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

Percentage Accurate: 100.0% → 100.0%

Time: 8.7s

Alternatives: 14

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: 85.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ \mathbf{if}\;t\_0 \leq 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-1, \mathsf{fma}\left(y, y, y\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_0 \leq 10^{+86}:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, x, x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))))
   (if (<= t_0 1e-5)
     (fma -1.0 (fma y y y) x)
     (if (<= t_0 2.0)
       1.0
       (if (<= t_0 1e+86) (/ x (- y)) (fma y (fma y x x) x))))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double tmp;
	if (t_0 <= 1e-5) {
		tmp = fma(-1.0, fma(y, y, y), x);
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else if (t_0 <= 1e+86) {
		tmp = x / -y;
	} else {
		tmp = fma(y, fma(y, x, 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 <= 1e-5)
		tmp = fma(-1.0, fma(y, y, y), x);
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	elseif (t_0 <= 1e+86)
		tmp = Float64(x / Float64(-y));
	else
		tmp = fma(y, fma(y, x, 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, 1e-5], N[(-1.0 * N[(y * y + y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, If[LessEqual[t$95$0, 1e+86], N[(x / (-y)), $MachinePrecision], N[(y * N[(y * x + x), $MachinePrecision] + x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 1.00000000000000008e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. mul-1-neg N/A

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

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

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. remove-double-neg N/A

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

      14. lower-+.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-fma.f64 84.7

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 84.6%

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

   if 1.00000000000000008e-5 < (/.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 97.3%

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

   if 2 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 1e86

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 97.6

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 69.4

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

   8. Applied rewrites 69.4%

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

   if 1e86 < (/.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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. lower-fma.f64 79.4

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

   8. Applied rewrites 79.4%

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

Alternative 3: 86.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ \mathbf{if}\;t\_0 \leq 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-1, \mathsf{fma}\left(y, y, y\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 2000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, x, x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))))
   (if (<= t_0 1e-5)
     (fma -1.0 (fma y y y) x)
     (if (<= t_0 2000000000.0) 1.0 (fma y (fma y x x) x)))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double tmp;
	if (t_0 <= 1e-5) {
		tmp = fma(-1.0, fma(y, y, y), x);
	} else if (t_0 <= 2000000000.0) {
		tmp = 1.0;
	} else {
		tmp = fma(y, fma(y, x, 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 <= 1e-5)
		tmp = fma(-1.0, fma(y, y, y), x);
	elseif (t_0 <= 2000000000.0)
		tmp = 1.0;
	else
		tmp = fma(y, fma(y, x, 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, 1e-5], N[(-1.0 * N[(y * y + y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$0, 2000000000.0], 1.0, N[(y * N[(y * x + x), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 1.00000000000000008e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. mul-1-neg N/A

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

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

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. remove-double-neg N/A

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

      14. lower-+.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-fma.f64 84.7

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 84.6%

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

   if 1.00000000000000008e-5 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 2e9

   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 96.4%

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

   if 2e9 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. lower-fma.f64 61.0

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

   8. Applied rewrites 61.0%

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

Alternative 4: 86.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ \mathbf{if}\;t\_0 \leq 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-1, \mathsf{fma}\left(y, y, y\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 2000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))))
   (if (<= t_0 1e-5)
     (fma -1.0 (fma y y y) x)
     (if (<= t_0 2000000000.0) 1.0 (* x (+ y 1.0))))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double tmp;
	if (t_0 <= 1e-5) {
		tmp = fma(-1.0, fma(y, y, y), x);
	} else if (t_0 <= 2000000000.0) {
		tmp = 1.0;
	} else {
		tmp = x * (y + 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(1.0 - y))
	tmp = 0.0
	if (t_0 <= 1e-5)
		tmp = fma(-1.0, fma(y, y, y), x);
	elseif (t_0 <= 2000000000.0)
		tmp = 1.0;
	else
		tmp = Float64(x * Float64(y + 1.0));
	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, 1e-5], N[(-1.0 * N[(y * y + y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$0, 2000000000.0], 1.0, N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 1.00000000000000008e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. mul-1-neg N/A

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

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

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. remove-double-neg N/A

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

      14. lower-+.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-fma.f64 84.7

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 84.6%

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

   if 1.00000000000000008e-5 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 2e9

   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 96.4%

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

   if 2e9 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 60.1

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

   8. Applied rewrites 60.1%

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

      1. distribute-lft1-in N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-+.f64 N/A

