) t_1)))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double t_1 = x / (1.0 - y);
	double tmp;
	if (t_0 <= -10.0) {
		tmp = t_1;
	} else if (t_0 <= 4e-7) {
		tmp = (y - x) * -1.0;
	} else if (t_0 <= 2.0) {
		tmp = y / (y + -1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x - y) / (1.0d0 - y)
    t_1 = x / (1.0d0 - y)
    if (t_0 <= (-10.0d0)) then
        tmp = t_1
    else if (t_0 <= 4d-7) then
        tmp = (y - x) * (-1.0d0)
    else if (t_0 <= 2.0d0) then
        tmp = y / (y + (-1.0d0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double t_1 = x / (1.0 - y);
	double tmp;
	if (t_0 <= -10.0) {
		tmp = t_1;
	} else if (t_0 <= 4e-7) {
		tmp = (y - x) * -1.0;
	} else if (t_0 <= 2.0) {
		tmp = y / (y + -1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / (1.0 - y)
	t_1 = x / (1.0 - y)
	tmp = 0
	if t_0 <= -10.0:
		tmp = t_1
	elif t_0 <= 4e-7:
		tmp = (y - x) * -1.0
	elif t_0 <= 2.0:
		tmp = y / (y + -1.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(1.0 - y))
	t_1 = Float64(x / Float64(1.0 - y))
	tmp = 0.0
	if (t_0 <= -10.0)
		tmp = t_1;
	elseif (t_0 <= 4e-7)
		tmp = Float64(Float64(y - x) * -1.0);
	elseif (t_0 <= 2.0)
		tmp = Float64(y / Float64(y + -1.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) / (1.0 - y);
	t_1 = x / (1.0 - y);
	tmp = 0.0;
	if (t_0 <= -10.0)
		tmp = t_1;
	elseif (t_0 <= 4e-7)
		tmp = (y - x) * -1.0;
	elseif (t_0 <= 2.0)
		tmp = y / (y + -1.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -10.0], t$95$1, If[LessEqual[t$95$0, 4e-7], N[(N[(y - x), $MachinePrecision] * -1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(y / N[(y + -1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < -10 or 2 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if -10 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 3.9999999999999998e-7

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. neg-sub0 N/A

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

      8. lift--.f64 N/A

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

      9. associate--r- N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. neg-sub0 N/A

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

      14. lift--.f64 N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

      17. associate--r+ N/A

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

      18. neg-sub0 N/A

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

      19. remove-double-neg N/A

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

      20. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 99.3%

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

   if 3.9999999999999998e-7 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 2

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. neg-sub0 N/A

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

      5. associate--r- N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq -10:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{elif}\;\frac{x - y}{1 - y} \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\left(y - x\right) \cdot -1\\ \mathbf{elif}\;\frac{x - y}{1 - y} \leq 2:\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ t_1 := \frac{x}{1 - y}\\ \mathbf{if}\;t\_0 \leq -10:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.05:\\ \;\;\;\;\left(y - x\right) \cdot -1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))) (t_1 (/ x (- 1.0 y))))
   (if (<= t_0 -10.0)
     t_1
     (if (<= t_0 0.05) (* (- y x) -1.0) (if (<= t_0 2.0) 1.0 t_1)))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double t_1 = x / (1.0 - y);
	double tmp;
	if (t_0 <= -10.0) {
		tmp = t_1;
	} else if (t_0 <= 0.05) {
		tmp = (y - x) * -1.0;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x - y) / (1.0d0 - y)
    t_1 = x / (1.0d0 - y)
    if (t_0 <= (-10.0d0)) then
        tmp = t_1
    else if (t_0 <= 0.05d0) then
        tmp = (y - x) * (-1.0d0)
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double t_1 = x / (1.0 - y);
	double tmp;
	if (t_0 <= -10.0) {
		tmp = t_1;
	} else if (t_0 <= 0.05) {
		tmp = (y - x) * -1.0;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / (1.0 - y)
	t_1 = x / (1.0 - y)
	tmp = 0
	if t_0 <= -10.0:
		tmp = t_1
	elif t_0 <= 0.05:
		tmp = (y - x) * -1.0
	elif t_0 <= 2.0:
		tmp = 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(1.0 - y))
	t_1 = Float64(x / Float64(1.0 - y))
	tmp = 0.0
	if (t_0 <= -10.0)
		tmp = t_1;
	elseif (t_0 <= 0.05)
		tmp = Float64(Float64(y - x) * -1.0);
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) / (1.0 - y);
	t_1 = x / (1.0 - y);
	tmp = 0.0;
	if (t_0 <= -10.0)
		tmp = t_1;
	elseif (t_0 <= 0.05)
		tmp = (y - x) * -1.0;
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -10.0], t$95$1, If[LessEqual[t$95$0, 0.05], N[(N[(y - x), $MachinePrecision] * -1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < -10 or 2 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if -10 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.050000000000000003

