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

Percentage Accurate: 100.0% → 100.0%

Time: 6.4s

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 99.9%

   \[\frac{x - y}{1 - y} \]
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: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ \mathbf{if}\;t\_0 \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(-1, \mathsf{fma}\left(y, y, y\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y, y, y\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))))
   (if (<= t_0 0.02)
     (fma -1.0 (fma y y y) x)
     (if (<= t_0 2.0) 1.0 (fma (fma y y y) x x)))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double tmp;
	if (t_0 <= 0.02) {
		tmp = fma(-1.0, fma(y, y, y), x);
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = fma(fma(y, y, 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 <= 0.02)
		tmp = fma(-1.0, fma(y, y, y), x);
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = fma(fma(y, y, 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, 0.02], N[(-1.0 * N[(y * y + y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, N[(N[(y * y + y), $MachinePrecision] * x + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.0200000000000000004

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-lft-identity N/A

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

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

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

      6. *-lft-identity N/A

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

      7. associate-*r* N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 86.3

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

   5. Applied rewrites 86.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 86.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 85.9%

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

   if 0.0200000000000000004 < (/.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. 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. Taylor expanded in y around 0

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

   7. Applied rewrites 2.0%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 97.1%

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

   if 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 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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-lft-identity N/A

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

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

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

      6. *-lft-identity N/A

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

      7. associate-*r* N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 73.8

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

   5. Applied rewrites 73.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 73.8%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(-1, \mathsf{fma}\left(y, y, y\right), x\right)\\ \mathbf{elif}\;\frac{x - y}{1 - y} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y, y, y\right), x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ \mathbf{if}\;t\_0 \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(-1 - y, y, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))))
   (if (<= t_0 0.02) (fma (- -1.0 y) y x) (if (<= t_0 2.0) 1.0 (fma y x x)))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double tmp;
	if (t_0 <= 0.02) {
		tmp = fma((-1.0 - y), y, x);
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = fma(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 <= 0.02)
		tmp = fma(Float64(-1.0 - y), y, x);
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = fma(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, 0.02], N[(N[(-1.0 - y), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, N[(y * x + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.0200000000000000004

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-lft-identity N/A

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

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

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

      6. *-lft-identity N/A

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

      7. associate-*r* N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 86.3

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

   5. Applied rewrites 86.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 85.9%

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

   if 0.0200000000000000004 < (/.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. 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. Taylor expanded in y around 0

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

   7. Applied rewrites 2.0%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 97.1%

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

   if 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 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. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-+.f64 71.6

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

   5. Applied rewrites 71.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 71.6%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(-1 - y, y, x\right)\\ \mathbf{elif}\;\frac{x - y}{1 - y} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ \mathbf{if}\;t\_0 \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(-1, y, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))))
   (if (<= t_0 0.02) (fma -1.0 y x) (if (<= t_0 2.0) 1.0 (fma y x x)))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double tmp;
	if (t_0 <= 0.02) {
		tmp = fma(-1.0, y, x);
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = fma(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 <= 0.02)
		tmp = fma(-1.0, y, x);
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = fma(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, 0.02], N[(-1.0 * y + x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, N[(y * x + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.0200000000000000004

   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. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-+.f64 85.4

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

   5. Applied rewrites 85.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 85.4%

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

   if 0.0200000000000000004 < (/.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. 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. Taylor expanded in y around 0

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

   7. Applied rewrites 2.0%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 97.1%

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

   if 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 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. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-+.f64 71.6

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

   5. Applied rewrites 71.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 71.6%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(-1, y, x\right)\\ \mathbf{elif}\;\frac{x - y}{1 - y} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.0)))
   (- 1.0 (/ (- x 1.0) y))
   (fma (+ -1.0 x) (fma y y y) x)))

C

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

Julia

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

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(1.0 - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 + x), $MachinePrecision] * N[(y * y + y), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   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. associate-+r+ N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 98.9

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

   5. Applied rewrites 98.9%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

      3. lower-fma.f64 N/A

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

      4. *-lft-identity N/A

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

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

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

      6. *-lft-identity N/A

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

      7. associate-*r* N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.9%

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

4. Final simplification 98.9%

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

Alternative 6: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.88 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1 + x, \mathsf{fma}\left(y, y, y\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -0.88) (not (<= y 1.0)))
   (- 1.0 (/ x y))
   (fma (+ -1.0 x) (fma y y y) x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -0.88) || !(y <= 1.0)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = fma((-1.0 + x), fma(y, y, y), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -0.88) || !(y <= 1.0))
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = fma(Float64(-1.0 + x), fma(y, y, y), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -0.88], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 + x), $MachinePrecision] * N[(y * y + y), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.880000000000000004 or 1 < y

   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. associate-+r+ N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 97.9%

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

   if -0.880000000000000004 < 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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-lft-identity N/A

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

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

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

      6. *-lft-identity N/A

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

      7. associate-*r* N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.88 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1 + x, \mathsf{fma}\left(y, y, y\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.78 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1 + x, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -0.78) (not (<= y 1.0))) (- 1.0 (/ x y)) (fma (+ -1.0 x) y x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -0.78) || !(y <= 1.0)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = fma((-1.0 + x), y, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -0.78) || !(y <= 1.0))
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = fma(Float64(-1.0 + x), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -0.78], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 + x), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.78000000000000003 or 1 < y

   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. associate-+r+ N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 97.9%

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

   if -0.78000000000000003 < 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. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-+.f64 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.78 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1 + x, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 10^{-11}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 1e-11) (- y) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 1e-11) {
		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)) <= 1d-11) 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)) <= 1e-11) {
		tmp = -y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 1e-11:
		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)) <= 1e-11)
		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)) <= 1e-11)
		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], 1e-11], (-y), 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 9.99999999999999939e-12

   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. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-+.f64 85.1

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 26.4%

      \[\leadsto -y \]

   if 9.99999999999999939e-12 < (/.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. 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. Taylor expanded in y around 0

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

   7. Applied rewrites 25.5%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 67.6%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 10^{-11}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.6% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) 1.0 (if (<= y 1.0) (fma (+ -1.0 x) 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((-1.0 + x), 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(-1.0 + x), 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[(-1.0 + x), $MachinePrecision] * y + x), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 99.9%

      \[\frac{x - y}{1 - y} \]
   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. Taylor expanded in y around 0

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

   7. Applied rewrites 1.7%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 76.2%

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

   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. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-+.f64 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 87.7%

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

Alternative 10: 74.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.54 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -0.54) (not (<= y 1.0))) 1.0 (fma y x x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -0.54) || !(y <= 1.0)) {
		tmp = 1.0;
	} else {
		tmp = fma(y, x, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -0.54) || !(y <= 1.0))
		tmp = 1.0;
	else
		tmp = fma(y, x, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -0.54], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], 1.0, N[(y * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.54000000000000004 or 1 < y

   1. Initial program 99.9%

      \[\frac{x - y}{1 - y} \]
   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. Taylor expanded in y around 0

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

   7. Applied rewrites 1.7%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 76.2%

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

   if -0.54000000000000004 < 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. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-+.f64 98.0

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 76.8%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.54 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 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 99.9%

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

Alternative 12: 39.7% 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 99.9%

   \[\frac{x - y}{1 - y} \]
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. Taylor expanded in y around 0

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

7. Applied rewrites 53.1%

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

8. Taylor expanded in y around inf

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

10. Applied rewrites 38.0%

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

11. Final simplification 38.0%

    \[\leadsto 1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024339 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, C"
  :precision binary64
  (/ (- x y) (- 1.0 y)))