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

Percentage Accurate: 100.0% → 100.0%

Time: 7.1s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 97.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ t_1 := \frac{x}{1 - y}\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right) - y\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))) (t_1 (/ x (- 1.0 y))))
   (if (<= t_0 -5.0)
     t_1
     (if (<= t_0 2e-23)
       (- (fma y x x) y)
       (if (<= t_0 2.0) (/ y (+ y -1.0)) t_1)))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double t_1 = x / (1.0 - y);
	double tmp;
	if (t_0 <= -5.0) {
		tmp = t_1;
	} else if (t_0 <= 2e-23) {
		tmp = fma(y, x, x) - y;
	} else if (t_0 <= 2.0) {
		tmp = y / (y + -1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(1.0 - y))
	t_1 = Float64(x / Float64(1.0 - y))
	tmp = 0.0
	if (t_0 <= -5.0)
		tmp = t_1;
	elseif (t_0 <= 2e-23)
		tmp = Float64(fma(y, x, x) - y);
	elseif (t_0 <= 2.0)
		tmp = Float64(y / Float64(y + -1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5.0], t$95$1, If[LessEqual[t$95$0, 2e-23], N[(N[(y * x + x), $MachinePrecision] - y), $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)) < -5 or 2 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 96.6

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

   5. Applied rewrites 96.6%

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

   if -5 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 1.99999999999999992e-23

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. associate--r+ N/A

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

      8. *-commutative N/A

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

      9. cancel-sign-sub N/A

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

      10. +-commutative N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 N/A

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

      14. sub-neg N/A

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

      15. remove-double-neg N/A

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

      16. *-rgt-identity N/A

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

      17. distribute-lft-out N/A

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

      18. distribute-rgt-out N/A

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

      19. *-lft-identity N/A

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

      20. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 1.99999999999999992e-23 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. neg-sub0 N/A

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

      5. associate--r- N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 96.7

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

   5. Applied rewrites 96.7%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq -5:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{elif}\;\frac{x - y}{1 - y} \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right) - y\\ \mathbf{elif}\;\frac{x - y}{1 - y} \leq 2:\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ t_1 := \frac{x}{1 - y}\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right) - y\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))) (t_1 (/ x (- 1.0 y))))
   (if (<= t_0 -5.0)
     t_1
     (if (<= t_0 0.05) (- (fma y x x) y) (if (<= t_0 2.0) 1.0 t_1)))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double t_1 = x / (1.0 - y);
	double tmp;
	if (t_0 <= -5.0) {
		tmp = t_1;
	} else if (t_0 <= 0.05) {
		tmp = fma(y, x, x) - y;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(1.0 - y))
	t_1 = Float64(x / Float64(1.0 - y))
	tmp = 0.0
	if (t_0 <= -5.0)
		tmp = t_1;
	elseif (t_0 <= 0.05)
		tmp = Float64(fma(y, x, x) - y);
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5.0], t$95$1, If[LessEqual[t$95$0, 0.05], N[(N[(y * x + x), $MachinePrecision] - y), $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)) < -5 or 2 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 96.6

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

   5. Applied rewrites 96.6%

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

   if -5 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. associate--r+ N/A

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

      8. *-commutative N/A

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

      9. cancel-sign-sub N/A

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

      10. +-commutative N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 N/A

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

      14. sub-neg N/A

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

      15. remove-double-neg N/A

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

      16. *-rgt-identity N/A

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

      17. distribute-lft-out N/A

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

      18. distribute-rgt-out N/A

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

      19. *-lft-identity N/A

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

      20. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 96.2%

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

Alternative 4: 73.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-81}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{elif}\;t\_0 \leq 0.05:\\ \;\;\;\;y \cdot \left(-1 - y\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))))
   (if (<= t_0 -2e-81)
     (fma y x x)
     (if (<= t_0 0.05) (* y (- -1.0 y)) (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 <= -2e-81) {
		tmp = fma(y, x, x);
	} else if (t_0 <= 0.05) {
		tmp = y * (-1.0 - y);
	} 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 <= -2e-81)
		tmp = fma(y, x, x);
	elseif (t_0 <= 0.05)
		tmp = Float64(y * Float64(-1.0 - y));
	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, -2e-81], N[(y * x + x), $MachinePrecision], If[LessEqual[t$95$0, 0.05], N[(y * N[(-1.0 - y), $MachinePrecision]), $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)) < -1.9999999999999999e-81 or 2 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 89.9

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

   5. Applied rewrites 89.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 57.9%

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

   if -1.9999999999999999e-81 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. neg-sub0 N/A

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

      5. associate--r- N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 58.4

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

   5. Applied rewrites 58.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 57.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 96.2%

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

Alternative 5: 73.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-81}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{elif}\;t\_0 \leq 0.05:\\ \;\;\;\;-y\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))))
   (if (<= t_0 -2e-81)
     (fma y x x)
     (if (<= t_0 0.05) (- y) (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 <= -2e-81) {
		tmp = fma(y, x, x);
	} else if (t_0 <= 0.05) {
		tmp = -y;
	} 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 <= -2e-81)
		tmp = fma(y, x, x);
	elseif (t_0 <= 0.05)
		tmp = Float64(-y);
	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, -2e-81], N[(y * x + x), $MachinePrecision], If[LessEqual[t$95$0, 0.05], (-y), 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)) < -1.9999999999999999e-81 or 2 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 89.9

