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

Percentage Accurate: 100.0% → 100.0%

Time: 4.2s

Alternatives: 12

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, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. lift--.f64 N/A

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

   3. div-sub N/A

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

   4. lower--.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 100.0

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{x}{1 - y} - \frac{y}{1 - y}} \]
5. Add Preprocessing

Alternative 2: 98.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{y - 1}\\ t_1 := \frac{x}{1 - y}\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(-1, y, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{y}{y - 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- y x) (- y 1.0))) (t_1 (/ x (- 1.0 y))))
   (if (<= t_0 -2.0)
     t_1
     (if (<= t_0 1e-6) (fma -1.0 y x) (if (<= t_0 2.0) (/ y (- y 1.0)) t_1)))))

C

double code(double x, double y) {
	double t_0 = (y - x) / (y - 1.0);
	double t_1 = x / (1.0 - y);
	double tmp;
	if (t_0 <= -2.0) {
		tmp = t_1;
	} else if (t_0 <= 1e-6) {
		tmp = fma(-1.0, 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(y - x) / Float64(y - 1.0))
	t_1 = Float64(x / Float64(1.0 - y))
	tmp = 0.0
	if (t_0 <= -2.0)
		tmp = t_1;
	elseif (t_0 <= 1e-6)
		tmp = fma(-1.0, 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[(y - x), $MachinePrecision] / N[(y - 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2.0], t$95$1, If[LessEqual[t$95$0, 1e-6], N[(-1.0 * y + 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)) < -2 or 2 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 99.9%

      \[\frac{x - y}{1 - y} \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

   if -2 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 9.99999999999999955e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   if 9.99999999999999955e-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. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 99.5

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

   5. Applied rewrites 99.5%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - 1} \leq -2:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{elif}\;\frac{y - x}{y - 1} \leq 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(-1, y, x\right)\\ \mathbf{elif}\;\frac{y - x}{y - 1} \leq 2:\\ \;\;\;\;\frac{y}{y - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{y - 1}\\ t_1 := \frac{x}{1 - y}\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.4:\\ \;\;\;\;\left(y - x\right) \cdot \left(-1 - y\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- y x) (- y 1.0))) (t_1 (/ x (- 1.0 y))))
   (if (<= t_0 -2.0)
     t_1
     (if (<= t_0 0.4) (* (- y x) (- -1.0 y)) (if (<= t_0 2.0) 1.0 t_1)))))

C

double code(double x, double y) {
	double t_0 = (y - x) / (y - 1.0);
	double t_1 = x / (1.0 - y);
	double tmp;
	if (t_0 <= -2.0) {
		tmp = t_1;
	} else if (t_0 <= 0.4) {
		tmp = (y - x) * (-1.0 - y);
	} 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 = (y - x) / (y - 1.0d0)
    t_1 = x / (1.0d0 - y)
    if (t_0 <= (-2.0d0)) then
        tmp = t_1
    else if (t_0 <= 0.4d0) then
        tmp = (y - x) * ((-1.0d0) - y)
    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 = (y - x) / (y - 1.0);
	double t_1 = x / (1.0 - y);
	double tmp;
	if (t_0 <= -2.0) {
		tmp = t_1;
	} else if (t_0 <= 0.4) {
		tmp = (y - x) * (-1.0 - y);
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y - x) / (y - 1.0)
	t_1 = x / (1.0 - y)
	tmp = 0
	if t_0 <= -2.0:
		tmp = t_1
	elif t_0 <= 0.4:
		tmp = (y - x) * (-1.0 - y)
	elif t_0 <= 2.0:
		tmp = 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y - x) / Float64(y - 1.0))
	t_1 = Float64(x / Float64(1.0 - y))
	tmp = 0.0
	if (t_0 <= -2.0)
		tmp = t_1;
	elseif (t_0 <= 0.4)
		tmp = Float64(Float64(y - x) * Float64(-1.0 - y));
	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 = (y - x) / (y - 1.0);
	t_1 = x / (1.0 - y);
	tmp = 0.0;
	if (t_0 <= -2.0)
		tmp = t_1;
	elseif (t_0 <= 0.4)
		tmp = (y - x) * (-1.0 - y);
	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[(y - x), $MachinePrecision] / N[(y - 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2.0], t$95$1, If[LessEqual[t$95$0, 0.4], N[(N[(y - x), $MachinePrecision] * N[(-1.0 - y), $MachinePrecision]), $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)) < -2 or 2 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 99.9%

      \[\frac{x - y}{1 - y} \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

   if -2 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.40000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. remove-double-neg N/A