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

      5. metadata-eval 60.2

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

   10. Applied rewrites 60.2%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-1, \mathsf{fma}\left(y, y, y\right), x\right)\\ \mathbf{elif}\;\frac{x - y}{1 - y} \leq 2000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ \mathbf{if}\;t\_0 \leq 10^{-5}:\\ \;\;\;\;x - y\\ \mathbf{elif}\;t\_0 \leq 2000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))))
   (if (<= t_0 1e-5) (- x y) (if (<= t_0 2000000000.0) 1.0 (* x (+ y 1.0))))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double tmp;
	if (t_0 <= 1e-5) {
		tmp = x - y;
	} else if (t_0 <= 2000000000.0) {
		tmp = 1.0;
	} else {
		tmp = x * (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x - y) / (1.0d0 - y)
    if (t_0 <= 1d-5) then
        tmp = x - y
    else if (t_0 <= 2000000000.0d0) then
        tmp = 1.0d0
    else
        tmp = x * (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double tmp;
	if (t_0 <= 1e-5) {
		tmp = x - y;
	} else if (t_0 <= 2000000000.0) {
		tmp = 1.0;
	} else {
		tmp = x * (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / (1.0 - y)
	tmp = 0
	if t_0 <= 1e-5:
		tmp = x - y
	elif t_0 <= 2000000000.0:
		tmp = 1.0
	else:
		tmp = x * (y + 1.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(1.0 - y))
	tmp = 0.0
	if (t_0 <= 1e-5)
		tmp = Float64(x - y);
	elseif (t_0 <= 2000000000.0)
		tmp = 1.0;
	else
		tmp = Float64(x * Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) / (1.0 - y);
	tmp = 0.0;
	if (t_0 <= 1e-5)
		tmp = x - y;
	elseif (t_0 <= 2000000000.0)
		tmp = 1.0;
	else
		tmp = x * (y + 1.0);
	end
	tmp_2 = 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, 1e-5], N[(x - y), $MachinePrecision], If[LessEqual[t$95$0, 2000000000.0], 1.0, N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 1.00000000000000008e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. mul-1-neg N/A

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

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

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. remove-double-neg N/A

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

      14. lower-+.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-fma.f64 84.7

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 84.6%

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

   9. Taylor expanded in y around 0

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

       1. mul-1-neg N/A

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

       2. sub-neg N/A

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

       3. lower--.f64 83.4

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

   11. Applied rewrites 83.4%

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

   if 1.00000000000000008e-5 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 2e9

   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 96.4%

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

   if 2e9 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 60.1

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

   8. Applied rewrites 60.1%

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

      1. distribute-lft1-in N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-+.f64 N/A

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

      5. metadata-eval 60.2

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

   10. Applied rewrites 60.2%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 10^{-5}:\\ \;\;\;\;x - y\\ \mathbf{elif}\;\frac{x - y}{1 - y} \leq 2000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ \mathbf{if}\;t\_0 \leq 10^{-5}:\\ \;\;\;\;x - y\\ \mathbf{elif}\;t\_0 \leq 2000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))))
   (if (<= t_0 1e-5) (- x y) (if (<= t_0 2000000000.0) 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 <= 1e-5) {
		tmp = x - y;
	} else if (t_0 <= 2000000000.0) {
		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 <= 1e-5)
		tmp = Float64(x - y);
	elseif (t_0 <= 2000000000.0)
		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, 1e-5], N[(x - y), $MachinePrecision], If[LessEqual[t$95$0, 2000000000.0], 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)) < 1.00000000000000008e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. mul-1-neg N/A

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

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

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. remove-double-neg N/A

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

      14. lower-+.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-fma.f64 84.7

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 84.6%

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

   9. Taylor expanded in y around 0

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

       1. mul-1-neg N/A

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

       2. sub-neg N/A

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

       3. lower--.f64 83.4

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

   11. Applied rewrites 83.4%

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

   if 1.00000000000000008e-5 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 2e9

   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 96.4%

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

   if 2e9 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 60.1

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

   8. Applied rewrites 60.1%

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

Alternative 7: 98.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      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 98.6

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

   5. Applied rewrites 98.6%

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

   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. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. mul-1-neg N/A

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

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

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. remove-double-neg N/A

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

      14. lower-+.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-fma.f64 99.4

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

   5. Applied rewrites 99.4%

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

4. Final simplification 99.0%

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

Alternative 8: 98.7% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -0.88)
   (/ (- y x) y)
   (if (<= y 1.0) (fma (+ x -1.0) (fma y y y) x) (- 1.0 (/ x y)))))

C

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

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[y, -0.88], N[(N[(y - x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 1.0], N[(N[(x + -1.0), $MachinePrecision] * N[(y * y + y), $MachinePrecision] + x), $MachinePrecision], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.880000000000000004

   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 98.5

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.4

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

   8. Applied rewrites 98.4%

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

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. distribute-frac-neg N/A

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

      7. neg-sub0 N/A

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

      8. associate-+l- N/A

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

      9. *-inverses N/A

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

      10. div-sub N/A

          \[\leadsto 0 - \color{blue}{\frac{x - y}{y}} \]

      11. lift--.f64 N/A

          \[\leadsto 0 - \frac{\color{blue}{x - y}}{y} \]

      12. neg-sub0 N/A

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

      13. distribute-frac-neg N/A

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

      14. lower-/.f64 N/A

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

      15. neg-sub0 N/A

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

      18. lift-neg.f64 N/A

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

      19. +-commutative N/A

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

      20. associate--r+ N/A

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

      21. neg-sub0 N/A

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

      22. lift-neg.f64 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 98.4

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

   10. Applied rewrites 98.4%

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

   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. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. mul-1-neg N/A

