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

Percentage Accurate: 100.0% → 100.0%

Time: 6.2s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 98.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ t_1 := \frac{x}{1 - y}\\ \mathbf{if}\;t\_0 \leq -40000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \mathsf{fma}\left(y, y, y\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{y}{y - 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))) (t_1 (/ x (- 1.0 y))))
   (if (<= t_0 -40000000.0)
     t_1
     (if (<= t_0 5e-5)
       (fma (- x 1.0) (fma y y y) x)
       (if (<= t_0 2.0) (/ y (- y 1.0)) t_1)))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double t_1 = x / (1.0 - y);
	double tmp;
	if (t_0 <= -40000000.0) {
		tmp = t_1;
	} else if (t_0 <= 5e-5) {
		tmp = fma((x - 1.0), fma(y, y, y), x);
	} else if (t_0 <= 2.0) {
		tmp = y / (y - 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(1.0 - y))
	t_1 = Float64(x / Float64(1.0 - y))
	tmp = 0.0
	if (t_0 <= -40000000.0)
		tmp = t_1;
	elseif (t_0 <= 5e-5)
		tmp = fma(Float64(x - 1.0), fma(y, y, y), x);
	elseif (t_0 <= 2.0)
		tmp = Float64(y / Float64(y - 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -40000000.0], t$95$1, If[LessEqual[t$95$0, 5e-5], N[(N[(x - 1.0), $MachinePrecision] * N[(y * y + y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(y / N[(y - 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < -4e7 or 2 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if -4e7 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 5.00000000000000024e-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. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-fma.f64 97.3

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

   5. Applied rewrites 97.3%

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

   if 5.00000000000000024e-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 x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. neg-sub0 N/A

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

      5. associate--r- N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 98.6

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

   5. Applied rewrites 98.6%

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

Alternative 3: 98.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ t_1 := \frac{x}{1 - y}\\ \mathbf{if}\;t\_0 \leq -40000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-1, \mathsf{fma}\left(y, y, y\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{y}{y - 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))) (t_1 (/ x (- 1.0 y))))
   (if (<= t_0 -40000000.0)
     t_1
     (if (<= t_0 5e-5)
       (fma -1.0 (fma y y y) x)
       (if (<= t_0 2.0) (/ y (- y 1.0)) t_1)))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double t_1 = x / (1.0 - y);
	double tmp;
	if (t_0 <= -40000000.0) {
		tmp = t_1;
	} else if (t_0 <= 5e-5) {
		tmp = fma(-1.0, fma(y, y, y), x);
	} else if (t_0 <= 2.0) {
		tmp = y / (y - 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(1.0 - y))
	t_1 = Float64(x / Float64(1.0 - y))
	tmp = 0.0
	if (t_0 <= -40000000.0)
		tmp = t_1;
	elseif (t_0 <= 5e-5)
		tmp = fma(-1.0, fma(y, y, y), x);
	elseif (t_0 <= 2.0)
		tmp = Float64(y / Float64(y - 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -40000000.0], t$95$1, If[LessEqual[t$95$0, 5e-5], N[(-1.0 * N[(y * y + y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(y / N[(y - 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < -4e7 or 2 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if -4e7 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 5.00000000000000024e-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. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-fma.f64 97.3

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.3%

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

   if 5.00000000000000024e-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 x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. neg-sub0 N/A

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

      5. associate--r- N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 98.6

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

   5. Applied rewrites 98.6%

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

Alternative 4: 62.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ \mathbf{if}\;t\_0 \leq -40000000 \lor \neg \left(t\_0 \leq 5 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \mathsf{fma}\left(y, y, y\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))))
   (if (or (<= t_0 -40000000.0) (not (<= t_0 5e-16)))
     (/ x (- 1.0 y))
     (fma -1.0 (fma y y y) x))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double tmp;
	if ((t_0 <= -40000000.0) || !(t_0 <= 5e-16)) {
		tmp = x / (1.0 - y);
	} else {
		tmp = fma(-1.0, fma(y, y, y), x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(1.0 - y))
	tmp = 0.0
	if ((t_0 <= -40000000.0) || !(t_0 <= 5e-16))
		tmp = Float64(x / Float64(1.0 - y));
	else
		tmp = fma(-1.0, fma(y, y, y), 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[Or[LessEqual[t$95$0, -40000000.0], N[Not[LessEqual[t$95$0, 5e-16]], $MachinePrecision]], N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(y * y + y), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < -4e7 or 5.0000000000000004e-16 < (/.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 48.6