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

      4. neg-mul-1 N/A

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

      5. associate-*l/ N/A

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

      6. distribute-lft-out N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. *-lft-identity N/A

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

      10. metadata-eval N/A

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

      11. lft-mult-inverse N/A

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

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

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. associate-*l* N/A

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

      16. associate-/l* N/A

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

      17. *-commutative N/A

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

      18. associate-*r/ N/A

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

      19. distribute-rgt-in N/A

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

      20. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 98.3%

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

   if 0.40000000000000002 < (/.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. 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. lower-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lower-/.f64 N/A

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

      6. neg-sub0 N/A

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

      7. lift--.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. neg-sub0 N/A

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

      12. remove-double-neg N/A

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

      13. lower--.f64 N/A

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

      14. neg-sub0 N/A

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

      15. lift--.f64 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

      18. associate--r+ N/A

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

      19. neg-sub0 N/A

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

      20. remove-double-neg N/A

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

      21. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 98.5%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - 1} \leq -2:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{elif}\;\frac{y - x}{y - 1} \leq 0.4:\\ \;\;\;\;\left(y - x\right) \cdot \left(-1 - y\right)\\ \mathbf{elif}\;\frac{y - x}{y - 1} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{y - 1}\\ \mathbf{if}\;t\_0 \leq 0.4:\\ \;\;\;\;\mathsf{fma}\left(-1, y, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- y x) (- y 1.0))))
   (if (<= t_0 0.4) (fma -1.0 y x) (if (<= t_0 2.0) 1.0 (fma -1.0 y x)))))

C

double code(double x, double y) {
	double t_0 = (y - x) / (y - 1.0);
	double tmp;
	if (t_0 <= 0.4) {
		tmp = fma(-1.0, y, x);
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = fma(-1.0, y, x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y - x) / Float64(y - 1.0))
	tmp = 0.0
	if (t_0 <= 0.4)
		tmp = fma(-1.0, y, x);
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = fma(-1.0, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y - x), $MachinePrecision] / N[(y - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.4], N[(-1.0 * y + x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, N[(-1.0 * y + x), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.40000000000000002 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 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. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 79.2

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 79.7%

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

   if 0.40000000000000002 < (/.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. 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. lower-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lower-/.f64 N/A

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

      6. neg-sub0 N/A

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

      7. lift--.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. neg-sub0 N/A

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

      12. remove-double-neg N/A

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

      13. lower--.f64 N/A

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

      14. neg-sub0 N/A

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

      15. lift--.f64 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

      18. associate--r+ N/A

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

      19. neg-sub0 N/A

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

      20. remove-double-neg N/A

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

      21. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 98.5%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - 1} \leq 0.4:\\ \;\;\;\;\mathsf{fma}\left(-1, y, x\right)\\ \mathbf{elif}\;\frac{y - x}{y - 1} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.22 \cdot 10^{+117}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\left(y - x\right) \cdot \left(-1 - y\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.22e+117)
   1.0
   (if (<= y -1.0) (/ (- x) y) (if (<= y 1.0) (* (- y x) (- -1.0 y)) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.22e+117) {
		tmp = 1.0;
	} else if (y <= -1.0) {
		tmp = -x / y;
	} else if (y <= 1.0) {
		tmp = (y - x) * (-1.0 - 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 <= (-1.22d+117)) then
        tmp = 1.0d0
    else if (y <= (-1.0d0)) then
        tmp = -x / y
    else if (y <= 1.0d0) then
        tmp = (y - x) * ((-1.0d0) - y)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.22e+117) {
		tmp = 1.0;
	} else if (y <= -1.0) {
		tmp = -x / y;
	} else if (y <= 1.0) {
		tmp = (y - x) * (-1.0 - y);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.22e+117:
		tmp = 1.0
	elif y <= -1.0:
		tmp = -x / y
	elif y <= 1.0:
		tmp = (y - x) * (-1.0 - y)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.22e+117)
		tmp = 1.0;
	elseif (y <= -1.0)
		tmp = Float64(Float64(-x) / y);
	elseif (y <= 1.0)
		tmp = Float64(Float64(y - x) * Float64(-1.0 - y));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.22e+117)
		tmp = 1.0;
	elseif (y <= -1.0)
		tmp = -x / y;
	elseif (y <= 1.0)
		tmp = (y - x) * (-1.0 - y);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.22e+117], 1.0, If[LessEqual[y, -1.0], N[((-x) / y), $MachinePrecision], If[LessEqual[y, 1.0], N[(N[(y - x), $MachinePrecision] * N[(-1.0 - y), $MachinePrecision]), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.22000000000000004e117 or 1 < y

