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

Percentage Accurate: 87.6% → 100.0%

Time: 5.2s

Alternatives: 6

Speedup: 1.1×

Specification

Math

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

FPCore

(FPCore (x y) :precision binary64 (/ (* x y) (+ y 1.0)))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return (x * y) / (y + 1.0)

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 87.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (/ (* x y) (+ y 1.0)))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return (x * y) / (y + 1.0)

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (* (/ y (+ y 1.0)) x))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return (y / (y + 1.0)) * x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.9%

   \[\frac{x \cdot y}{y + 1} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 99.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 71.6%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 99.0

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

   5. Applied rewrites 99.0%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-out N/A

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

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

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

      6. distribute-rgt1-in N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-in N/A

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

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

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

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

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

      15. distribute-neg-in N/A

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

      16. metadata-eval N/A

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

      17. distribute-lft1-in N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-neg.f64 98.7

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

   5. Applied rewrites 98.7%

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

   7. Applied rewrites 98.7%

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

Alternative 3: 98.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x \cdot 1\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y, -x, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) (* x 1.0) (if (<= y 0.75) (* y (fma y (- x) x)) (* x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x * 1.0;
	} else if (y <= 0.75) {
		tmp = y * fma(y, -x, x);
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(x * 1.0);
	elseif (y <= 0.75)
		tmp = Float64(y * fma(y, Float64(-x), x));
	else
		tmp = Float64(x * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], N[(x * 1.0), $MachinePrecision], If[LessEqual[y, 0.75], N[(y * N[(y * (-x) + x), $MachinePrecision]), $MachinePrecision], N[(x * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.75 < y

   1. Initial program 71.6%

      \[\frac{x \cdot y}{y + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 97.1%

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

   if -1 < y < 0.75

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-out N/A

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

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

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

      6. distribute-rgt1-in N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-in N/A

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

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

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

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

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

      15. distribute-neg-in N/A

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

      16. metadata-eval N/A

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

      17. distribute-lft1-in N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-neg.f64 98.7

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

   5. Applied rewrites 98.7%

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

   7. Applied rewrites 98.7%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x \cdot 1\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y, -x, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x \cdot 1\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, -y, y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) (* x 1.0) (if (<= y 0.75) (* x (fma y (- y) y)) (* x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x * 1.0;
	} else if (y <= 0.75) {
		tmp = x * fma(y, -y, y);
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(x * 1.0);
	elseif (y <= 0.75)
		tmp = Float64(x * fma(y, Float64(-y), y));
	else
		tmp = Float64(x * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], N[(x * 1.0), $MachinePrecision], If[LessEqual[y, 0.75], N[(x * N[(y * (-y) + y), $MachinePrecision]), $MachinePrecision], N[(x * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.75 < y

   1. Initial program 71.6%

      \[\frac{x \cdot y}{y + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 97.1%

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

   if -1 < y < 0.75

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-out N/A

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

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

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

      6. distribute-rgt1-in N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-in N/A

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

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

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

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

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

      15. distribute-neg-in N/A

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

      16. metadata-eval N/A

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

      17. distribute-lft1-in N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-neg.f64 98.7

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

   5. Applied rewrites 98.7%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x \cdot 1\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, -y, y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 71.6%

      \[\frac{x \cdot y}{y + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 97.1%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 97.5%

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

Alternative 6: 49.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ y \cdot x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return y * x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.9%

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

3. Taylor expanded in y around 0

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

   1. lower-*.f64 47.7

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

5. Applied rewrites 47.7%

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

6. Final simplification 47.7%

   \[\leadsto y \cdot x \]
7. Add Preprocessing

Developer Target 1: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot y} - \left(\frac{x}{y} - x\right)\\ \mathbf{if}\;y < -3693.8482788297247:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 6799310503.41891:\\ \;\;\;\;\frac{x \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (/ x (* y y)) (- (/ x y) x))))
   (if (< y -3693.8482788297247)
     t_0
     (if (< y 6799310503.41891) (/ (* x y) (+ y 1.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = (x / (y * y)) - ((x / y) - x);
	double tmp;
	if (y < -3693.8482788297247) {
		tmp = t_0;
	} else if (y < 6799310503.41891) {
		tmp = (x * y) / (y + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x / (y * y)) - ((x / y) - x)
    if (y < (-3693.8482788297247d0)) then
        tmp = t_0
    else if (y < 6799310503.41891d0) then
        tmp = (x * y) / (y + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x / (y * y)) - ((x / y) - x);
	double tmp;
	if (y < -3693.8482788297247) {
		tmp = t_0;
	} else if (y < 6799310503.41891) {
		tmp = (x * y) / (y + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x / (y * y)) - ((x / y) - x)
	tmp = 0
	if y < -3693.8482788297247:
		tmp = t_0
	elif y < 6799310503.41891:
		tmp = (x * y) / (y + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x / Float64(y * y)) - Float64(Float64(x / y) - x))
	tmp = 0.0
	if (y < -3693.8482788297247)
		tmp = t_0;
	elseif (y < 6799310503.41891)
		tmp = Float64(Float64(x * y) / Float64(y + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x / (y * y)) - ((x / y) - x);
	tmp = 0.0;
	if (y < -3693.8482788297247)
		tmp = t_0;
	elseif (y < 6799310503.41891)
		tmp = (x * y) / (y + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[Less[y, -3693.8482788297247], t$95$0, If[Less[y, 6799310503.41891], N[(N[(x * y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024232 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -36938482788297247/10000000000000) (- (/ x (* y y)) (- (/ x y) x)) (if (< y 679931050341891/100000) (/ (* x y) (+ y 1)) (- (/ x (* y y)) (- (/ x y) x)))))

  (/ (* x y) (+ y 1.0)))