Data.Random.Distribution.Normal:normalF from random-fu-0.2.6.2

Percentage Accurate: 100.0% → 100.0%

Time: 8.4s

Alternatives: 8

Speedup: 1.0×

• 26.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ e^{\left(x \cdot y\right) \cdot y} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return math.exp(((x * y) * y))

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{\left(x \cdot y\right) \cdot y} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return math.exp(((x * y) * y))

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\mathsf{E}\left(\right)}^{\left(\left(y \cdot x\right) \cdot y\right)} \end{array} \]

FPCore

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

Derivation

1. Initial program 100.0%

   \[e^{\left(x \cdot y\right) \cdot y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f64 N/A

      \[\leadsto \color{blue}{e^{\left(x \cdot y\right) \cdot y}} \]

   2. remove-double-neg N/A

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

   3. sinh---cosh-rev N/A

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

   4. cosh-neg-rev N/A

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

   5. sinh-neg-rev N/A

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

   6. lower--.f64 N/A

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

   7. lower-cosh.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. sinh-neg-rev N/A

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

   12. lower-sinh.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*l* N/A

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

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

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

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

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

   20. lower-*.f64 N/A

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

   21. lower-neg.f64 77.2

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

4. Applied rewrites 77.2%

   \[\leadsto \color{blue}{\cosh \left(\left(y \cdot x\right) \cdot y\right) - \sinh \left(\left(\left(-y\right) \cdot y\right) \cdot x\right)} \]
5. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-cosh.f64 N/A

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

   3. cosh-neg-rev N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

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

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

   7. lift-neg.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-sinh.f64 N/A

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

   13. sinh---cosh-rev N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

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

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

   17. lift-*.f64 N/A

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

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

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

   19. lift-neg.f64 N/A

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

   20. remove-double-neg N/A

       \[\leadsto e^{x \cdot \left(\color{blue}{y} \cdot y\right)} \]

   21. lift-*.f64 N/A

       \[\leadsto e^{x \cdot \color{blue}{\left(y \cdot y\right)}} \]

6. Applied rewrites 100.0%

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

   1. lift-exp.f64 N/A

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

   2. exp-1-e N/A

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

   3. lower-E.f64 100.0

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

8. Applied rewrites 100.0%

   \[\leadsto {\color{blue}{\mathsf{E}\left(\right)}}^{\left(\left(y \cdot x\right) \cdot y\right)} \]
9. Add Preprocessing

Alternative 2: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{\left(x \cdot y\right) \cdot y} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return math.exp(((x * y) * y))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{\left(x \cdot y\right) \cdot y} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 3: 61.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-111}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(y \cdot y\right) \cdot \left(x \cdot x\right)\right) \cdot 0.5, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5e-111)
   1.0
   (if (<= y 1.85e+153)
     (fma (* (* (* y y) (* x x)) 0.5) (* y y) 1.0)
     (fma (* y y) x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5e-111) {
		tmp = 1.0;
	} else if (y <= 1.85e+153) {
		tmp = fma((((y * y) * (x * x)) * 0.5), (y * y), 1.0);
	} else {
		tmp = fma((y * y), x, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5e-111)
		tmp = 1.0;
	elseif (y <= 1.85e+153)
		tmp = fma(Float64(Float64(Float64(y * y) * Float64(x * x)) * 0.5), Float64(y * y), 1.0);
	else
		tmp = fma(Float64(y * y), x, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5e-111], 1.0, If[LessEqual[y, 1.85e+153], N[(N[(N[(N[(y * y), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.0000000000000003e-111

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 63.1%

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

   if 5.0000000000000003e-111 < y < 1.8500000000000001e153

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\left(x \cdot y\right) \cdot y}} \]

      2. remove-double-neg N/A

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

      3. sinh---cosh-rev N/A

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

      4. cosh-neg-rev N/A

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

      5. sinh-neg-rev N/A

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

      6. lower--.f64 N/A

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

      7. lower-cosh.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. sinh-neg-rev N/A

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

      12. lower-sinh.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

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

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

      20. lower-*.f64 N/A

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

      21. lower-neg.f64 74.4

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

   4. Applied rewrites 74.4%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 62.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot \left(\left(\left(y \cdot y\right) \cdot x\right) \cdot x - \left(\left(-x\right) - x\right)\right), y \cdot y, 1\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \left({x}^{2} \cdot {y}^{2}\right), \color{blue}{y} \cdot y, 1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 61.2%

       \[\leadsto \mathsf{fma}\left(\left(\left(\left(y \cdot y\right) \cdot x\right) \cdot x\right) \cdot 0.5, \color{blue}{y} \cdot y, 1\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 60.9%

       \[\leadsto \mathsf{fma}\left(\left(\left(y \cdot y\right) \cdot \left(x \cdot x\right)\right) \cdot 0.5, y \cdot y, 1\right) \]

   if 1.8500000000000001e153 < y

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot {y}^{2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot {y}^{2} + 1} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{{y}^{2} \cdot x} + 1 \]

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 65.4

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

   5. Applied rewrites 65.4%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-111}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(y \cdot y\right) \cdot \left(x \cdot x\right)\right) \cdot 0.5, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.5 \cdot \left(\mathsf{fma}\left(y \cdot y, x, 2\right) \cdot x\right), y \cdot y, 1\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (fma (* 0.5 (* (fma (* y y) x 2.0) x)) (* y y) 1.0))