   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. lower-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lower-/.f64 N/A

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

      6. neg-sub0 N/A

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

      7. lift--.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. neg-sub0 N/A

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

      12. remove-double-neg N/A

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

      13. lower--.f64 N/A

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

      14. neg-sub0 N/A

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

      15. lift--.f64 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

      18. associate--r+ N/A

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

      19. neg-sub0 N/A

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

      20. remove-double-neg N/A

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

      21. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 85.3%

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

   if -1.22000000000000004e117 < y < -1

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. sub-neg N/A

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

      5. div-sub N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 93.8

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

   5. Applied rewrites 93.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 56.8%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. remove-double-neg N/A

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

      4. neg-mul-1 N/A

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

      5. associate-*l/ N/A

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

      6. distribute-lft-out N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. *-lft-identity N/A

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

      10. metadata-eval N/A

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

      11. lft-mult-inverse N/A

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

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

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. associate-*l* N/A

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

      16. associate-/l* N/A

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

      17. *-commutative N/A

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

      18. associate-*r/ N/A

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

      19. distribute-rgt-in N/A

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

      20. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 97.7%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.22 \cdot 10^{+117}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\left(y - x\right) \cdot \left(-1 - y\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double t_0 = ((1.0 - x) / y) - -1.0;
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = (y - x) * (-1.0 - y);
	} else {
		tmp = t_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 = ((1.0d0 - x) / y) - (-1.0d0)
    if (y <= (-1.0d0)) then
        tmp = t_0
    else if (y <= 1.0d0) then
        tmp = (y - x) * ((-1.0d0) - y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. sub-neg N/A

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

      5. div-sub N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. remove-double-neg N/A

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

      4. neg-mul-1 N/A

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

      5. associate-*l/ N/A

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

      6. distribute-lft-out N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. *-lft-identity N/A

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

      10. metadata-eval N/A

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

      11. lft-mult-inverse N/A

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

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

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. associate-*l* N/A

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

      16. associate-/l* N/A

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

      17. *-commutative N/A

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

      18. associate-*r/ N/A

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

      19. distribute-rgt-in N/A

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

      20. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 97.7%

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

4. Final simplification 98.2%

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

Alternative 7: 50.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - 1} \leq 0.4:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- y x) (- y 1.0)) 0.4) (- y) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((y - x) / (y - 1.0)) <= 0.4) {
		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 - x) / (y - 1.0d0)) <= 0.4d0) 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 - x) / (y - 1.0)) <= 0.4) {
		tmp = -y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((y - x) / (y - 1.0)) <= 0.4:
		tmp = -y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(y - x) / Float64(y - 1.0)) <= 0.4)
		tmp = Float64(-y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((y - x) / (y - 1.0)) <= 0.4)
		tmp = -y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(y - x), $MachinePrecision] / N[(y - 1.0), $MachinePrecision]), $MachinePrecision], 0.4], (-y), 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.40000000000000002

   1. Initial program 99.9%

      \[\frac{x - y}{1 - y} \]
   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. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 83.3

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

   5. Applied rewrites 83.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 27.9%

      \[\leadsto -y \]

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   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. lower-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lower-/.f64 N/A

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

      6. neg-sub0 N/A

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

      7. lift--.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. neg-sub0 N/A

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

      12. remove-double-neg N/A

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

      13. lower--.f64 N/A

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

      14. neg-sub0 N/A

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

      15. lift--.f64 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

      18. associate--r+ N/A

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

      19. neg-sub0 N/A

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

      20. remove-double-neg N/A

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

      21. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 63.5%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - 1} \leq 0.4:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.8% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) 1.0 (if (<= y 1.0) (* (- y x) (- -1.0 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 = (y - x) * (-1.0 - 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 <= (-1.0d0)) then
        tmp = 1.0d0
    else if (y <= 1.0d0) then
        tmp = (y - x) * ((-1.0d0) - y)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = 1.0
	elif y <= 1.0:
		tmp = (y - x) * (-1.0 - 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(Float64(y - x) * Float64(-1.0 - y));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