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

   5. Applied rewrites 89.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 57.9%

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

   if -1.9999999999999999e-81 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. neg-sub0 N/A

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

      5. associate--r- N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 58.4

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

   5. Applied rewrites 58.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 57.0%

      \[\leadsto -y \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 96.2%

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

Alternative 6: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.82:\\ \;\;\;\;1 + \frac{-x}{y}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right) - y\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1 - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -0.82)
   (+ 1.0 (/ (- x) y))
   (if (<= y 1.0) (- (fma y x x) y) (+ 1.0 (/ (- 1.0 x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -0.82) {
		tmp = 1.0 + (-x / y);
	} else if (y <= 1.0) {
		tmp = fma(y, x, x) - y;
	} else {
		tmp = 1.0 + ((1.0 - x) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -0.82)
		tmp = Float64(1.0 + Float64(Float64(-x) / y));
	elseif (y <= 1.0)
		tmp = Float64(fma(y, x, x) - y);
	else
		tmp = Float64(1.0 + Float64(Float64(1.0 - x) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -0.82], N[(1.0 + N[((-x) / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(N[(y * x + x), $MachinePrecision] - y), $MachinePrecision], N[(1.0 + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.819999999999999951

   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

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. div-sub N/A

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

      5. lower-+.f64 N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 98.2

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 98.2%

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

   if -0.819999999999999951 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. associate--r+ N/A

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

      8. *-commutative N/A

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

      9. cancel-sign-sub N/A

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

      10. +-commutative N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 N/A

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

      14. sub-neg N/A

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

      15. remove-double-neg N/A

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

      16. *-rgt-identity N/A

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

      17. distribute-lft-out N/A

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

      18. distribute-rgt-out N/A

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

      19. *-lft-identity N/A

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

      20. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

   if 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

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. div-sub N/A

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

      5. lower-+.f64 N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 98.3

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

   5. Applied rewrites 98.3%

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

Alternative 7: 98.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{-x}{y}\\ \mathbf{if}\;y \leq -0.82:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right) - y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (- x) y))))
   (if (<= y -0.82) t_0 (if (<= y 1.0) (- (fma y x x) y) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (-x / y);
	double tmp;
	if (y <= -0.82) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = fma(y, x, x) - y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(Float64(-x) / y))
	tmp = 0.0
	if (y <= -0.82)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = Float64(fma(y, x, x) - y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[((-x) / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.82], t$95$0, If[LessEqual[y, 1.0], N[(N[(y * x + x), $MachinePrecision] - y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.819999999999999951 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

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. div-sub N/A

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

      5. lower-+.f64 N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 97.9%

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

   if -0.819999999999999951 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. associate--r+ N/A

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

      8. *-commutative N/A

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

      9. cancel-sign-sub N/A

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

      10. +-commutative N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 N/A

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

      14. sub-neg N/A

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

      15. remove-double-neg N/A

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

      16. *-rgt-identity N/A

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

      17. distribute-lft-out N/A

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

      18. distribute-rgt-out N/A

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

      19. *-lft-identity N/A

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

      20. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

Alternative 8: 85.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+78}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.7:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right) - y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6e+78)
   1.0
   (if (<= y -1.7) (/ x (- y)) (if (<= y 1.0) (- (fma y x x) y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6e+78) {
		tmp = 1.0;
	} else if (y <= -1.7) {
		tmp = x / -y;
	} else if (y <= 1.0) {
		tmp = fma(y, x, x) - y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6e+78)
		tmp = 1.0;
	elseif (y <= -1.7)
		tmp = Float64(x / Float64(-y));
	elseif (y <= 1.0)
		tmp = Float64(fma(y, x, x) - y);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6e+78], 1.0, If[LessEqual[y, -1.7], N[(x / (-y)), $MachinePrecision], If[LessEqual[y, 1.0], N[(N[(y * x + x), $MachinePrecision] - y), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.99999999999999964e78 or 1 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 73.8%

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

   if -5.99999999999999964e78 < y < -1.69999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 69.6

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

   5. Applied rewrites 69.6%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 64.2%

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

   if -1.69999999999999996 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. associate--r+ N/A

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

      8. *-commutative N/A

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

      9. cancel-sign-sub N/A

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

      10. +-commutative N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 N/A

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

      14. sub-neg N/A

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

      15. remove-double-neg N/A

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

      16. *-rgt-identity N/A

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

      17. distribute-lft-out N/A

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

      18. distribute-rgt-out N/A

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

      19. *-lft-identity N/A

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

      20. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

Alternative 9: 50.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. neg-sub0 N/A

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

      5. associate--r- N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 32.6

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

   5. Applied rewrites 32.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 32.4%

      \[\leadsto -y \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 67.2%

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

Alternative 10: 85.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 67.8%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. associate--r+ N/A

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

      8. *-commutative N/A

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

      9. cancel-sign-sub N/A

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

      10. +-commutative N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 N/A

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

      14. sub-neg N/A

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

      15. remove-double-neg N/A

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

      16. *-rgt-identity N/A

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

      17. distribute-lft-out N/A

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

      18. distribute-rgt-out N/A

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

      19. *-lft-identity N/A

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

      20. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

Alternative 11: 38.5% accurate, 18.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around inf

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

5. Applied rewrites 35.4%

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

Reproduce

herbie shell --seed 2024226 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, C"
  :precision binary64
  (/ (- x y) (- 1.0 y)))