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. neg-sub0 N/A

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

      8. lift--.f64 N/A

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

      9. associate--r- N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. neg-sub0 N/A

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

      14. lift--.f64 N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

      17. associate--r+ N/A

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

      18. neg-sub0 N/A

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

      19. remove-double-neg N/A

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

      20. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 95.8%

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

   if 0.050000000000000003 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 2

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 100.0%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq -10:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{elif}\;\frac{x - y}{1 - y} \leq 0.05:\\ \;\;\;\;\left(y - x\right) \cdot -1\\ \mathbf{elif}\;\frac{x - y}{1 - y} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ \mathbf{if}\;t\_0 \leq 0.05:\\ \;\;\;\;\left(y - x\right) \cdot -1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(x, y, x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))))
   (if (<= t_0 0.05)
     (* (- y x) -1.0)
     (if (<= t_0 2.0) 1.0 (fma y (fma x y x) x)))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double tmp;
	if (t_0 <= 0.05) {
		tmp = (y - x) * -1.0;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = fma(y, fma(x, y, x), x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(1.0 - y))
	tmp = 0.0
	if (t_0 <= 0.05)
		tmp = Float64(Float64(y - x) * -1.0);
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = fma(y, fma(x, y, x), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.05], N[(N[(y - x), $MachinePrecision] * -1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, N[(y * N[(x * y + x), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.050000000000000003

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. neg-sub0 N/A

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

      8. lift--.f64 N/A

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

      9. associate--r- N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. neg-sub0 N/A

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

      14. lift--.f64 N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

      17. associate--r+ N/A

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

      18. neg-sub0 N/A

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

      19. remove-double-neg N/A

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

      20. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 79.8%

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

   if 0.050000000000000003 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 2

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 100.0%

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

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

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 72.5%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.05:\\ \;\;\;\;\left(y - x\right) \cdot -1\\ \mathbf{elif}\;\frac{x - y}{1 - y} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(x, y, x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if -1 < y < 1

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. neg-sub0 N/A

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

      8. lift--.f64 N/A

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

      9. associate--r- N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. neg-sub0 N/A

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

      14. lift--.f64 N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

      17. associate--r+ N/A

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

      18. neg-sub0 N/A

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

      19. remove-double-neg N/A

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

      20. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. mul-1-neg N/A

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

      8. distribute-lft-neg-in N/A

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

      9. unpow2 N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-in N/A

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

      12. +-commutative N/A

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

      13. unsub-neg N/A

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

      14. lower--.f64 N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. lower-fma.f64 98.8

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

   7. Applied rewrites 98.8%

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

Alternative 7: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{-y}\\ \mathbf{if}\;y \leq -0.85:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(x + -1, \mathsf{fma}\left(y, y, y\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- y))))
   (if (<= y -0.85) t_0 (if (<= y 1.0) (fma (+ x -1.0) (fma y y y) x) t_0))))

C

double code(double x, double y) {
	double t_0 = (x - y) / -y;
	double tmp;
	if (y <= -0.85) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = fma((x + -1.0), fma(y, y, y), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(-y))
	tmp = 0.0
	if (y <= -0.85)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = fma(Float64(x + -1.0), fma(y, y, y), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / (-y)), $MachinePrecision]}, If[LessEqual[y, -0.85], t$95$0, If[LessEqual[y, 1.0], N[(N[(x + -1.0), $MachinePrecision] * N[(y * y + y), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.849999999999999978 or 1 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if -0.849999999999999978 < y < 1

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. mul-1-neg N/A

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

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

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. remove-double-neg N/A

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

      14. lower-+.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-fma.f64 98.3

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

   5. Applied rewrites 98.3%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.85:\\ \;\;\;\;\frac{x - y}{-y}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(x + -1, \mathsf{fma}\left(y, y, y\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{-y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+25}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -3.8:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right) - y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6e+25)
   1.0
   (if (<= y -3.8) (/ x (- y)) (if (<= y 1.0) (- (fma y x x) y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6e+25) {
		tmp = 1.0;
	} else if (y <= -3.8) {
		tmp = x / -y;
	} else if (y <= 1.0) {
		tmp = fma(y, x, x) - y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6e+25)
		tmp = 1.0;
	elseif (y <= -3.8)
		tmp = Float64(x / Float64(-y));
	elseif (y <= 1.0)
		tmp = Float64(fma(y, x, x) - y);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6e+25], 1.0, If[LessEqual[y, -3.8], N[(x / (-y)), $MachinePrecision], If[LessEqual[y, 1.0], N[(N[(y * x + x), $MachinePrecision] - y), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.00000000000000011e25 or 1 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 74.3%