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

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], 1.0, If[LessEqual[y, 1.0], N[(N[(y - x), $MachinePrecision] * N[(-1.0 - y), $MachinePrecision]), $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. 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. lower-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lower-/.f64 N/A

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

      6. neg-sub0 N/A

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

      7. lift--.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. neg-sub0 N/A

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

      12. remove-double-neg N/A

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

      13. lower--.f64 N/A

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

      14. neg-sub0 N/A

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

      15. lift--.f64 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

      18. associate--r+ N/A

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

      19. neg-sub0 N/A

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

      20. remove-double-neg N/A

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

      21. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 76.1%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. remove-double-neg N/A

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

      4. neg-mul-1 N/A

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

      5. associate-*l/ N/A

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

      6. distribute-lft-out N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. *-lft-identity N/A

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

      10. metadata-eval N/A

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

      11. lft-mult-inverse N/A

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

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

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. associate-*l* N/A

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

      16. associate-/l* N/A

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

      17. *-commutative N/A

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

      18. associate-*r/ N/A

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

      19. distribute-rgt-in N/A

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

      20. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 97.7%

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

4. Final simplification 87.0%

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

Alternative 9: 86.6% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) 1.0 (if (<= y 1.0) (fma (- x 1.0) y x) 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((x - 1.0), y, x);
	} 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 = fma(Float64(x - 1.0), y, x);
	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[(x - 1.0), $MachinePrecision] * y + x), $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. 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. lower-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lower-/.f64 N/A

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

      6. neg-sub0 N/A

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

      7. lift--.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. neg-sub0 N/A

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

      12. remove-double-neg N/A

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

      13. lower--.f64 N/A

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

      14. neg-sub0 N/A

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

      15. lift--.f64 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

      18. associate--r+ N/A

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

      19. neg-sub0 N/A

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

      20. remove-double-neg N/A

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

      21. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 76.1%

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

   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. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 97.5

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

   5. Applied rewrites 97.5%

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

Alternative 10: 74.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.25:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -0.25) 1.0 (if (<= y 1.0) (fma y x x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -0.25) {
		tmp = 1.0;
	} else if (y <= 1.0) {
		tmp = fma(y, x, x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -0.25)
		tmp = 1.0;
	elseif (y <= 1.0)
		tmp = fma(y, x, x);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -0.25], 1.0, If[LessEqual[y, 1.0], N[(y * x + x), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -0.25 or 1 < y

   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. lower-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lower-/.f64 N/A

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

      6. neg-sub0 N/A

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

      7. lift--.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. neg-sub0 N/A

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

      12. remove-double-neg N/A

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

      13. lower--.f64 N/A

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

      14. neg-sub0 N/A

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

      15. lift--.f64 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

      18. associate--r+ N/A

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

      19. neg-sub0 N/A

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

      20. remove-double-neg N/A

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

      21. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 75.7%

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

   if -0.25 < 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. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 78.9%

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

Alternative 11: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 12: 38.8% 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. 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. lower-/.f64 N/A

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

   4. frac-2neg N/A

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

   5. lower-/.f64 N/A

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

   6. neg-sub0 N/A

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

   7. lift--.f64 N/A

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

   8. sub-neg N/A

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

   9. +-commutative N/A

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

   10. associate--r+ N/A

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

   11. neg-sub0 N/A

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

   12. remove-double-neg N/A

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

   13. lower--.f64 N/A

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

   14. neg-sub0 N/A

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

   15. lift--.f64 N/A

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

   16. sub-neg N/A

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

   17. +-commutative N/A

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

   18. associate--r+ N/A

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

   19. neg-sub0 N/A

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

   20. remove-double-neg N/A

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

   21. lower--.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in y around inf

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

7. Applied rewrites 39.9%

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

Reproduce

herbie shell --seed 2024308 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, C"
  :precision binary64
  (/ (- x y) (- 1.0 y)))