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

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

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. remove-double-neg N/A

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

      14. lower-+.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-fma.f64 99.4

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

   5. Applied rewrites 99.4%

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

   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 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.2

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

   8. Applied rewrites 98.2%

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

      1. distribute-frac-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 98.2

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

   10. Applied rewrites 98.2%

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

4. Final simplification 98.8%

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

Alternative 9: 98.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.8:\\ \;\;\;\;\frac{y - x}{y}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right) - y\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -0.8)
   (/ (- y x) y)
   (if (<= y 1.0) (- (fma y x x) y) (- 1.0 (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -0.8) {
		tmp = (y - x) / y;
	} else if (y <= 1.0) {
		tmp = fma(y, x, x) - y;
	} else {
		tmp = 1.0 - (x / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -0.8)
		tmp = Float64(Float64(y - x) / y);
	elseif (y <= 1.0)
		tmp = Float64(fma(y, x, x) - y);
	else
		tmp = Float64(1.0 - Float64(x / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -0.8], N[(N[(y - x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 1.0], N[(N[(y * x + x), $MachinePrecision] - y), $MachinePrecision], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.80000000000000004

   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 98.5

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.4

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

   8. Applied rewrites 98.4%

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

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. distribute-frac-neg N/A

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

      7. neg-sub0 N/A

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

      8. associate-+l- N/A

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

      9. *-inverses N/A

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

      10. div-sub N/A

          \[\leadsto 0 - \color{blue}{\frac{x - y}{y}} \]

      11. lift--.f64 N/A

          \[\leadsto 0 - \frac{\color{blue}{x - y}}{y} \]

      12. neg-sub0 N/A

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

      13. distribute-frac-neg N/A

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

      14. lower-/.f64 N/A

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

      15. neg-sub0 N/A

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

      18. lift-neg.f64 N/A

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

      19. +-commutative N/A

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

      20. associate--r+ N/A

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

      21. neg-sub0 N/A

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

      22. lift-neg.f64 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 98.4

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

   10. Applied rewrites 98.4%

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

   if -0.80000000000000004 < 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 98.3

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

   5. Applied rewrites 98.3%

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

   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 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.2

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

   8. Applied rewrites 98.2%

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

      1. distribute-frac-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 98.2

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

   10. Applied rewrites 98.2%

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

Alternative 10: 98.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ \mathbf{if}\;y \leq -0.8:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right) - y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))))
   (if (<= y -0.8) 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 <= -0.8) {
		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(x / y))
	tmp = 0.0
	if (y <= -0.8)
		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, -0.8], 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 < -0.80000000000000004 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 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.3

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

   8. Applied rewrites 98.3%

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

      1. distribute-frac-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 98.3

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

   10. Applied rewrites 98.3%

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

   if -0.80000000000000004 < 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 98.3

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

   5. Applied rewrites 98.3%

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

Alternative 11: 50.6% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 1e-5) (- y) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 1e-5) {
		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-5) 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-5) {
		tmp = -y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 1e-5:
		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-5)
		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-5)
		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-5], (-y), 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 1.00000000000000008e-5

   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 30.5

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

   5. Applied rewrites 30.5%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 29.3

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

   8. Applied rewrites 29.3%

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

   if 1.00000000000000008e-5 < (/.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 69.7%

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

Alternative 12: 86.6% accurate, 0.8× speedup

Math

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

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 72.1%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. 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 98.3

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

   5. Applied rewrites 98.3%

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

Alternative 13: 86.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4000000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4000000000000.0) 1.0 (if (<= y 1.0) (- x y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4000000000000.0) {
		tmp = 1.0;
	} else if (y <= 1.0) {
		tmp = x - y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-4000000000000.0d0)) then
        tmp = 1.0d0
    else if (y <= 1.0d0) then
        tmp = x - y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4000000000000.0) {
		tmp = 1.0;
	} else if (y <= 1.0) {
		tmp = x - y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4000000000000.0:
		tmp = 1.0
	elif y <= 1.0:
		tmp = x - y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4000000000000.0)
		tmp = 1.0;
	elseif (y <= 1.0)
		tmp = Float64(x - y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4000000000000.0)
		tmp = 1.0;
	elseif (y <= 1.0)
		tmp = x - y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4000000000000.0], 1.0, If[LessEqual[y, 1.0], N[(x - y), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -4e12 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 73.1%

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

   if -4e12 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. mul-1-neg N/A

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

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

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. remove-double-neg N/A

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

      14. lower-+.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-fma.f64 97.9

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 96.5%

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

   9. Taylor expanded in y around 0

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

       1. mul-1-neg N/A

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

       2. sub-neg N/A

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

       3. lower--.f64 95.6

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

   11. Applied rewrites 95.6%

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

Alternative 14: 39.3% 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 38.3%

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

Reproduce

herbie shell --seed 2024216 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, C"
  :precision binary64
  (/ (- x y) (- 1.0 y)))