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

   if -6.00000000000000011e25 < y < -3.7999999999999998

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 91.7

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

   5. Applied rewrites 91.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 74.1%

      \[\leadsto \frac{x}{-y} \]

   if -3.7999999999999998 < y < 1

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. associate--r+ N/A

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

      8. *-commutative N/A

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

      9. cancel-sign-sub N/A

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

      10. +-commutative N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 N/A

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

      14. sub-neg N/A

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

      15. remove-double-neg N/A

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

      16. *-rgt-identity N/A

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

      17. distribute-lft-out N/A

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

      18. distribute-rgt-out N/A

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

      19. *-lft-identity N/A

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

      20. lower-fma.f64 97.2

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

   5. Applied rewrites 97.2%

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

Alternative 9: 72.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+25}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-59}:\\ \;\;\;\;-1 \cdot \left(-x\right)\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+22}:\\ \;\;\;\;-\mathsf{fma}\left(y, y, y\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.8e+25)
   1.0
   (if (<= y 7.5e-59) (* -1.0 (- x)) (if (<= y 2.6e+22) (- (fma y y y)) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.8e+25) {
		tmp = 1.0;
	} else if (y <= 7.5e-59) {
		tmp = -1.0 * -x;
	} else if (y <= 2.6e+22) {
		tmp = -fma(y, y, y);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.8e+25)
		tmp = 1.0;
	elseif (y <= 7.5e-59)
		tmp = Float64(-1.0 * Float64(-x));
	elseif (y <= 2.6e+22)
		tmp = Float64(-fma(y, y, y));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.8e+25], 1.0, If[LessEqual[y, 7.5e-59], N[(-1.0 * (-x)), $MachinePrecision], If[LessEqual[y, 2.6e+22], (-N[(y * y + y), $MachinePrecision]), 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.79999999999999992e25 or 2.6e22 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 76.3%

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

   if -4.79999999999999992e25 < y < 7.50000000000000019e-59

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. neg-sub0 N/A

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

      8. lift--.f64 N/A

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

      9. associate--r- N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. neg-sub0 N/A

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

      14. lift--.f64 N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

      17. associate--r+ N/A

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

      18. neg-sub0 N/A

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

      19. remove-double-neg N/A

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

      20. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 93.2%

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

   8. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 78.2

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

   10. Applied rewrites 78.2%

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

   if 7.50000000000000019e-59 < y < 2.6e22

   1. Initial program 99.9%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. mul-1-neg N/A

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

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

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. remove-double-neg N/A

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

      14. lower-+.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-fma.f64 77.0

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

   5. Applied rewrites 77.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 57.6%

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

Alternative 10: 50.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.05:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.05) (- y) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.05) {
		tmp = -y;
	} 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 - y)) <= 0.05d0) then
        tmp = -y
    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 - y)) <= 0.05) {
		tmp = -y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.05:
		tmp = -y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.05)
		tmp = Float64(-y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x - y) / (1.0 - y)) <= 0.05)
		tmp = -y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.05], (-y), 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.050000000000000003

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. neg-sub0 N/A

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

      5. associate--r- N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 27.0

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

   5. Applied rewrites 27.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 24.9%

      \[\leadsto -y \]

   if 0.050000000000000003 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 63.9%

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

Alternative 11: 86.5% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) 1.0 (if (<= y 1.0) (- (fma y x x) y) 1.0)))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 70.6%