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

   5. Applied rewrites 48.6%

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

   if -4e7 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 5.0000000000000004e-16

   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. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-fma.f64 97.2

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.2%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq -40000000 \lor \neg \left(\frac{x - y}{1 - y} \leq 5 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \mathsf{fma}\left(y, y, y\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. sub-neg N/A

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

      5. div-sub N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 98.4

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

   5. Applied rewrites 98.4%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-fma.f64 98.5

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

   5. Applied rewrites 98.5%

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

4. Final simplification 98.4%

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

Alternative 6: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.86 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\frac{-x}{y} - -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \mathsf{fma}\left(y, y, y\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -0.86) (not (<= y 1.0)))
   (- (/ (- x) y) -1.0)
   (fma (- x 1.0) (fma y y y) x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -0.86) || !(y <= 1.0)) {
		tmp = (-x / y) - -1.0;
	} else {
		tmp = fma((x - 1.0), fma(y, y, y), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -0.86) || !(y <= 1.0))
		tmp = Float64(Float64(Float64(-x) / y) - -1.0);
	else
		tmp = fma(Float64(x - 1.0), fma(y, y, y), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -0.86], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(N[((-x) / y), $MachinePrecision] - -1.0), $MachinePrecision], N[(N[(x - 1.0), $MachinePrecision] * N[(y * y + y), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.859999999999999987 or 1 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. sub-neg N/A

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

      5. div-sub N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 97.7%

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

   if -0.859999999999999987 < 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. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-fma.f64 98.5

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

   5. Applied rewrites 98.5%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.86 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\frac{-x}{y} - -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \mathsf{fma}\left(y, y, y\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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 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 28.3

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

   5. Applied rewrites 28.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 26.8%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-fma.f64 98.5

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.4%

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

4. Final simplification 60.1%

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

Alternative 8: 43.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-113} \lor \neg \left(x \leq 1.3 \cdot 10^{-178}\right):\\ \;\;\;\;\mathsf{fma}\left(x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.02e-113) (not (<= x 1.3e-178))) (fma x y x) (- y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.02e-113) || !(x <= 1.3e-178)) {
		tmp = fma(x, y, x);
	} else {
		tmp = -y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.02e-113) || !(x <= 1.3e-178))
		tmp = fma(x, y, x);
	else
		tmp = Float64(-y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.02e-113], N[Not[LessEqual[x, 1.3e-178]], $MachinePrecision]], N[(x * y + x), $MachinePrecision], (-y)]

Derivation

1. Split input into 2 regimes

2. if x < -1.02e-113 or 1.29999999999999999e-178 < x

   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 62.0

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

   5. Applied rewrites 62.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 42.8%

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

   if -1.02e-113 < x < 1.29999999999999999e-178

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 50.1

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

   5. Applied rewrites 50.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 38.4%

      \[\leadsto -y \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-113} \lor \neg \left(x \leq 1.3 \cdot 10^{-178}\right):\\ \;\;\;\;\mathsf{fma}\left(x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-1, y, x\right) \end{array} \]

FPCore

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

C

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

Julia

function code(x, y)
	return fma(-1.0, y, x)
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

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

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. +-commutative N/A

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

   8. distribute-neg-in N/A

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

   9. remove-double-neg N/A

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

   10. sub-neg N/A

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

   11. lower--.f64 46.6

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

5. Applied rewrites 46.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 47.1%

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

Alternative 10: 14.6% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ -y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return -y

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := (-y)

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

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

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. +-commutative N/A

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

   8. distribute-neg-in N/A

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

   9. remove-double-neg N/A

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

   10. sub-neg N/A

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

   11. lower--.f64 46.6

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

5. Applied rewrites 46.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 14.8%

   \[\leadsto -y \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024324 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, C"
  :precision binary64
  (/ (- x y) (- 1.0 y)))