C

double code(double x, double y) {
	return fma((0.5 * (fma((y * y), x, 2.0) * x)), (y * y), 1.0);
}

Julia

function code(x, y)
	return fma(Float64(0.5 * Float64(fma(Float64(y * y), x, 2.0) * x)), Float64(y * y), 1.0)
end

Wolfram

code[x_, y_] := N[(N[(0.5 * N[(N[(N[(y * y), $MachinePrecision] * x + 2.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{\left(x \cdot y\right) \cdot y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f64 N/A

      \[\leadsto \color{blue}{e^{\left(x \cdot y\right) \cdot y}} \]

   2. remove-double-neg N/A

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

   3. sinh---cosh-rev N/A

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

   4. cosh-neg-rev N/A

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

   5. sinh-neg-rev N/A

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

   6. lower--.f64 N/A

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

   7. lower-cosh.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. sinh-neg-rev N/A

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

   12. lower-sinh.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*l* N/A

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

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

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

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

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

   20. lower-*.f64 N/A

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

   21. lower-neg.f64 77.2

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

4. Applied rewrites 77.2%

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

5. Taylor expanded in y around 0

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

7. Applied rewrites 73.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot \left(\left(\left(y \cdot y\right) \cdot x\right) \cdot x - \left(\left(-x\right) - x\right)\right), y \cdot y, 1\right)} \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \left(x \cdot \left(2 + x \cdot {y}^{2}\right)\right), y \cdot y, 1\right) \]
9. Step-by-step derivation

10. Applied rewrites 73.1%

    \[\leadsto \mathsf{fma}\left(0.5 \cdot \left(\mathsf{fma}\left(y \cdot y, x, 2\right) \cdot x\right), y \cdot y, 1\right) \]

11. Final simplification 73.1%

    \[\leadsto \mathsf{fma}\left(0.5 \cdot \left(\mathsf{fma}\left(y \cdot y, x, 2\right) \cdot x\right), y \cdot y, 1\right) \]
12. Add Preprocessing

Alternative 5: 71.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(\left(\left(y \cdot y\right) \cdot x\right) \cdot x\right) \cdot 0.5, y \cdot y, 1\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (fma (* (* (* (* y y) x) x) 0.5) (* y y) 1.0))

C

double code(double x, double y) {
	return fma(((((y * y) * x) * x) * 0.5), (y * y), 1.0);
}

Julia

function code(x, y)
	return fma(Float64(Float64(Float64(Float64(y * y) * x) * x) * 0.5), Float64(y * y), 1.0)
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{\left(x \cdot y\right) \cdot y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f64 N/A

      \[\leadsto \color{blue}{e^{\left(x \cdot y\right) \cdot y}} \]

   2. remove-double-neg N/A

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

   3. sinh---cosh-rev N/A

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

   4. cosh-neg-rev N/A

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

   5. sinh-neg-rev N/A

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

   6. lower--.f64 N/A

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

   7. lower-cosh.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. sinh-neg-rev N/A

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

   12. lower-sinh.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*l* N/A

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

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

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

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

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

   20. lower-*.f64 N/A

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

   21. lower-neg.f64 77.2

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

4. Applied rewrites 77.2%

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

5. Taylor expanded in y around 0

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

7. Applied rewrites 73.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot \left(\left(\left(y \cdot y\right) \cdot x\right) \cdot x - \left(\left(-x\right) - x\right)\right), y \cdot y, 1\right)} \]

8. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \left({x}^{2} \cdot {y}^{2}\right), \color{blue}{y} \cdot y, 1\right) \]
9. Step-by-step derivation

10. Applied rewrites 72.7%

    \[\leadsto \mathsf{fma}\left(\left(\left(\left(y \cdot y\right) \cdot x\right) \cdot x\right) \cdot 0.5, \color{blue}{y} \cdot y, 1\right) \]

11. Final simplification 72.7%

    \[\leadsto \mathsf{fma}\left(\left(\left(\left(y \cdot y\right) \cdot x\right) \cdot x\right) \cdot 0.5, y \cdot y, 1\right) \]
12. Add Preprocessing

Alternative 6: 66.6% accurate, 9.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{\left(x \cdot y\right) \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + x \cdot {y}^{2}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{x \cdot {y}^{2} + 1} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{{y}^{2} \cdot x} + 1 \]

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 67.5

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

5. Applied rewrites 67.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot y, x, 1\right)} \]
6. Add Preprocessing

Alternative 7: 63.4% accurate, 9.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{\left(x \cdot y\right) \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + x \cdot {y}^{2}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{x \cdot {y}^{2} + 1} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{{y}^{2} \cdot x} + 1 \]

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 67.5

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

5. Applied rewrites 67.5%

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

7. Applied rewrites 65.3%

   \[\leadsto \mathsf{fma}\left(y \cdot x, \color{blue}{y}, 1\right) \]
8. Add Preprocessing

Alternative 8: 50.8% accurate, 111.0× 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%

   \[e^{\left(x \cdot y\right) \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Applied rewrites 50.3%

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

Reproduce

herbie shell --seed 2024337 
(FPCore (x y)
  :name "Data.Random.Distribution.Normal:normalF from random-fu-0.2.6.2"
  :precision binary64
  (exp (* (* x y) y)))