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

   if -1 < y < 1

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. associate--r+ N/A

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

      8. *-commutative N/A

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

      9. cancel-sign-sub N/A

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

      10. +-commutative N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 N/A

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

      14. sub-neg N/A

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

      15. remove-double-neg N/A

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

      16. *-rgt-identity N/A

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

      17. distribute-lft-out N/A

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

      18. distribute-rgt-out N/A

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

      19. *-lft-identity N/A

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

      20. lower-fma.f64 97.2

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

   5. Applied rewrites 97.2%

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

Alternative 12: 72.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+25}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-59}:\\ \;\;\;\;-1 \cdot \left(-x\right)\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+22}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.8e+25)
   1.0
   (if (<= y 7.5e-59) (* -1.0 (- x)) (if (<= y 2.6e+22) (- y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.8e+25) {
		tmp = 1.0;
	} else if (y <= 7.5e-59) {
		tmp = -1.0 * -x;
	} else if (y <= 2.6e+22) {
		tmp = -y;
	} 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 (y <= (-4.8d+25)) then
        tmp = 1.0d0
    else if (y <= 7.5d-59) then
        tmp = (-1.0d0) * -x
    else if (y <= 2.6d+22) then
        tmp = -y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.8e+25) {
		tmp = 1.0;
	} else if (y <= 7.5e-59) {
		tmp = -1.0 * -x;
	} else if (y <= 2.6e+22) {
		tmp = -y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.8e+25:
		tmp = 1.0
	elif y <= 7.5e-59:
		tmp = -1.0 * -x
	elif y <= 2.6e+22:
		tmp = -y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.8e+25)
		tmp = 1.0;
	elseif (y <= 7.5e-59)
		tmp = Float64(-1.0 * Float64(-x));
	elseif (y <= 2.6e+22)
		tmp = Float64(-y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.8e+25)
		tmp = 1.0;
	elseif (y <= 7.5e-59)
		tmp = -1.0 * -x;
	elseif (y <= 2.6e+22)
		tmp = -y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.8e+25], 1.0, If[LessEqual[y, 7.5e-59], N[(-1.0 * (-x)), $MachinePrecision], If[LessEqual[y, 2.6e+22], (-y), 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.79999999999999992e25 or 2.6e22 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 76.3%

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

   if -4.79999999999999992e25 < y < 7.50000000000000019e-59

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. neg-sub0 N/A

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

      8. lift--.f64 N/A

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

      9. associate--r- N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. neg-sub0 N/A

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

      14. lift--.f64 N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

      17. associate--r+ N/A

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

      18. neg-sub0 N/A

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

      19. remove-double-neg N/A

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

      20. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 93.2%

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

   8. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 78.2

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

   10. Applied rewrites 78.2%

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

   if 7.50000000000000019e-59 < y < 2.6e22

   1. Initial program 99.9%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. neg-sub0 N/A

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

      5. associate--r- N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 56.4

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

   5. Applied rewrites 56.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 56.7%

      \[\leadsto -y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 73.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.01:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-59}:\\ \;\;\;\;\mathsf{fma}\left(x, y, x\right)\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+22}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -0.01)
   1.0
   (if (<= y 7.5e-59) (fma x y x) (if (<= y 2.6e+22) (- y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -0.01) {
		tmp = 1.0;
	} else if (y <= 7.5e-59) {
		tmp = fma(x, y, x);
	} else if (y <= 2.6e+22) {
		tmp = -y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -0.01)
		tmp = 1.0;
	elseif (y <= 7.5e-59)
		tmp = fma(x, y, x);
	elseif (y <= 2.6e+22)
		tmp = Float64(-y);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -0.01], 1.0, If[LessEqual[y, 7.5e-59], N[(x * y + x), $MachinePrecision], If[LessEqual[y, 2.6e+22], (-y), 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0100000000000000002 or 2.6e22 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 71.4%

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

   if -0.0100000000000000002 < y < 7.50000000000000019e-59

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 82.9

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

   5. Applied rewrites 82.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 82.8%

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

   if 7.50000000000000019e-59 < y < 2.6e22

   1. Initial program 99.9%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. neg-sub0 N/A

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

      5. associate--r- N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 56.4

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

   5. Applied rewrites 56.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 56.7%

      \[\leadsto -y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+25}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\left(y - x\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.8e+25) 1.0 (if (<= y 1.0) (* (- y x) -1.0) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.8e+25) {
		tmp = 1.0;
	} else if (y <= 1.0) {
		tmp = (y - x) * -1.0;
	} 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 (y <= (-4.8d+25)) then
        tmp = 1.0d0
    else if (y <= 1.0d0) then
        tmp = (y - x) * (-1.0d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.8e+25) {
		tmp = 1.0;
	} else if (y <= 1.0) {
		tmp = (y - x) * -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.8e+25:
		tmp = 1.0
	elif y <= 1.0:
		tmp = (y - x) * -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.8e+25)
		tmp = 1.0;
	elseif (y <= 1.0)
		tmp = Float64(Float64(y - x) * -1.0);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.8e+25)
		tmp = 1.0;
	elseif (y <= 1.0)
		tmp = (y - x) * -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.8e+25], 1.0, If[LessEqual[y, 1.0], N[(N[(y - x), $MachinePrecision] * -1.0), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -4.79999999999999992e25 or 1 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 74.3%

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

   if -4.79999999999999992e25 < y < 1

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. neg-sub0 N/A

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

      8. lift--.f64 N/A

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

      9. associate--r- N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. neg-sub0 N/A

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

      14. lift--.f64 N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

      17. associate--r+ N/A

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

      18. neg-sub0 N/A

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

      19. remove-double-neg N/A

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

      20. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 93.1%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+25}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\left(y - x\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 38.7% accurate, 18.0× 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 100.0%

   \[\frac{x - y}{1 - y} \]
2. Add Preprocessing

3. Taylor expanded in y around inf

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

5. Applied rewrites 34.9%

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

Reproduce

herbie shell --seed 2024223 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, C"
  :precision binary64
  (/ (- x y) (- 1.0 y)))